SAT Preparation Class Lessons:
January 30 - November 20, 2021


November Lessons - 2021

SAT Math - Questions from Fall 2021, Week 9 (November 20, 2021)

Absolute Value Equations

The absolute value of a number is the distance of that number from 0 on the number line. The symbol |x| is used to designate the absolute value of x. Since distance is nonnegative, the absolute value of a number is always positive or zero. Thus, |7| = 7 because 7 is seven units away from 0 on the number line. Also, |– 6| = 6 because 6 is six units away from 0 on the number line.

Some equations contain an absolute value expression, which take the form |ax + b| = c. To solve an absolute value equation, it is necessary to consider the possibility that “ax + b” may be positive and “ax + b” may be negative. To solve for x in the equation, we solve both for “ax + b = c” and “ax + b = – c”. Thus there may be two solutions for the equation. It is always necessary to check the solutions in the original equation to verify the solutions.

Examine examples 1, 2, 3, and 4 below; solve each of the equations.

Example 1.                |2x + 3| = 15
                                     2x + 3 = 15                2x + 3 = – 15
                                           2x = 12                       2x = – 18
                                              x = 6                           x = – 9

We have two answers, 6 and – 9. Now we can check our answers for each value of x in the original equation.
 
           x = 6                                                        x = – 9

          |2x + 3| = 15                                        |2x + 3| = 15
        |2(6) + 3| = 15                                  |2(– 9) + 3| = 15
          |12 + 3| = 15                                    |– 18 + 3| = 15
                |15| = 15                                          |– 15| = 15
                  15 = 15                                                15 = 15                 Our answers check successfully.

Example 2.                 4|6 – 2x| – 5 = 27
To solve the absolute value equation, the absolute value term must be alone on one side of the equation. Thus we need to add 5 to both sides of the equation and then divide both sides by 4.
          4|6 – 2x| = 32
            |6 – 2x| = 8
              6 – 2x = 8                                           6 – 2x = – 8
                 – 2x = 2                                              – 2x = – 14
                      x = – 1                                                 x = 7

We have two answers, – 1 and 7. We can check our answers for each value of x in the original equation.
                       x = – 1                                                x = 7

     4|6 – 2x| – 5 = 27                             4|6 – 2x| – 5 = 27
 4|6 – 2(–1)| – 5 = 27                          4|6 – 2(7)| – 5 = 27
        4|6 +2| – 5 = 27                            4|6 – 14| – 5 = 27
             4|8| – 5 = 27                                  4|– 8| – 5 = 27
             4(8) – 5 = 27                                      4(8) – 5 = 27
               32 – 5 = 27                                        32 – 5 = 27
                      27 = 27                                             27 = 27                Our answers check successfully.

Example 3.                        |3x + 9| + 4 = 0
               |3x + 9| = – 4

This equation has no solution. Since the absolute value of a number is always positive or zero, there is no value of x that would make this sentence true.

Example 4.                             |x – 2| = 2x – 10

               |x – 2| = 2x – 10                                   |x – 2| = 2x – 10
                 x – 2 = 2x – 10                                      x – 2 = – (2x – 10)
                    – 2 = x                                        – 10 x – 2 = – 2x + 10
                       8 = x                                                     3x = 12
                                                                                      x = 4
We have two answers, 8 and 4. We can check our answers for each value of x in the original equation.
                    x = 8                                                          x = 4

                  |x – 2| = 2x – 10                                  |x – 2| = 2x – 10
                  |8 – 2| = 2(8) – 10                               |4 – 2| = 2(4) – 10
                        |6| = 16 – 10                                        |2| = 8 – 10
                          6 = 6                                                     2 = – 2          NO

In this case, only one value of x works; x = 8 works, but x = 4 does not. The only solution is x = 8.

Keeping in mind the information above, solve the following problems.

  1. |x – 3| = 17

  2. |x + 11| = 42

  3. 11|x – 9| = 121

  4. |2x + 7| = x – 4

  5. 8|x – 3| = 88

  6. 3|x + 6| = 9x – 6

SAT Math - Answers to Questions from Fall 2021, Week 9 (November 20, 2021)

    1. |x – 3| = 17

                        x – 3 = 17                                                x – 3 = – 17
                               x = 20                                                      x = – 14
      We have two answers, 20 and – 14. We can check our answers for each value of x in the original equation.

                                x = 20                                                 x = – 14
                        |x – 3| = 17                                        |x – 3| = 17
                 
                       |20 – 3| = 17                                | – 14 – 3| = 17
                             |17| = 17                                      | – 17| = 17
                               17 = 17                                             17 = 17         Our answers check successfully.

    2. |x + 11| = 42

                           x + 11 = 42                                       x + 11 = – 42
                                   x = 31                                                x = – 53
      We have two answers, 31 and – 53. We can check our answers for each value of x in the original equation.

                                     x = 31                                               x = – 53

                           |x + 11| = 42                                    |x + 11| = 42
                        |31 + 11| = 42                              | – 53 + 11| = 42
                                |42| = 42                                       | – 42| = 42
                                  42 = 42                                              42 = 42      Our answers check successfully.

    3. 11|x – 9| = 121

                           |x – 9| = 11
                             x – 9 = 11                                           x – 9 = – 11
                                   x = 20                                                 x = – 2

      We have two answers, 20 and – 2. We can check our answers for each value of x in the original equation.

                                    x = 20                                                 x = – 2

                        11|x – 9| = 121                                  11|x – 9| = 121
                     11|20 – 9| = 121                              11| – 2 – 9| = 121
                           11|11| = 121                                  11| – 11| = 121
                          11(11)| = 121                                      11(11) = 121
                                121 = 121                                         121 = 121    Our answers check successfully.

    4. |2x + 7| = x – 4

      There is no solution for this equation. There is no value of x that will make the two sides of the
      equation equal. The absolute value of 2x + 7 will always be greater than x – 4.

    5. 8|x – 3| = 88

                             |x – 3| = 11
                               x – 3 = 11                                           x – 3 = – 11
                                     x = 14                                                 x = – 8

      We have two answers, 14 and – 8. We can check our answers for each value of x in the original equation.


                                        x = 14                                              x = – 8

                              8|x – 3| = 88                                      8|x – 3| = 88
                           8|14 – 3| = 88                                  8| – 8 – 3| = 88
                                 8|11| = 88                                      8| – 11| = 88
                                 8(11) = 88                                           8(11) = 88
                                     88 = 88                                               88 = 88  Our answers check successfully.

    6. 3|x + 6| = 9x – 6

                                 |x + 6| = 3x – 2
                                   x + 6 = 3x – 2                                       x + 6 = – (3x – 2)
                                         8 = 2x                                             x + 6 = – 3x + 2
                                         4 = x                                                   4x = – 4
                                                                                                      x = – 1

      We have two answers, 4 and – 1. We can check our answers for each value of x in the original equation.

                                             x = 4                                                  x = – 1
                                   3|x + 6| = 9x – 6                               3|x + 6| = 9x – 6
                                   3|4 + 6| = 9(4) – 6                        3| – 1 + 6| = 9(– 1) – 6
                                       3|10| = 36 – 6                                    3|5| = – 9 – 6
                                       3(10) = 30                                           3(5) = – 15
                                           30 = 30                                             15 = – 15            NO

      In this case, only one value of x works; x = 4 works, but x = – 1 does not. The only solution is x = 4.

SAT Verbal - Questions from Fall 2021, Week 9 (November 20, 2021)

SAT QUICK CHALLENGE R21
Parallelism

What Is Parallelism? Parallelism is defined as the repetition of a specific grammatical construction within a sentence or paragraph. This writing strategy makes writing clear and easy for the reader to follow. Its use indicates that the ideas involved are equally important. For example, in a listing of two or more items, each item must appear in the same format, as in all verb phrases, all infinitives, all phrases, etc. Note the error and corrections in Sentences A-C, which follow.

Sentence A. Flo likes flying better than to drive. Incorrect: mixes a gerund (flying) with an infinitive  (to drive)

Sentence B
.
Flo likes flying better than driving. Correct: contains two gerunds (flying and driving)

Sentence C. Flo likes to fly better than to drive. Correct: contains two infinitives (to fly and to drive)

Do Not Mix Phrases and Clauses in a Series in the Same Sentence. Parallelism requires that similar items in a sentence be expressed in the same way-- whether those items are clauses, phrases, or even individual words. Note Sentence D, which follows. Sentence D. Fred says that he would enjoy playing football, running track, and he wants to sing in the choir. This sentence is not parallel because the two phrases in it ("playing football" and "running track") are followed by a clause ("he wants to sing in the choir"). However, all three of the elements in the series must follow the same format, as illustrated in Sentences E and F, which follow. Sentence E. Fred says that he would enjoy playing football, running track, and singing in the choir. This sentence is correct because all three elements are expressed as gerund phrases. Sentence F. Willie says that he would like to play football, run track, and sing in the choir. (Note that the infinitive "to" does not have to be repeated for each infinitive phrase; the sentence means the same even when the word "to" is not repeated.) This sentence is also correct because all three items in the series are expressed as infinitive phrases.

Now, keeping in mind the information above, complete the Quick Challenge exercise below.

SAT Quick Challenge R21 - Parallelism

Directions. Replace the underlined word(s) in each question below with the answer choice that corrects the error in the sentence. If there is no error, select choice A -- NO CHANGE.

1.  Whether you coordinate the kitchen activities, oversee the cleaning crew, or if you are managing the financial responsibilities, we truly appreciate your help with our annual "Fall Frolic."

  1. NO CHANGE
  2. manage
  3. operating
  4. if you operate

2.   Our family will go to New York to visit popular landmarks, dine in famous restaurants, and we would love to watch live plays in the theaters there.

  1. NO CHANGE
  2. shows in the theaters would be great
  3. enjoy watching live theater productions
  4. going to see theater shows

3.  Little Leon says that he would love playing basketball, running track, and to race sports cars when he grows up.

  1. NO CHANGE
  2. racing sports cars
  3. he'd enjoy racing sports cars
  4. he loves sports car races

4.  People form opinions of you based on what you say and your actions.

  1. NO CHANGE
  2. promises made by you
  3. your promises
  4. what you do

SAT Verbal - Answers to Questions from Fall 2021, Week 9 (November 20, 2021)

  1. B
  2. C
  3. B
  4. D


SAT Math - Questions from Fall 2021, Week 8 (November 13, 2021)

Linear Equations Review

When solving linear equations, sometimes there is a need to solve for one variable in terms of another variable. For example, an SAT problem may ask you to solve for “a” in terms of “b”, or solve for “x” in terms of “y and z”. Some students who can readily solve linear equations where there is only one variable are not sure what to do when they are asked to solve a problem involving one variable in terms of another variable.

Actually, the process for determining one variable in terms of another variable is similar to solving an equation where there is only one variable. We isolate the variable for which we are solving on one side of the equal sign and put everything else on the other side of the equal sign, remembering that whatever we do to one side of the equation, we must do the same for the other side of the equation. We can perform several operations to both sides of the equation: add, subtract, multiply, divide, raise to a power, or take a root.


Now examine examples 1, 2, and 3.

Example 1.
Solve for x in terms of m:     2x – 3 = m + x

Solve for x in terms of m                                                                        Similar problem with one variable
2x – 3 = m + x                                                                                              2x – 3 = 4 + x
   Subtract x from both sides:          x – 3 = m                                              x – 3 = 4
   Add 3 to both sides: x = m + 3                                                                   x = 4 + 3 = 7


Example 2.     
Solve for x in terms of y: (5 + 6x)/2 = y + x
                                         
 Solve for x in terms of y                                                                            Similar problem with one variable
     (5 + 6x)/2 = y + x                                                                                        (5 + 6x)/2 = 3 + x
     Multiply both sides by 2:  5 + 6x = 2(y + x)                                              5 + 6x = 2(3 + x)
                                            5 + 6x = 2y + 2x                                                  5 + 6x = 2(3) + 2x
 Subtract 2x from both sides: 5 + 4x = 2y                                                     5 + 4x = 2(3)
 Subtract 5 from both sides:          4x = 2y – 5                                                    4x = 2(3) – 5
 Divide both sides by 4:                    x = (2y – 5)/4                                                 x = (2(3) – 5)/4 = (6 – 5)/4 = 1/4

Example 3. Solve for b in terms of a, c, and x: a(b - 2)/(c – 3) = x
           Solve for b in terms of a, c, and x                                                     Similar problem with one variable
                         a(b - 2)/(c – 3) = x                                                                   4(b - 2)/(7 – 3) = 9
Multiply both sides by (c – 3):   a(b-2) = x(c-3)                                             4(b-2) = 9(7 – 3)
Divide both sides by a:           b – 2 = x(c-3)/a                                                  b – 2 = 9(7-3)/4
Add 2 to both sides:                    b = x(c-3)/a + 2                                             b = 9(7-3)/4 + 2 = 9(4)/4 + 2 = 9+2=11


Keeping in mind the information above, solve the following problems.

  1. Solve for x in terms of y: (x + 2y)/3 = 6 + 3x
  2. Solve for a in terms of x and b: x = – b/2a
  3. Solve for x in terms of a and n: (3ax – n)/5 = – 4
  4. Solve for y in terms of b and c: (by + 2)/3 = c
  5. Solve for x in terms of e, y, and z: ex – 2y = 3z
  6. Solve for b in terms of k, m, and x: k = (x/2)(b + m)

SAT Math - Answers to Questions from Fall 2021, Week 8 (November 13, 2021)

    1. Solve for x in terms of y: (x + 2y)/3 = 6 + 3x
      Multiply both sides by 3: x + 2y = 3(6 + 3x)
                                                x + 2y = 18 + 9x
      Subtract x from both sides:   2y = 18 + 8x
      Subtract 18 from both sides: 2y - 18 = 8x
      Divide both sides by 8: (2y – 18)/8 = x
                              x = (2y – 18)/8 = (y – 9)/4
    2. Solve for a in terms of x and b: x = – b/2a 

      Multiply both sides by 2a:    2ax = – b
      Divide both sides by 2x:            a = – b//2x

    3. Solve for x in terms of a and n: (3ax – n)/5 = – 4

      Multiply both sides by 5:    3ax – n = – 20
      Add n to both sides: 3ax = n – 20
      Divide both sides by 3a: x = (n – 20)/3a

    4. Solve for y in terms of b and c: (by + 2)/3 = c

      Multiply both sides by 3: by + 2 = 3c
      Subtract 2 from both sides: by = 3c – 2
      Divide both sides by b: y = (3c – 2)/b

    5. Solve for x in terms of e, y, and z: ex – 2y = 3z

      Add 2y to both sides: ex = 3z + 2y
      Divide both sides by e: x = (3z + 2y)/e

    6. Solve for b in terms of k, m, and x: k = (x/2)(b + m)

      Multiply both sides by 2: 2k = x(b + m)
      Divide both sides by x: 2k/x = b + m
      Subtract m from both sides: 2k/x – m = b

SAT Verbal - Questions from Fall 2021, Week 8 (November 13, 2021)

SAT QUICK CHALLENGE Q21
The Comma and Nonessential vs. Essential Clauses

Essential and Nonessential Information. Information that is needed so that the reader can understand clearly what the writer is saying is called essential information. However, nonessential information gives the reader "extra" insight about the writer's point, but the sentence will still make sense without that extra information. Its removal will not affect a sentence's grammatical structure or its essential meaning. Since the added information is not a part of the writer's main point, that information must be separated from the rest of the sentence with commas. Note Sentences A - D, which follow.

The Words "Who" and "Which": Essential or Nonessential Information? To address this question, let's examine Sentences A - D below, beginning with Sentences A and B. Sentence A. My sister who lives in Florida likes to spend her vacations skiing in the mountains. Sentence B. My sister, who lives in Florida, likes to spend her vacations skiing in the mountains. Note that both sentences talk about a sister who likes skiing, but in Sentence A, the clause "who lives in Florida" is not cut off from the rest of that sentence because the clause modifies "sister" by telling which sister likes skiing in the mountains. However, in Sentence B, commas do separate "who lives in Florida" from the rest of the sentence because (1) the writer has only one sister, and (2) the clause simply provides extra information about her; it does not affect the sentence's main point, and if the quote is removed, the sentence will still make sense.

Note that the logic just discussed also applies to the word "which," as used in Sentences C and D, which follow. Sentence C. Because of allergies, Ava can't drink cocoa which is made from cow's milk. Sentence D. Many traditional foods, which are now hard to get because of supply shortages, may be missing from holiday meals this year.

Essential Clauses and the Word "That." Generally speaking, clauses beginning with the word "that" are always essential, and the word "that" should not have commas or any other form of punctuation in front of or behind it. However, an interrupter must always be separated from a sentence because an interrupter is not critical to the meaning of a sentence -- even when that interrupter follows the word "that," as it does in Sentence E, which follows. Sentence E: We discovered that, as we had expected, our star player couldn't play because of a serious injury.

While a comma does follow the word "that" in Sentence E, the purpose of the comma is just to help separate the interrupter (as we had expected) from the rest of the sentence. The interrupter simply happens to follow the word "that."

Now, using the information above, complete the exercise below.

SAT Quick Challenge Q21 - The Comma and Nonessential vs. Essential Clauses

Directions. . For each question below, select the answer choice which shows where the comma(s) must be placed in order for the sentence to be punctuated correctly. Select choice A -- no CHANGE -- if you think that the sentence is already punctuated correctly.

1.   Mrs. Evans who owns several day care centers provides reading classes for all her students.

  1. NO CHANGE
  2. after "Evans" and after "centers"
  3. after "who" and after "centers"
  4. before "who" and after "classes"

2.   Chocolate is the ice cream flavor that my mom likes best.

  1. NO CHANGE
  2. after "Chocolate" and after "flavor"
  3. after "that"
  4. after "flavor"

3.  Several kinds of apple pie which is Little Al's favorite dessert were served at his birthday party.

  1. NO CHANGE
  2. after "apple pie"
  3. after "served"
  4. after "apple pie" and after "dessert"

SAT Verbal - Answers to Questions from Fall 2021, Week 8 (November 13, 2021)

  1. B
  2. A
  3. D


SAT Math - Questions from Fall 2021, Week 7 (November 6, 2021)

INEQUALITIES: Word Problems Continued

The following symbols are used in inequalities:

≠    is not equal to
>    is greater than
<    is less than
≥     is greater than or equal to
≤    is less than or equal to

Some word problems involving inequalities require that you only write the inequality; it is not necessary to solve the inequality. These problems are usually in a multiple choice context, where you must select the appropriate choice.

It is important to read the problem carefully, identify the relevant information, and use the appropriate direction of the inequality signs.

Examine examples 1, 2, and 3.

  1. An architect in an arid region determines that a building’s current landscaping uses $1700 worth of water monthly. The architect plans to replace the current landscaping with arid zone landscaping at a cost of $18,000, which will reduce the monthly watering cost to $820. Which of the following inequalities can be used to find x, the number of months after replacement that the savings in water costs will be at least as much as the cost of replacing the landscaping?

    1. 820x ≥ 18,000
    2. (1,700 – 820)x ≤ 18,000
    3. 820x < 18,000
    4. (1,700 – 820)x ≥ 18,000

    We want to know how long it will take for our total savings to be greater than our cost of replacement. Our monthly savings will be (1700 – 820). Eliminate choices A and C since they do not contain our savings. Our total savings will be (1700 – 820)x. We want this value to be greater than or equal to the replacement cost of 18,000. Thus our answer is D.

  2. Mr. Cox is an insurance agent who sells two types of life insurance policies: a $40,000 policy and a $200,000 policy. Last month his goal was to sell at least 70 insurance policies. While he did not meet his goal, the total value of the policies he sold was over $4,000,000. Which of the following inequalities describes x, the possible number of $40,000 policies, and y, the possible number of $200,000 policies, that Mr. Cox sold last month?

    1. x + y < 70
      40,000x + 200,000y < 4,000,000

    2. x + y > 70
      40,000x + 200,000y > 4,000,000

    3. x + y < 70
      40,000x + 200,000y > 4,000,000

    4. x + y > 70
      40,000x + 200,000y < 4,000,000

    He did not meet his goal of selling at least 70 policies; thus eliminate choices B and D. The total value of the policies he sold was over $4,000,000. Eliminate choice A. Thus our answer is C.

  3. Delores needs to hire at least 10 staff members for an upcoming project. The staff members will be made up of junior directors, who will be paid $640 per week, and senior directors, who will be paid $880 per week. Her budget for hiring the staff members is no more than $9,700 per week. She must hire at least 3 junior directors and at least 1 senior director. Which of the following inequalities represent the conditions described if x is the number of junior directors and y is the number of senior directors?
    1. 640x + 880y ≥ 9,700
      x + y ≤ 10
      x ≥ 3
      y ≥ 1

    2. 640x + 880y ≤ 9,700
      x + y ≥ 10
      x ≥ 3
      y ≥ 1

    3. 640x + 880y ≥ 9,700
      x + y ≥ 10
      x ≤ 3
      y ≤ 1

    4. 640x + 880y ≤ 9,700
      x + y ≤ 10
      x ≤ 3
      y ≤ 1

    The amount to be paid to the staff members, 640x + 880y, can be no more than 9,700. Eliminate A and C. Delores will hire at least 10 staff members. Eliminate D. The answer is B, which meets all of the inequality conditions.

Keeping in mind the information above, solve the following problems.

  1. The average annual energy cost for a certain home is $4,800. The homeowner plans to spend $30,000 to install a geothermal heating system. The homeowner estimates that the average annual cost will be reduced to $3,100. Which of the following inequalities can be used to find x, the number of years after installation at which the total amount of energy cost savings will exceed the installation cost?

    1. 3,100x > 30,000
    2. 3,100x < 30,000 – 4,800
    3. 4,800 – 3,100)x > 30,000
    4. 4,800 – 3,100)x < 30,000

  2. A psychologist set up an experiment to study the tendency of a person to select the first item when presented with a series of items. In the experiment, 300 people were presented with a set of five pictures arranged in random order. Each person was asked to choose the most appealing picture. Of the first 150 participants, 36 choose the first picture in the set. Among the remaining 150 participants, x people chose the first picture in the set. If more than 20% of all participants chose the first picture in the set, which of the following inequalities best describes the possible values of x?

    1. x > 0.20(300 – 36), where x ≤ 150
    2. x > 0.20(300 + 36), where x ≤ 150
    3. x -36 > 0.20(300), where x ≤ 150
    4. x + 36 > 0.20(300), where x ≤ 150

  3. Jackie has two summer jobs. She works as a tutor, which pays $12 per hour and she works as a lifeguard, which pays $8 per hour. She can work no more than 20 hours per week, but she wants to earn at least $220 per week. Which of the following inequalities represent this situation in terms of x and y, where x is the number of hours she tutors and y is the number of hours she works as a lifeguard?

    1. 12x + 8y ≤ 220
      x + y ≥ 20

    2. 12x + 8y ≤ 220
      x + y ≤ 20

    3. 12x + 8y ≥ 220
      x+ y ≤ 20

    4. 12x + 8y ≥ 220
      x + y ≥ 20

  4. Dogs need 8.5 to 17 ounces of water each day for every 10 pounds of their weight. Ronald has two dogs – Rover is a 35 pound black lab mix, and Rex is a 55 pound beagle. Which of the following ranges represents the approximate total number of ounces of water , x, that Rover and Rex need in a week?

    1. 77 ≤ w ≤ 153
    2. 109 ≤ w ≤ 218
    3. 536 ≤ w ≤ 1,071
    4. 765 ≤ w ≤ 1,530

  5. A laundry service is buying detergent and fabric softener from its supplier. The supplier will deliver no more than 300 pounds in a shipment. Each container of detergent weighs 7.35 pounds and each container of fabric softener weighs 6.2 pounds. The service wants to buy at least twice as many containers of detergent as containers of fabric softener. Let d represent the number of containers of detergent, and let s represent the number of containers of fabric softener, where d and s are nonnegative integers. Which of the following inequalities best represents this situation?

    1. 7.35d + 6.2s ≤ 300
      d ≥ 2s

    2. 7.35d + 6.2s ≤ 300
      2d ≥ s

    3. 4,800 – 3,100)x > 30,000
      d ≥ 2s

    4. 4,800 – 3,100)x < 30,000
      2d ≥ s
  6. David must buy at least 100 shares of stock for his portfolio. The shares he buys will be from Stock X, which costs $22 per share and Stock Y, which costs $35 per share. His budget for buying stock is no more than $4,500. He must buy at least 20 shares of Stock X and 15 shares of Stock Y. Which of the following represents the condition described if x is the number of shares of Stock X purchased and y is the number of shares of Stock Y purchased?

    1. 22x + 35y ≤ 4,500
      x + y ≥ 100
      x ≤ 20
      y ≤ 15

    2. 22x + 35y ≤ 4,500
      x + y ≤ 100
      x ≤ 20
      y ≤ 15

    3. 22x + 35y ≤ 4,500
      x + y ≤ 100
      x ≥ 20
      y ≥ 15

    4. 22x + 35y ≤ 4,500
      x + y ≥ 100
      x ≥ 20
      y ≥ 15


SAT Math - Answers to Questions from Fall 2021, Week 7 (November 6, 2021)

    1. The average annual energy cost for a certain home is $4,800. The homeowner plans to spend $30,000 to install a geothermal heating system. The homeowner estimates that the average annual cost will be reduced to $3,100. Which of the following inequalities can be used to find x, the number of years after installation at which the total amount of energy cost savings will exceed the installation cost?

      1. 3,100x > 30,000
      2. 3,100x < 30,000 – 4,800
      3. 4,800 – 3,100)x > 30,000
      4. 4,800 – 3,100)x < 30,000

      We want to know how long it will take for our total savings to be greater than our installation cost. Our annual savings will be (4,800 – 3,100). Eliminate choices A and B since they do not contain our savings. Our total savings will be (4,800 – 3,100)x. We want this value to be equal to or greater than the installation cost of 30,000. Thus our answer is C.

    2. A psychologist set up an experiment to study the tendency of a person to select the first item when presented with a series of items. In the experiment, 300 people were presented with a set of five pictures arranged in random order. Each person was asked to choose the most appealing picture. Of the first 150 participants, 36 choose the first picture in the set. Among the remaining 150 participants, x people chose the first picture in the set. If more than 20% of all participants chose the first picture in the set, which of the following inequalities best describes the possible values of x?

      1. x > 0.20(300 – 36), where x ≤ 150
      2. x > 0.20(300 + 36), where x ≤ 150
      3. x -36 > 0.20(300), where x ≤ 150
      4. x + 36 > 0.20(300), where x ≤ 150

      More than 20% of all participants chose the first picture in the set. 20% of 300 = 0.20(300)= 60. 36 of the first 150 participants chosen the first picture in the set. From the remaining 150, some amount, x, chose the first picture, such that x + 36 is greater than 60. Thus x + 6 > 60, or x + 36 > 0.20(300). The answer is D.

    3. Jackie has two summer jobs. She works as a tutor, which pays $12 per hour and she works as a life guard, which pays $8 per hour. She can work no more than 20 hours per week, but she wants to earn at least $220 per week. Which of the following inequalities represent this situation in terms of x and y, where x is the number of hours she tutors and y is the number of hours she works as a lifeguard?

      1. 12x + 8y ≤ 220
        x + y ≥ 20

      2. 3,100x < 30,000 – 4,800
        x + y ≤ 20

      3. 12x + 8y ≥ 220
        x+ y ≤ 20

      4. 12x + 8y ≥ 220
        x + y ≥ 20

      Jackie can work no more than 20 hours per week; eliminate A and D. She wants to earn at least $220 per week; eliminate B. Thus the answer is C.

    4. Dogs need 8.5 to 17 ounces of water each day for every 10 pounds of their weight. Ronald has two dogs – Rover is a 35 pound black lab mix, and Rex is a 55 pound beagle. Which of the following ranges represents the approximate total number of ounces of water , x, that Rover and Rex need in a week?

      1. 77 ≤ w ≤ 153
      2. 109 ≤ w ≤ 218
      3. 536 ≤ w ≤ 1,071
      4. 765 ≤ w ≤ 1,530

      The dogs need 8.5 to 17 ounces of water for every 10 pounds of weight. Together the dogs weigh 90 pounds (35 + 55 = 90). There are 9 ten pounds in 90. Thus the range would be 8.5(9) = 76.5 and 17(9) = 153. This is A: 77 ≤ w ≤ 153. But this is a trap answer that many students would choose if they did not read the question carefully. What is the trap? The range in A is for a day; the question asks for the range in a week. Thus the correct range is 76.5(7) = 535.5 and 153(7) = 1,071 (since there are 7 days in a week). The answer is C.

    5. A laundry service is buying detergent and fabric softener from its supplier. The supplier will deliver no more than 300 pounds in a shipment. Each container of detergent weighs 7.35 pounds and each container of fabric softener weighs 6.2 pounds. The service wants to buy at least twice as many containers of detergent as containers of fabric softener. Let d represent the number of containers of detergent, and let s represent the number of containers of fabric softener, where d and s are nonnegative integers. Which of the following inequalities best represents this situation?

      1. 7.35d + 6.2s ≤ 300
        d ≥ 2s
      2. 7.35d + 6.2s ≤ 300
        2d ≥ s
      3. 4,800 – 3,100)x > 30,000
        d ≥ 2s 4,800 – 3,100)x < 30,000
        2d ≥ s

      The service wants to buy at least twice as many containers of detergent as containers of fabric softener: d ≥ 2s. Eliminate B and D. 7.35 is multiplied by d and 6.2 is multiplied by s. Eliminate C. the answer is A.

    6. David must buy at least 100 shares of stock for his portfolio. The shares he buys will be from Stock X, which costs $22 per share and Stock Y, which costs $35 per share. His budget for buying stock is no more than $4,500. He must buy at least 20 shares of Stock X and 15 shares of Stock Y. Which of the following represents the condition described if x is the number of shares of Stock X purchased and y is the number of shares of Stock Y purchased?

      1. 22x + 35y ≤ 4,500
        x + y ≥ 100
        x ≤ 20
        y ≤ 15

      2. 22x + 35y ≤ 4,500
        x + y ≤ 100
        x ≤ 20
        y ≤ 15

      3. 22x + 35y ≤ 4,500
        x + y ≤ 100
        x ≥ 20
        y ≥ 15

      4. 22x + 35y ≤ 4,500
        x + y ≥ 100
        x ≥ 20
        y ≥ 15


      The total number of shares is equal to or greater than 100. Eliminate B and C. He must buy at least 20 shares of Stock X and 15 shares of Stock Y. Eliminate A. The answer is D.

SAT Verbal - Questions from Fall 2021, Week 7 (November 6, 2021)

SAT QUICK CHALLENGE P21
Review -- Using Commas Correctly

Some SAT questions test to see how well students remember how to use commas correctly, This exercise will help you maintain mastery of some of the comma skills tested frequently on the SAT.

Separating Items in a List or Series. A very common use for commas is to separate items in a list or series, as in Sentences A1 and A2, which follow.

Sentence A1: Missy wants shrimp, green beans, candied yams, and rolls for her birthday dinner.
Sentence A2: Missy wants shrimp, green beans, candied yams and rolls for her birthday dinner.

Introductory Words and Phrases. An introductory word/phrase (1) comes at the beginning of a sentence, (2) sets the tone for what the sentence will say, and (3) needs a comma after it. Examples of those words/phrases include the following: in fact, by the way, as you might expect, as you know, yes, no, initially, before, and however. Note the bold, underlined introductory phrase at the beginning of Sentence B.
Sentence B: As we had anticipated,
Lee took a long nap after he ran in the marathon.

Interrupters. An interrupter (1) comes within a sentence, (2) shows emotion, tone, or emphasis by temporarily breaking the flow of the thought the sentence is expressing, and (3) must be enclosed in commas. Many expressions routinely used as introductory words/phrases (in fact, indeed, initially, however, as you might expect, for example, etc.) are also used as interrupters. Even a person's name can be used as an interrupter. Note the interrupters in Sentences C and D, which follow.
Sentence C
: What did you do, Roger, when you realized that you were locked out of your house?
Sentence D: After Lee ran in the marathon, as we had anticipated, he took a very long nap.

Now, keeping in mind the information above, complete the exercise below.

SAT Quick Challenge P21
Using Commas Correctly

Directions. . For each statement below, select the answer choice which shows where the comma9s0 must be placed in order for the sentence to be punctuated correctly. Select choice A -- no CHANGE -- if you think that the sentence is already punctuated correctly. Then use the answer key in the dropdown below to check your work.

1.   We found the keys Lynn exactly where you said they would be.

  1. NO CHANGE
  2. before and after "Lynn"
  3. after "Lynn"
  4. after "said"

2.   As you know people who go to see the play must wear masks during the entire event.

  1. NO CHANGE
  2. after "know"
  3. before "must" and after "masks"
  4. after "play"

3.  After the art lesson, paint brushes, paint smears, markers, glue, and scraps of paper were scattered all over the floor.

  1. NO CHANGE
  2. after "scattered"
  3. after "and"
  4. after "paper"

SAT Verbal - Answers to Questions from Fall 2021, Week 7 (November 6, 2021)

  1. B
  2. B
  3. A

October Lessons


SAT Math - Questions from Fall 2021, Week 6 (October 23, 2021)

INEQUALITIES: Word Problems

The following symbols are used in inequalities:

≠    is not equal to
>    is greater than
<    is less than
≥     is greater than or equal to
≤    is less than or equal to

Just as some word problems can be approached by solving equations (see Linear Equations: Word Problems, Week 3, Fall 2021), some other word problems are solved by inequalities.

The following steps can help you solve inequality word problems:

  1. Read the problem carefully and be sure that you understand key words and concepts.

  2. Determine how the key words and concepts are related mathematically by
    converting the words to math; write the inequality in mathematical terms.

    1. It is important to determine what the variable is; express the unknown as a
      mathematical symbol (x or any other latter).

    2. It is also important to determine accurately the direction of the inequality sign:
      is it greater than or is it less than?

    3. Determine what goes on either side of the inequality sign.
  3. Solve the problem and interpret the solution.

  4. Be sure that the answer is reasonable and answers the question that was asked.

Examples;
Now examine examples 1, 2, and 3.

  1. Six more than twice a number is at least 60. What is the minimum value of the number?

    Our unknown, our variable, is a number. Let’s call it x. Twice x plus 6 is greater than 60; it could be 80, or 300, or 20,000, or any number greater than 60. But we want the smallest possible value that meets our conditions. Thus the direction of the inequality sign is greater than or equal to. We want the smallest number that is greater than or equal to 27.

    2x + 6 ≥ 60
    2x ≥ 54
    x ≥ 27 Our answer is 27.

  2. The Petty family is considering renting a boat for a fun ride on the Dan River. Booker Boats charges $260 per week, plus $2 per hour, for use of its boats. A competitor, Saulter Ships, charges $40 per week, plus $8 per hour, for use of its boats. Boats of both companies are similar. How much would the Petty family have to use the boat in a week in order for Booker Boats to be the better deal?

    Because of the small weekly cost, $40 vs $260, Saulter Ships is the better deal if we use the boat just a few hours per week. But if we use the boat for many hours during the week, Booker Boats is the better choice because of its lower per hour charge. Our question, then, is when does Booker Boats become the better deal?

    Our variable is the number of hours per week; let’s call it x. We want to find x when the total cost for Booker Boats is less than the total cost for Saulter Ships. Thus the direction for the inequality is less than.

    260 + 2h < 40 + 8h
    220 < 6h
    36.667 < h

    Therefore the Petty family would have to use the boat for 37 or more hours per week for Booker Boats to be the better deal.

  3. Tickets for A&T’s winter concert cost $4.00 for students and $10 for adults. If Mr. Harrison spends at least $42 but no more than $70 on x student tickets and 3 adult tickets, what is one possible value of x?

    We have a compound inequality: an amount is between two values. For example, if x is an integer and we have

    5 < x ≤ 9 then x could be 6, 7, 8, or 9.

    In our ticket example, we have a lower amount, $42, and a higher amount, $70, and the variable is the number of student tickets.

    42 ≤ 4x + 3(10) ≤ 70
    42 ≤ 4x + 30 ≤ 70
    12 ≤ 4x ≤ 40
    3 ≤ x ≤ 10

    Thus the number of student tickets could be any number between 3 and 10, including 3 and 10.

    Keeping in mind the information above, answer the following questions.

Keeping in mind the information above, solve the following problems.

  1. Dave and Ron are the top scorers on their high school basketball team. If the team is to have any chance of winning its next game, Dave and Ron together must score at least 50 points. What is the minimum number of points that Dave must score if Ron scores 4 points less than twice the number of Dave’s points?

  2. I have 200 shares of IBM stock. My uncle, who recently retired, has 2000 shares of IBM stock. He has substantial investments in other companies, and wants to give the IBM stock to his five nephews. Starting tomorrow, he will give each of his five nephews, including me, one share of his IMB stock every day. How many shares of IBM stock will I have on the first day that I have more shares of the stock than my uncle has?

  3. Marcus rented a motor cycle. The rental cost $15 per hour, and he also had to pay for a helmet that costs $20. In total, he spent more than $90 for the rental and helmet. If the motor cycle was available for only a whole number of hours, what was the minimum number of hours that Marcus could have rented the motor cycle?

  4.  C = 25x + 8y

    The formula above gives the monthly cost C, in dollars, of operating a printer when a technician works a total of x hours and when y reams of paper are used. If, in July it costs no more than $5,800 to operate the printer and 125 reams of paper were used, what is the maximum number of hours the technician could have worked?

    1. 144
    2. 160
    3. 192
    4. 240
  5.  A restaurant is making its weekly purchase of beef and chicken from its supplier. The supplier will deliver no more than 250 pounds in a shipment. Each package of beef
    weighs 20 pounds and each package of chicken weighs 15 pounds. The restaurant wants to purchase 2 times as many packages of chicken as packages of beef, and the restaurant wants to purchase at least 100 pounds of meat. What is the minimum number of packages of beef that should be purchased?

  6. The average monthly cable television cost for the Adams family is $120. The family plans to spend $3,000 to install a satellite television system. The family estimates that the average annual satellite television cost will be $1,100. How many years after installation of the satellite system will the total amount of television cost savings exceed
    the installation cost?

SAT Math - Answers to Questions from Fall 2021, Week 6 (October 23, 2021)

    1. Dave and Ron are the top scorers on their high school basketball team. If the team is to have any chance of winning its next game, Dave and Ron together must score at least 50 points. What is the minimum number of points that Dave must score if Ron scores 4 points less than twice the number of Dave’s points?

      The variable is the number of points that Dave must score. The direction of the inequality sign is greater than. Dave’s score is x and Ron’s score is 2x – 4. Thus we have
      x + 2x – 4 ≥ 50
      3x ≥ 54
      x ≥ 18

      Thus Dave must score at least 18 points.

    2. I have 200 shares of IBM stock. My uncle, who recently retired, has 2000 shares of IBM stock. He has substantial investments in other companies, and wants to give the IBM stock to his five nephews. Starting tomorrow, he will give each of his five nephews, including me, one share of his IMB stock every day. How many shares of IBM stock will I have on the first day that I have more shares of the stock than my uncle has?

      Each day my uncle gives away share of stock, I gain one share. After he has given away stock on x days, I have 200 + x shares of stock. Each day my uncle gives away stock, he loses 5 shares; thus he has given away 5x shares after x days. This leaves him with 2000 – 5x shares of IBM stock. We want to know when I will have more shares than he has, so we want to know when

      200 + x > 2000 – 5x

      We now solve for x.
      200 + x > 2000 – 5x
      6x > 1800
      x > 300

      The first time I will have more shares of IBM stock than my uncle is just after he has given away shares of stock 301 times (301 is the smallest number that is greater than 300). At that point, I will have 200 + 301 = 501 shares, and he will have 2000 – 5(301) = 2000 – 1505 = 495 shares.

      Therefore I will have 501 shares of stock the first time I have more shares than my uncle has.

    3. Marcus rented a motor cycle. The rental cost $15 per hour, and he also had to pay for a helmet that costs $20. In total, he spent more than $90 for the rental and helmet. If the motor cycle was available for only a whole number of hours, what was the minimum number of hours that Marcus could have rented the motorcycle?

      The variable, x, is the number of hours Marcus rents the motorcycle.

      There is a cost of $12 per hour; if he rents the motorcycle of x hours, he will spend 12x dollars plus an initial one time payment of $20 for the helmet. Thus the total cost of the rental and helmet is 12x + 20.

      The amount he will spend is more than 90; thus we have this inequality:
      12x + 20 > 90

      Now solve for x:
      12x + 20 > 90
      12x > 70
      x > 70/12 = 5 10/12 = 5 5/6

      Since he must rent for a whole number of hours, the minimum number of hours is 6.

    4.  C = 25x + 8y

      The formula above gives the monthly cost C, in dollars, of operating a printer when a technician works a total of x hours and when y reams of paper are used. If, in July it costs no more than $5,800 to operate the printer and 125 reams of paper were used, what is the maximum number of hours the technician could have worked?

      1. 144
      2. 160
      3. 192
      4. 240

        The variable is the number of hours the technician could have worked; the direction of the inequality is less than. Thus we have the following inequality:

        25x + 8y ≤ 5,800
        25x + 8(125) ≤ 5,800
        25x + 1,000 ≤ 5,800
        25x ≤ 4,800
        x ≤ 192 -------> C

    5.  A restaurant is making its weekly purchase of beef and chicken from its supplier. The supplier will deliver no more than 250 pounds in a shipment. Each package of beef weighs 20 pounds and each package of chicken weighs 15 pounds. The restaurant wants to purchase 2 times as many packages of chicken as packages of beef, and the restaurant wants to purchase at least 100 pounds of meat. What is the minimum number of packages of beef that should be purchased?

      We have a compound inequality: an amount is between two values, 100 and 250. Our variable, x, is the number of packages of beef; the number of packages of chicken is 2x.

      100 ≤ 20x + 15(2x) ≤ 250
      100 ≤ 20x + 30x ≤ 250
      100 ≤ 50x ≤ 250
      2 ≤ x ≤ 5

      The minimum number of packages of beef is 2.

    6. The average monthly cable television cost for the Adams family is $120. The family plans to spend $3,000 to install a satellite television system. The family estimates that the average annual satellite television cost will be $1,100. How many years after installation of the satellite system will the total amount of television cost savings exceed
      the installation cost?

      Our variable, x, is the number of years it takes for the savings to exceed the installation cost. There will be annual savings since the annual cost after installation, $1,100, is less than the family’s current annual cost of $1,440 ($120(12) = $1,440). The direction of the inequality sign is greater than since the total savings will exceed the installation
      cost. Thus the inequality is:
      (1440 – 1100)x > 3,000
      340x > 3,000
      x > 8.82

      It will take 9 years of savings for the family to recoup the installation cost.

SAT Verbal - Questions from Fall 2021, Week 6 (October 23, 2021)

SAT QUICK CHALLENGE
Exercise O21 -- Placing Modifiers Correctly

The Modifier and the Introductory Modifying Phrase (IMP). You may recall that a modifier is a word/word group that provides information about another word/word group in a sentence and must be placed as close as possible to the word/word group it modifies. For example, the introductory modifying phrase (IMP) comes at the beginning of a sentence, is followed by a comma, and describes the subject of the sentence -- which comes right after the IMP, as in Sentence A, which follows.

Sentence A. Excited that the fair starts today, crowds began lining up at the gates early this morning.
Note the bold, underlined IMP at the beginning of the sentence, the comma that follows it, and the bold, italicized subject right after the comma. As stated in the paragraph above, the IMP modifies that subject.

The Modifier Followed by Parentheses Instead of a Comma. Some people ignore information in parentheses, including modifiers, because they erroneously assume that anything in parentheses is unimportant. However, some students taking the SAT have lost points because they have made that same mistake. Do not let that happen to you! Remember that whether enclosed in commas or parentheses, a modifier that is placed as close as possible to the word it modifies is always placed correctly, and that kind of answer choice could be the correct answer. Note Sentences B and C, which follow.

Sentence B. Mrs. Hall must take her baby to the pediatrician, a doctor who specializes in treating children.
Sentence C. Mrs. Hall must take her baby to the pediatrician (a doctor who specializes in treating children).

Although the modifier in Sentence B follows a comma, and the modifier in Sentence C is in parentheses, each one modifies the noun that it follows ("pediatrician"). Accordingly, each modifier is in the right place. Now, keeping in mind the information above, complete the exercise below.

SAT Quick Challenge O21
Placing Modifiers Correctly

Directions. . For each statement below, select the letter of the answer choice which places the underlined modifier in the right place. If you think that the modifier is already in the right place, select choice A -- NO CHANGE. Then use the answer key in the dropdown below to check your work.

1.  At First Community Church, Freedom School is held during the summer (a local church that promotes better education for low income children,.

  1. NO CHANGE
  2. after "during the summer"
  3. after "Freedom School is held"
  4. after "At First Community Church"

2.  So that interested people can find good jobs, job fairs are held on a quarterly basis at Albany's Mt. Zion (a local church that helps workers get
     good jobs).

  1. NO CHANGE
  2. after "So that interested people"
  3. after "job fairs are held on a quarterly basis"
  4. after "can find good jobs,"

3. For Halloween, children and adults love to dress up in costumes (a time when people wear disguises and have fun with their families and friends). .

  1. NO CHANGE
  2. after "love to dress up"
  3. after "Halloween"
  4. after "children and adults"

SAT Verbal - Answers to Questions from Fall 2021, Week 6 (October 23, 2021)

  1. D
  2. A
  3. C

SAT Math - Questions from Fall 2021, Week 5 (October 16, 2021)

INEQUALITIES: No Solution; All Real Numbers Solution
As with linear equations, it is possible for inequalities to have no solution. And it is also possible that the solution to an inequality can include all real numbers.

The following symbols are used in inequalities:

≠    is not equal to
>    is greater than
<    is less than
≥     is greater than or equal to
≤    is less than or equal to

An inequality will have no solution if the “solution” states something false or gives a contradiction. For example, if we work through an inequality and arrive either at the following as our solution, there is no solution.

4 < -4
6 < x < -6

Statements similar to these are false and each inequality has no solution.

We may have other inequalities where all real numbers are solutions to an inequality. This is the case when it is true for any value of the variable, whether it is 0, less than 0, or greater than 0. For example, if we work through an inequality and arrive something similar to either of the following as our solution, the inequality is true for all real numbers:

-3x + 7 ≥ 7 -3x
-1 ≤ x2 -1

Examples;
Now examine examples 1, 2, and 3.

  1. What values of x satisfy 5x + 2 < 6x < 5x - 2
    Subtract 5x from each section: 5x + 2 < 6x < 5x – 2 becomes 2 < x < - 2

    We cannot pick a value of x that is greater than 2 and less than -2. Thus this inequality has no solution.

  2. What values of x satisfy 7x -11x + 3 ≥ 3 -4x

    Combine like terms:
    7x -11x + 3 ≥ 3 -4x becomes -4x + 3 ≥ 3 -4x

    Add 4x to both sides:
    -4x + 3 ≥ 3 -4x becomes 3 ≥ 3

    This statement is true for all real numbers.

  3. What values of x satisfy 2(3x – 4) < 4 + 6x - 15
    Remove parentheses:
    2(3x – 4) < 4 + 6x – 15 becomes 6x – 8 < 4 + 6x - 15
    Combine like terms:
    6x – 8 < 4 + 6x - 15 becomes 6x – 8 < 6x - 11

    Subtract 6x from both sides: 6x – 8 < 6x - 11 becomes – 8 < - 11

    This statement is false because – 8 is not less than –11. Therefore the inequality has no solution.

Keeping in mind the information above, solve the following problems.

  1. What values of x satisfy 9 + 2x – 5x ≥ –x + 12 – 2x ?

  2. What values of x satisfy 10(x -2) < 6x?

  3. What values of c satisfy 2 – 3c ≥ 6c – 3 – 9c?

  4. What values of m satisfy 9m + 5 – 12m ≥ 7 + 3m +10?

  5. What values of y satisfy 9 + 6(y + 1) ≥ 18 + 3(3y – 1) – 3y?

  6. What values of x satisfy 5(x – 1) +7 ≤ 2(x – 4) +3x + 1?

SAT Math - Answers to Questions from Fall 2021, Week 5 (October 16, 2021)

  1. What values of x satisfy 9 + 2x – 5x ≥ –x + 12 – 2x ?
    Combine like terms:
    9 + 2x – 5x ≥ –x + 12 – 2x becomes 9 – 3x ≥ 12 – 3x

    Add 3x to both sides:
    9 – 3x ≥ 12 – 3x becomes 9 ≥ 12

    This statement is false because 9 is not greater than or equal to 12. Therefore the inequality has no solution.

  2. What values of x satisfy 10(x -2) < 6x?
    Remove parentheses:
    10(x -2) < 6x becomes 10x - 20 < 6x

    Subtract 6x from both sides and add 20 to both sides:
    10x -20 < 6x becomes 4x < 20

    Solve for x: x < 5

    This is a true statement. Here x could be any value less than 5, not including 5.

  3. What values of c satisfy 2 – 3c ≥ 6c – 3 – 9c?
    Combine like terms:
    2 – 3c ≥ 6c – 3 – 9c becomes 2 – 3c ≥ – 3 – 3c

    Add 3c to both sides:
    2 – 3c ≥ – 3 – 3c becomes 2 ≥ – 3

    This statement is true for all real numbers of c. Here c could be any real number to solve the inequality.

  4. What values of m satisfy 9m + 5 – 12m ≥ 7 + 3m +10?
    Combine like terms:
    9m + 5 – 12m ≥ 7 + 3m +10 becomes 5 – 3m ≥ 17 + 3m

    Subtract 17 from both sides and add 3m to both sides:
    5 – 3m ≥ 17 + 3m becomes – 12 ≥ 6m

    Solve for m: – 2 ≥ m
    This is a true statement. Here m could be any value less than or equal to 2, including 2.

  5. What values of y satisfy 9 + 6(y + 1) ≥ 18 + 3(3y – 1) – 3y?
    Remove parentheses:
    9 + 6(y + 1) ≥ 18 + 3(3y – 1) – 3y becomes 9 + 6y +6 ≥ 18 + 9y –3 – 3y

    Combine like terms:
    9 + 6y +6 ≥ 18 + 9y –3 – 3y becomes 15 + 6y ≥ 15 + 6y

    Subtract 6y from both sides: 15 + 6y ≥ 15 + 6y becomes 15 ≥ 15

    This statement is true for all real numbers of y. Here y could be any real number to solve the inequality.

  6. What values of x satisfy 5(x – 1) +7 ≤ 2(x – 4) +3x + 1?
    Remove parentheses: 5(x – 1) +7 ≤ 2(x – 4) +3x + 1 becomes 5x – 5 +7 ≤ 2x – 8+3x +1

    Combine like terms:
    5x – 5 +7 ≤ 2x – 8+3x +1 becomes 5x +2 ≤ 5x –7

    Subtract 5x from both sides:
    5x +2 ≤ 5x –7 becomes 2 ≤ –7

    This statement is false because 2 is not less than or equal to -7. Therefore the inequality has no solution.

SAT Verbal - Questions from Fall 2021, Week 5 (October 16, 2021)

SAT QUICK CHALLENGE
Exercise N21 -- Dangling Modifiers

The Modifier. A modifier is a word/word group that adds extra information about another word/word group in a sentence. The modifier must be placed as close as possible to the word/word group being modified. An introductory modifying phrase (IMP) comes at the beginning of a sentence; is followed by a comma; and describes the subject of the sentence, which must come right after the IMP.

The Dangling Modifier. When a sentence has a modifier in it, but does not have the word that is being modified, that modifier is said to be "dangling," and the sentence's message can be illogical, unclear, or absurd, as you will note in Sentence A, which follows.

Sentence A: After undergoing extensive physical therapy, Grandma's balance improved greatly. Note that the IMP at the beginning of this sentence tells us that Grandma's balance (the subject of the sentence) has had extensive physical therapy. Of course, that idea does not make sense. However, we can tell (1) that it was actually Grandma -- not her balance -- that had physical therapy, and (2) that her balance improved after the physical therapy. Hence, the IMP is a dangling modifier.

Correcting Dangling Modifiers. Sentences B and C below show two ways of correcting Sentence A's dangling modifier. One correction strategy is to ask yourself who/what the IMP logically refers to. (In this case, the answer is "Grandma.") Then, place that word right into the IMP. Finally, following the comma after the IMP, complete the point being made about that word -- as in Sentence B, which follows. Sentence (B). After Grandma had extensive physical therapy, her balance improved tremendously. Now, there is no dangling modifier, and the sentence makes sense.

Another correction strategy is to leave intact the IMP, place the subject right after the comma that follows the IMP (in this case, "Grandma"), and end the sentence with an appropriately revised main clause that tells about the results of the physical therapy, as in Sentence C, which follows: 
Sentence (C). After undergoing extensive physical therapy, Grandma found that her balance had improved tremendously. Again, the dangling modifier is gone, and the sentence now makes sense.


Keeping in mind the information above, complete the exercise below.




SAT Quick Challenge N21
Misplaced Modifiers

Directions. . For each statement below, select the letter of the answer choice which corrects the underlined part of that statement. If you believe that the underlined part is already correct, select Choice A -- NO CHANGE. After completing this activity, use the answer key in the dropdown below to check your work.

1. Watching the evening news, the smoke alarm went off, and smoke spread throughout the house.

  1. NO CHANGE
  2. With the evening news
  3. While Mr. Carter watched the evening news
  4. Coughing as smoke filled the air

2.  To become such a great singer, regular, faithful practice daily was for Ava

  1. NO CHANGE
  2. regular, faithful practice was every day for Ava.
  3. Ava's dedicated practice was on a daily basis.
  4. Ava practiced faithfully each day.

3. Flying along the road in the new sports car, the world seemed to whiz by rapidly.

  1. NO CHANGE
  2. As Olivia raced
  3. Olivia speeding
  4. Dashing and zipping

SAT Verbal - Answers to Questions from Fall 2021, Week 5 (October 16, 2021)

  1. C
  2. D
  3. B


SAT Math - Questions from Fall 2021, Week 4 (October 9, 2021)

INEQUALITIES

In an equation, one side equals the other. In an inequality, the two sides are not equal.

The following symbols are used in inequalities:

≠    is not equal to
>    is greater than
<    is less than
≤    is less than or equal to
≥     is greater than or equal to

You can usually work with inequalities in exactly the same way you work with equations. You can collect similar terms, and you can simplify by doing the same thing to both sides: adding, subtracting, multiplying, dividing, raising to a power, or taking a root. An important caution: multiplying both sides of an inequality by a negative number reverses the direction of the inequality.

Examine examples 1, 2, and 3.

1. x > y    
    a. Add 2 to both sides
    b. Multiply both sides by 10
    c. Multiply both sides by -2

    a. x > y becomes x + 2 > y + 2
    b. x > y becomes 10x > 10y
    c. x > y becomes -2x < -2y

2. If 2x < 3 and 3x > 4, what is one possible value of x?
    The best approach here is to solve for x and then express each fraction as a decimal.
    2x < 3                                       3x > 4
      x < 3/2                                      x > 4/3
      x < 1.5                                       x > 1.33333

       Thus x is between 1.33333 and 1.5, not including 1.33333 and 1.5. Any value in this range would be acceptable.

3.   What values of x satisfy 7 + 2x – 5x ≥ – x + 13 – 4x ?

      Solve for x by adding, subtracting, multiplying, dividing, raising to a power, or taking a root.
      -3x + 7 ≥ –5x + 13
        2x ≥ 6
          x ≥ 3

          Thus x is any value greater than or equal to 3, including 3.

   Keeping in mind the information above, solve the following problems.

  1. 6x - 9y > 12
    Which of the following inequalities is equivalent to the inequality above?

    1. x - y > 2
    2. 2x - 3y > 4
    3. 3x - 2y > 4
    4. 3y - 2x > 2

  2. If -4 < x < -2, which of the following could be the value of 3x?

    1. -2.5
    2. -3.5
    3. -4.5
    4. -7.5
    5. -14.5

  3. What values of x satisfy 7x + 3 – 10x ≥ 4 + 2x + 14?

  4. If x + 6 > 0 and 1 – 2x > - 1, then x could equal each of the following EXCEPT
    1. -6
    2. -4
    3. 0
    4. 1/2

  5. If -6 < -4x + 10 ≤ 2, what is the least possible value of 4x + 3?
    1. 2
    2. 5
    3. 8
    4. 11

  6. If -8 < -(3/5)r + 1 ≤ -(16/5) , what is one possible value of r?
    1. 3
    2. 5
    3. 8
    4. 16

SAT Math - Answers to Questions from Fall 2021, Week 4 (October 9, 2021)

      1. 6x - 9y > 12
        Which of the following inequalities is equivalent to the inequality above?

        1. x - y > 2
        2. 2x - 3y > 4
        3. 3x - 2y > 4
        4. 3y - 2x > 2

        Divide each term by 3: 6x – 9x > 12 becomes 2x – 3y > 4                  --------> B
      2. If -4 < x < -2, which of the following could be the value of 3x?

        1. -2.5
        2. -3.5
        3. -4.5
        4. -7.5
        5. -14.5

        Multiply each term by 3: -4 < x < -2 becomes -12 < 3x < -6 It could be -7.5.             --------> D
      3. What values of x satisfy 7x + 3 – 10x ≥ 4 + 2x + 14?

        Solve for x: 7x + 3 – 10x ≥ 4 + 2x + 14
                                     3 – 3x ≥ 2x + 18
                                        – 5x ≥ 15
                                          – x ≥ 3
                                            x ≤ -3

      4. If x + 6 > 0 and 1 – 2x > - 1, then x could equal each of the following EXCEPT
          1. -6
          2. -4
          3. 0
          4. 1/2

        Solve each inequality and determine the range of values for x.
        x + 6 > 0                                  1 – 2x > - 1
        x > -6                                           –2x > - 2
                                                                 x < 1

        Here x could be any value between -6 and 1, not including -6 and 1. Thus x could be -4, 0, or ½. It could not be -6. The answer is A.

      5. If -6 < -4x + 10 ≤ 2, what is the least possible value of 4x + 3?
          1. 2
          2. 5
          3. 8
          4. 11

        What do we do here? We want to add, subtract, multiply, or divide so that -4x + 10 becomes 4x + 3.
        The first step is to change the -4x to 4x.
        We can make this change by multiplying each section by -1:
        -6 < -4x + 10 ≤ 2 becomes
         6 > 4x - 10 ≥ - 2

        Next we need to change the -10 to +3. We can make this change by adding 13 to each section:
        6 > 4x - 10 ≥ - 2 becomes
        19 > 4x +3 ≥ 11

        Thus 4x + 3 is between 11 and 19, not including 19. The lowest value is 11.     --------> D

      6. If -6 < -4x + 10 ≤ 2, what is the least possible value of 4x + 3?
        1. 3
        2. 5
        3. 8
        4. 16

                            We will solve for r. We will begin by multiplying each section by -1:
                            -8 < -(3/5)r + 1 ≤ -(16/5) becomes
                              8 > (3/5)r - 1 ≥ (16/5)

                              Now we multiply each section by 5 (to get rid of the 5 in the denominator).
                              8 > (3/5)r - 1 ≥ (16/5) becomes
                              40 > 3r - 5 ≥ 16

                              Now we add 5 to each section.
                              40 > 3r - 5 ≥ 16 becomes
                              45 > 3r ≥ 21

                              Finally we divide each section by 3. 45 > 3r ≥ 21
                              becomes 15 > r ≥ 7

                              Thus r is any value between 7 and 15, not including 15.

SAT Verbal - Questions from Fall 2021, Week 4 (October 9, 2021)

SAT QUICK CHALLENGE
Exercise M21 -- The Misplaced Modifier

A modifier is a word or word group that describes someone or something. If the modifier is not placed next to (or as close as possible to) the word/word group it should describe, it will be misplaced, and the sentence will not convey the meaning intended. The "possessive noun" and "descriptive aside" modifier errors are often tested on the SAT. Both are explained below.

The Possessive Noun Error. A possessive noun error is a situation in which a possessive noun is placed where a noun that is not possessive should be. Note the following sentence: Using endless patience and encouragement, Mrs. White's clear, understandable explanations helped me learn to solve math problems I had never been able to solve before.

EXPLANATION
: The closest noun phrase to the introductory modifier ("Using endIess patience and encouragement") is "Mrs. White's clear, understandable explanations." Accordingly, the sentence says that Mrs. White's explanations were using endless patience and encouragement to help me learn to do math, but explanations do not use endless patience and encouragement, and they do not teach; teachers do those things. Hence, the sentence has a possessive noun error.  
Note the following correction: "Using endless patience and encouragement, as well as clear, understandable explanations, Mrs. White helped me learn to solve math problems I had never been able to solve before." Now, the modifying phrases correctly modify "Mrs. White," and the sentence makes sense.

The Misplaced "Descriptive Aside" Modifier. Sometimes, a modifier that is not part of a sentence's main idea is added to a sentence, as follows: "Diana Carter does breathtaking stunts, the drum major, at the games." Enclosed in commas because it is extra, nonessential information, "the drum major" is interpreted as describing the noun it follows -- "stunts." Accordingly, the sentence says that "stunts" are the drum major. Obviously, the description is not in the right place, and the point that it actually makes where it is does not make sense because it identifies the word "stunts" as the drum major. Placed after "Diana Carter," the proper noun that it should modify, the modifier would say correctly that Diana is the drum major, as indicated in the sentence which follows: "Diana Carter, the drum major, does breathtaking stunts at the games." Now, keep in mind the information above, and complete Exercise M21 below. 


SAT Quick Challenge M21
The Misplaced Modifier

Directions. Select the letter of the answer choice which corrects the underlined part of each statement below. If you believe that the underlined part is already correct, select Choice A -- NO CHANGE. After completing this activity, use the answer key to check your work.

1. Practicing two hours every weekday and four hours every Saturday and Sunday, Ava's saxophone skills improved a great deal in just
    a few months.. _______

  1. NO CHANGE
  2. Ava improved her saxophone skills
  3. the saxophone skills of Ava improved
  4. improvement was noticed in Ava's playing

2. Granny Hawkins mixes her own spices, an energetic little lady in her late eighties, for her delicious, award-winning apple pies. To make this sentence grammatically correct, the underlined phrase should be placed

  1. NO CHANGE
  2. after the word "mixes"
  3. after the word "pies"
  4. after the word "Hawkins"

3. Overjoyed about finally beating their longtime competitor, the crowd's cheers and celebrations continued long after the game had ended.  ______

  1. NO CHANGE
  2. cheering and celebrating continued
  3. fans continued to cheer and celebrate
  4. the stadium was filled with cheering crowds

SAT Verbal - Answers to Questions from Fall 2021, Week 4 (October 9, 2021)

  1. B
  2. D
  3. C


SAT Math - Questions from Fall 2021, Week 3 (October 2, 2021)

Linear Equations: Word Problems

In recent administrations of the SAT there has been an increase in the number of word problems included. The following steps can help you solve word problems:

  1. Read the problem carefully and avoid misreading anything important.
    1. It is necessary to define the variables and create equations to represent relationships. State in words what the unknown is.
    2. Carefully determine the equality: identify what goes on the left side of the equal sign and what goes on the right side of the equal sign. Identify the key values and determine how they are related mathematically by converting the words to math.
  2. Solve the problem and interpret the solution.
  3. Make sure that the answer is reasonable. Ask yourself: Does it make sense? Does it answer the question that was asked?

Now examine examples 1, 2, and 3.

Example 1:
A computer that costs $800 is to be purchased with a down payment of $80 and weekly payments of $40. How many weekly payments will be necessary to complete the purchase, assuming that there are no taxes or fees?

To answer this question, we need to determine the elements of the equation: what the variable is, what goes on the right side of the equal sign, and what goes on the left side of the equation.
  1. The variable is the number of weekly payments necessary to complete the purchase.
  2. The cost of the computer, $800, goes on the right side of the equal sign.
  3. The amount needed to purchase the computer, the down payment plus the weekly payments, goes on the left side of the equal sign.
    80 + 40x = 800
    40x = 720
    x = 18                                    18 weekly payments are needed.

Example 2:
Harold and his roommate are buying a printer for the computer that they share. Since Harold will use the printer much more often than his roommate, Harold will pay $70 more than his roommate. If the printer costs $260, how much will the roommate pay?
  1. The variable is the amount the roommate will have to pay.
  2. The cost of the printer, $260, goes on the right side of the equal sign.
  3. The amount the roommate will pay, x, plus the amount Harold will pay, $70 + x, goes on the left side of the equal sign.
    x + 70 + x = 260
    70 + 2x = 260
    2x = 190
    x = 95                         The roommate will pay $95 and Harold will pay $165.

Example 3:
Oliver is selling photographs as part of a project for his entrepreneurship class. He sells the first 20 photographs for $10 each. Because the first 20 photographs sold so quickly, he raised the price of the photographs to $15 each for the rest of the project. After his expenses, Oliver earns a profit of 80% of the revenues from his sales. How many photographs must he sell for the rest of the project to earn a profit of $400?
  1. 18
  2. 20
  3. 24
  4. 32
The variable is the number of additional photographs he must sell. The profit of $400 goes on the right side of the equal sign. The total revenues times 80% goes on the left side of the equal sign. The total revenues = $10 times 20 + $15 times x. Thus we have:

0.8(200 + 15x) = 400
160 + 12x = 400
12x = 240
x = 20 ----------------------> B



Keeping in mind the information above, answer the following questions.

  1. A customer paid $53.00 for a jacket after a 6% sales tax was added. What was the price of the jacket before the sales tax was added?

    1. $47.60
    2. $50.00 
    3. $52.60
    4. $52.84
  2. Lynn has $8.00 to spend on apples and oranges. Apples cost $0.65 each, and oranges cost $0.75 each. If there is no sales tax on this purchase and she buys 5 apples, what is the maximum number of whole oranges she can buy?

  3. Sam plans to rent a boat. The boat rental costs $60 per hour plus a water safety course that costs $10. Sam has budgeted $280 for the rental and the course. If the boat rental is available only for a whole number of hours, what is the maximum number of hours for which Sam can rent the boat?

  4. A school district is forming a committee to discuss plans for the construction of a new high school. Of those invited to join the committee, 15% are parents of students, 45% are teachers from the current high school, 25% are school and district administrators, and 6 individuals are students. How many more teachers were invited to join the committee than school and district administrators?

  5. The mean score of 8 players in a basketball game was 14.5 points. If the highest individual score is removed, the mean score of the remaining 7 players becomes 12 points. What was the highest score?
    1. 20
    2. 24
    3. 32
    4. 36

  6. Max is working this summer as part of a crew on a farm. He earned $8 per hour for the first 10 hours he worked this week. Because of his performance, his crew leader raised his salary to $10 per hour for the rest of the week. Max saves 90% of his earnings from each week. How many hours must he work the rest of the week to save $270 for the week?
    1. 16
    2. 22
    3. 33
    4. 38


SAT Math - Answers to Questions from Fall 2021, Week 3 (October 2, 2021)

  1. A customer paid $53.00 for a jacket after a 6% sales tax was added. What was the price of the jacket before the sales tax was added?

    1. $47.60
    2. $50.00 
    3. $52.60
    4. $52.84

    The variable is the price of the jacket before the sales tax is added. The amount the customer paid of $53.00 goes on the right side of the equal sign. The initial price plus the sales tax goes on the left side of the equal sign.

    x + .06x = 53
    1.06x = 53
    x = 50 ----------------> B
  2. Lynn has $8.00 to spend on apples and oranges. Apples cost $0.65 each, and oranges cost $0.75 each. If there is no sales tax on this purchase and she buys 5 apples, what is the maximum number of whole oranges she can buy?

    The variable is the number of oranges she can buy. The amount she can spend, $8.00, goes on the right side of the equal sign. The amount spent on apples and oranges goes on the left side of the equal sign.

    0.65 times 5 + 0.75 times x = 8.00
    3.25 + 0.75x = 8
    0.75x = 4.75
    x = 6.33                          She can buy 6 whole oranges.

  3. Sam plans to rent a boat. The boat rental costs $60 per hour plus a water safety course that costs $10. Sam has budgeted $280 for the rental and the course. If the boat rental is available only for a whole number of hours, what is the maximum number of hours for which Sam can rent the boat?

    The variable is the number of hours he can rent the boat. The amount he has budgeted, $280, goes on the right side of the equal sign. The rental cost plus the cost of the water safety course goes on the left side of the equal sign.

    Thus our equation is:
    60x + 10 = 280
    60x = 270
    x = 4.5
    He can rent a boat for a maximum of 4 hours since the boat can be rented only for a whole number of hours.

  4. A school district is forming a committee to discuss plans for the construction of a new high school. Of those invited to join the committee, 15% are parents of students, 45% are teachers from the current high school, 25% are school and district administrators, and 6 individuals are students. How many more teachers were invited to join the committee than school and district administrators?

    The variable is the number of people invited to join the committee. The total number of people invited to join the committee goes on the right side of the equal sign. The total of the individual groups goes on the left side of the equal sign.

    Thus our equation is:
    0.15x + 0.45x + 0.25x + 6 = x
    0.85x + 6 = x
    6 = 0.15x
    x = 6 / 0.15 = 40
    40 people were invited to join the committee.

    Teachers:                                                  0.45(40) = 18
    School and district administrators:        0.25(40) = 10
                                                                                        ---
                                                                                           8    Our answer is 8.

  5. The mean score of 8 players in a basketball game was 14.5 points. If the highest individual score is removed, the mean score of the remaining 7 players becomes 12 points. What was the highest score?
    1. 20
    2. 24
    3. 32
    4. 36

    The variable is the highest individual score. The total number of points scored by the remaining 7 players goes on the right side of the equal sign. The total number of points scored by all 8 players minus the highest individual score goes on the left side of the equal sign.

    Remember that in a mean (average) problem the first thing to do is to find the total. There are two ways to find the total: 1) add all of the items; 2) multiply the mean by the number of items. On the SAT the second method is used more often.

    The total of the seven players = 7(12) = 84
    The total of the eight players = 8(14.5) = 116

    The equation is:
    116 - x = 84
    -x = -32
    x = 32 --------------------> C

  6. Max is working this summer as part of a crew on a farm. He earned $8 per hour for the first 10 hours he worked this week. Because of his performance, his crew leader raised his salary to $10 per hour for the rest of the week. Max saves 90% of his earnings from each week. How many hours must he work the rest of the week to save $270 for the week?
      1. 16
      2. 22
      3. 33
      4. 38

    The variable is the number of additional hours he must work. The savings of $270 goes on the right side of the equal sign. The total earnings times 90% goes on the left side of the equal sign. The total earnings = $8 times 10 + $10 times x. Thus we have: 0.9(80 + 10x) = 270

    72 + 9x = 270
    9x = 198
    x = 22 --------------------> B

SAT Verbal - Questions from Fall 2021, Week 3 (October 2, 2021)

SAT QUICK CHALLENGE
Exercise L21 -- Noun Agreement Errors and Faulty Comparison

Then or Than for Comparisons? (1) Use "then" to indicate "when," as in the following: "First, do your homework. Then, you can get on social media." Getting on social media must wait until after the homework has been done. Did you note (a) that "then" and "when" rhyme and (b) that both words end with "en"? (2) Use "than" to make a comparison, as in the following: "Cindy runs faster than Diana." Did you note that both "comparison" and "than" have the letter "a" in them? Use than to compare!

Consistency -- Noun and Pronoun Synonyms. When a writer uses a noun repeatedly, a synonymous noun or pronoun should replace some of those repetitions. Moreover, to maintain consistency, the writer must replace singular nouns with singular counterparts and plural nouns with plural counterparts.

Using "That of" and "Those of." Both "that of" and "those of" (1) show ownership and (2) help you avoid being repetitive. For example, to compare the success of two singers, you could say, "Singer A's annual income is higher than the annual income of Singer B." However, to be concise, you could say, "Singer A's annual income is higher than that of Singer B." For plural nouns, use the pronoun "those" in place of "that." Hence, you could say, "Singer A's annual sales are higher than those of Singer B."

Comparing Similar Things. Because there is no basis for comparison without recognized similarities or differences, writers must compare the same kinds of things: living things with living things, cars with cars, sports with sports, etc. In fact, a common kind of comparison error on the SAT is a statement which does not compare the same kinds of things. Therefore, you must be able to recognize and correct such errors, often referred to as "faulty comparisons." Note the faulty comparison error in the next sentence, as well as various corrections beneath the sentence: Tamela Mann's music is less popular than Kirk Franklin.

EXPLANATION AND CORRECTIONS: That sentence does not compare the same kinds of things. It compares music (Tamela Mann's music) with a person (singer Kirk Franklin). Note corrections 1-3 below.

  1. Tamela Mann's music is less popular than the music of Kirk Franklin. (compares music with music)
  2. Tamela Mann's music is less popular than Kirk Franklin. (compares music with music)
  3. Tamela Mann's music is less popular than that of Kirk Franklin. (compares music with music; the singular pronoun "that" takes the place of the singular noun phrase "the music")

    Now, using the information above, complete the exercise below. Then use the answer key in the dropdown below to check your work.

SAT QUICK CHALLENGE Exercise L21
Noun Agreement Errors and Faulty Comparisons

Directions. For each sentence below, select the answer choice that corrects the error in the underlined part. If you think the underlined part is already correct, select choice A -- NO CHANGE. When you have finished the exercise, use the answer key to check your work.

1. Cindy Davis said that her cousin's jokes are funnier than Jeff Allen. _______

  1. NO CHANGE
  2. than those of Jeff Allen's
  3. than those of Jeff Allen
  4. as those of Jeff Allen's

2. Mr. Stuart's trained service dogs have instincts of a mature adult. ____

  1. NO CHANGE
  2. similar to those of a mature adult
  3. of a young adult
  4. similar to a young adult

3. After weeks of private coaching sessions, Olivia now sings better than anyone else in her choir. ______

  1. NO CHANGE
  2. now sings better then anybody else
  3. now sings better then anyone
  4. now sings better than anyone else

SAT Verbal - Answers to Questions from Fall 2021, Week 3 (October 2, 2021)

  1. C
  2. B
  3. D

September Lessons


SAT Math - Questions from Fall 2021, Week 2 (September 25, 2021)

Linear Equations: one solution, no solution, and an infinite number of solutions

Linear equations may have one solution, no solution, or an infinite number of solutions. On the SAT most of the linear equation problems will have one solution. It is important to recognize, however, that there will occasionally be problems where there is no solution or an infinite number of solutions.

There will be one solution when we can isolate the variable for which we are solving on one side of the equal sign and the other parts of the equation on the other side. By adding, subtracting, multiplying, dividing, raising to a power, or taking a root, we arrive at a situation such as “x = 4” or “x = 17”.

There will be no solution when we make our algebraic manipulations and arrive at a situation such as “2x + 7 = 2x + 5” or “ + 7 = + 5”. The coefficient of the x term on the left side of the equation is the same as the coefficient of the x term on right side of the equation, and the constant on the left side of the equation is different from the constant on the right side of the equation. There is no value of x that will make both sides of the equation equal, and + 7 can never equal + 5.

There will be an infinite number of solutions when we make our algebraic manipulations and arrive at a situation such as “5x + 3 = 5x + 3”. The coefficient of the x term on the left side of the equation is the same as the coefficient of the x term on the right side of the equation, and the constant on the left side of the equation is the same as the constant on the right side of the equation. All possible values of x will make both sides of the equation equal.

Consider the following example (example 1):

Solve the following equation: 7x – 6 –x = 12 + 2x + 10
Add or subtract to combine like terms: 6x – 6 = 22 + 2x

Subtract 2x from both sides and add 6 to both sides: 4x = 28
Divide both sides by 4: x = 7. There is one solution for example 1.


Consider another example (example 2):

Solve the following equation: x + 5 + 3x = 6x – 2 – 6
Add or subtract to combine like terms: 4x + 5 = 4x – 8
There is no value of x that will make both sides of the equation equal, and + 5 can never equal – 8.
There is no solution for example 2.



Consider another example (example 3):


Solve the following equation: 3x – 9 + 5x + 4 = 11x – 5 – 3x
Add or subtract to combine like terms: 8x – 5 = 8x – 5
In example 3 there is an infinite number of solutions. All possible values of x will make both sides of the equation equal.


Keeping in mind the information above, solve the following problems.
  1. Solve the following equation: 3(x + 2) + 5x – 1 = 7x – 3 + x

  2. If 2x + 8 = 16, what is the value of x + 4?

  3. 2(9x – 6) - 12 = 3(9x – 6)
    Based on the equation above, what is the value of 3x – 2?

  4. Solve the following equation: 3(2x + 1) – 2x = 8x + 5 – 4x – 2

  5. 9ax + 9b – 6 = 21
    Based on the equation above, what is the value of ax + b?

    1. 3
    2. 6
    3. 8
    4. 12
  6. 2ax – 15 = 3(x + 5) + 5(x – 1)
    In the equation above, a is a constant. If no value of x satisfies the equation, what is the value of a?

    1. 1
    2. 2
    3. 4
    4. 8

  7. a(x + b) = 4x + 10
    In the equation above, a and b are constants. If the equation has infinitely many solutions for x, what is the value of b?

SAT Math - Answers to Questions from Fall 2021, Week 2 (September 25, 2021)

  1. Solve the following equation: 3(x + 2) + 5x – 1 = 7x – 3 + x
    3x + 6 + 5x – 1 = 6x – 3
    8x + 5 = 8x – 3
    There is no solution.

  2. If 2x + 8 = 16, what is the value of x + 4?

    Solve for x; then solve for x + 4.   
    2x + 8   =  16
    2x  =  8
     x  =  4

    x + 4  =  4 + 4  =  8      This is our answer.


    Recognizing that (2x + 8) is a multiple of (x + 4), there is an alternative method for finding our answer. We can divide both sides by 2.
    2x + 8 = 16

    Dividing both sides by 2, we get x + 4 = 8.
    This is our answer: x + 4 = 8, which is the same as the answer above.

  3. 2(9x – 6) - 12 = 3(9x – 6)
    Based on the equation above, what is the value of 3x – 2?

    18x – 12 - 12 = 27x – 18
    18x – 24 = 27x - 18
    -9x = 6
    -x = 6/9 = 2/3
    x = -2/3
    3x – 2 = 3(-2/3) – 2 = - 6/3 – 2 = - 2 – 2 = -4 This is our answer.


    Recognizing that (9x – 6) is a multiple of (3x – 2), there is an alternative method for finding our answer. We can subtract 2(9x – 6) from both sides.
    2(9x – 6) - 12 = 3(9x – 6)

    Subtracting 2(9x – 6) from both sides, we get -12 = 9x – 6.
    Divide both sides by 3: -4 = 3x – 2. This is our answer: 3x – 2 = -4, which is the same as the answer above.

  4. Solve the following equation: 3(2x + 1) – 2x = 8x + 5 – 4x – 2

    6x + 3 – 2x = 4x + 3
    4x + 3 = 4x + 3
    There is an infinite number of solutions.

  5. 9ax + 9b – 6 = 21
    Based on the equation above, what is the value of ax + b?

    1. 3
    2. 6
    3. 8
    4. 12


    Some students give up on problems of this type. They want to get the value of a, multiply it by the value of x, and add the value of b. Since they cannot find the values of these variables, they do not know what to do. Actually, we cannot determine the value of a, we cannot determine the value of x, and we cannot determine the value of b, but we can find the value of ax + b.

    9ax + 9b - 6 - 21
    Add 6 to both sides:  9ax + 9b = 27
    Divide both sides by 9:  ax + b = 3 ----------> A     This is our answer.

  6. 2ax – 15 = 3(x + 5) + 5(x – 1)
    In the equation above, a is a constant. If no value of x satisfies the equation, what is the value of a?
    1. 1
    2. 2
    3. 4
    4. 8

    For no solution, we want to arrive at a situation where the coefficient of the x term on the left side of the equation is the same as the coefficient of the x term on right side of the equation, and the constant on the left side of the equation is different from the constant on the right side of the equation.

    2ax – 15 = 3(x + 5) + 5(x – 1)
    2ax – 15 = 3x + 15 + 5x – 5
    2ax – 15 = 8x + 10

    The constant on the left side of the equation ( -15) is different from the constant on the right side of the equation (+10). Now we need the coefficients of the x terms to be the same. Therefore 2a must equal 8. 2a = 8.
    a = 4 ----------> C

  7. a(x + b) = 4x + 10
    In the equation above, a and b are constants. If the equation has infinitely many solutions for x, what is the value of b?

    For there to be infinitely many solutions, we want to arrive at a situation where the coefficient of the x term on the left side of the equation is the same as the coefficient of the x term on right side of the equation, and the constant on the left side of the equation is the same as the constant on the right side of the equation.

    a(x + b) = 4x + 10
    ax + ab = 4x + 10

    ax = 4x and ab = 10
    Thus a must equal 4 and ab must equal 10.

    a = 4
    ab = 10
    Since a = 4, 4b = 10
    b = 10/4 = 5/2 = 2.5

SAT Verbal - Questions from Fall 2021, Week 2 (September 25, 2021)

SAT Quick Challenge K21 - Faulty Comparisons

Some SAT questions test to see if you know how to use correctly word pairs routinely used to point out similarities and differences. Note the commonly used word pairs in the chart below. Remember that you must always use the pairs together; do not mix or match them with one another, or with any other words. After studying the information, complete the exercise that follows.


WORD PAIRS ROUTINELY USED FOR MAKING COMPARISONS

Comparison Word Pair What the Comparison Does Sample Sentence(s) Explanation of What Job Word Pair Does
1. As...as Indicates that two people or things are equal Ann runs as fast as Doris does. Indicates that Ann and Doris run equally fast.
2. Not only...but also  Points out two different qualities of a person on thing Beyonce is not only a great singer, but also a fantastic dancer. Names two of Beyonce's great characteristics
3. More/...er than Shows that one of two people or things has a larger or (smaller) portion or amount of a characteristic or quality (1) than that person or thing had at another time or (2) than another person or thing has or had This movie was more frightening than the other one.

This chair is stronger than that one.

Jeff Bezos is much wealthier than Richard Branson.

Tells which of two movies frightened people more.

Tells which of two chairs has more strength.

Tells which of two billionaires has more wealth.

4. Neither...nor Indicates that none of two or more possible participants will actually participate. Neither Victoria nor her mother will go to see the play. Indicates that not even one of two possible participants -- Victoria and her mother -- will go to see the play.

SAT QUICK CHALLENGE EXERCISE K21 -- Faulty Comparisons

Directions. Keeping in mind the information above, select the answer choice that corrects the underlined part in each question. If you think that part is already correct, select choice A (NO CHANGE) as your answer.

  1. The two little boys were arguing about which of them has the most strongest muscles.
    1. NO CHANGE
    2. most strong
    3. stronger
    4. strongest

  2. It's only spring, but last week's temperatures were as hottest than the hottest days last summer.
    1. NO CHANGE
    2. more hot than
    3. as hotter than
    4. as hot as

  3. Neither the schools or the churches will be open for regular activities for the next three months
    because of the pandemic that is causing so much severe illness and death.
    1. NO CHANGE
    2. the schools nor
    3. the schools and
    4. the schools yet

SAT Verbal - Answers to Questions from Fall 2021, Week 2 (September 25, 2021)

  1. C
  2. D
  3. B

SAT Math - Questions from Fall 2021, Week 1 (September 18, 2021)

Linear Equations

Understanding how to solve linear equations is essential for solving higher level math problems. Performing well on many of the problems on the SAT will require a mastery of equation manipulation.

An equation will show that two expressions are equal. To solve the equation, we isolate the variable for which we are solving (usually x, but it can be any letter) on one side of the equal sign and the other parts of the equation on the other side.

Important note: whatever we do to one side of the equation, we must do the same for the other side of the equation. We can perform several operations to both sides of the equation: add, subtract, multiply, divide, raise to a power, or take a root. For example, if we add 3x to one side of the equation, we must add 3x to the other side of the equation; if we divide one side of the equation by 2, we must also divide the other side by 2.

Keeping in mind the information above, solve the following problems.


Solve the following equations:

  1. 4x + 8 = 36

  2. 2x – 7 = 3x + 4

  3. 6x + 3 = 4x – 9

  4. 4(b – 5) = 3(4b +4)

  5. (8x +5)/4 = (3x - 9)/2

  6. Solve for x in terms of a, b, and c.
    5a = (3x + b)/4c

  7. For what value of m do the equations 3x – 4 = 11 and mx – 3 = 27 have the same value of x?


SAT Math - Answers to Questions from Fall 2021, Week 1 (September 18, 2021)

  1. 4x + 8  =  36 
    4x  =  28 
    x  =   7 

  2. 2x – 7  =  3x + 4   
    – x   =  11         
    x =  –11  

  3. 6x + 3  =  4x – 9
    2x  =  – 12 
    x   =  – 6

  4. 4(x – 5)  =  3(4x +4)
    4x – 20  =  12x + 12   
    – 8x  =  32         
    x  = – 4

  5. (6x +5)/4  =  (3x - 9)/2
    Cross multiply:  2(8x + 5)  =  4(3x – 9)                             
                               16x + 10  =  12x – 36
                               4x  =  – 46    
                                 x  =  – 46/4  =  – 23/2

  6. Solve for x in terms of a, b, and c.  
    5a  =  (3x + b)/4c
    20ac  =  3x + b
    20ac – b  =  3x           
                x  =  (20ac – b)/3  

  7. For what value of m do the equations 3x – 4  =  11   and   mx – 3  =  27   have the same value of x?

    Solve for x in the first equation; then substitute this value of x in the second equation and solve for m.           
    3x – 4  =  11                 
          3x  =  15                   
            x  =  5

     mx – 3  =  27              
     5m – 3  =  27                    
     5m  =  30                      
       m  =  6

SAT Verbal - Questions from Fall 2021, Week 1 (September 18, 2021)

SAT Quick Challenge J21 - Using the Correct Word

Deciding which of two commonly confused words should be used may not be as difficult as one might think
initially. The tips below explain how to use some words in that category correctly.

Words in Question A: "less" vs. "fewer"
RULE: If the word in question is followed by a singular noun and refers to something that cannot
be counted, use the word "less." If the word in question is followed by a plural noun and refers to
something that can be counted, use the word fewer.

Sentence 1
: During meals, Mrs. Parker usually has _____ food on her plate than her son has on
his. Explanation 1: A singular noun (food) follows the word in question and names something
that cannot be counted. (NOTE: You can count individual food items, but not the concept of
"food.") Therefore, the missing word must be "less." Correct Sentence: Mrs. Parks usually
has less food on her plate than her son has on his.

Sentence 2
:
When I take my time, I make _____ mistakes than I do when I rush. Explanation 2: A
plural noun (mistakes) follows the word in question and refers to something that can be
counted. Therefore, the missing word is "fewer." Correct Sentence: When I take my time, I
make fewer mistakes than I do when I rush.

Words in Question B: "much" vs. "many"
RULE: If the word in question is followed by a singular noun and refers to something that cannot
be counted, use the word "much." If the word in question is followed by a plural noun and refers
to something that can be counted, use the word many.
Sentence 3: During meals, Mrs. Parker's son usually has _____ more food on his plate than she
has on hers. Explanation 3: A singular noun (food) follows the word in question and names
something that cannot be counted. (NOTE: You can count individual food items, but not the
concept of "food.") Therefore, the missing word is "much." Correct Sentence: Mrs. Hall's
son usually has much more food on his plate than she has on hers.
Sentence 4: When I rush, I make _____ more mistakes than I do when I take my time. Explanation
4: A plural noun ("mistakes") follows the word in question and refers to something that can be
counted. Therefore, the missing word is "many." Correct Sentence: When I rush, I make many
more mistakes than I do when I take my time.


Keeping in mind the information above, complete Exercise J21 below.

Exercise J21

Directions. On the line that follows each statement below, place the letter of the answer choice that corrects any error in the statement. If there is no error, mark choice "A" as your answer. When you have finished the exercise, use the answer key to check your work.

1. When Ella works carefully, she makes far less mistakes than she makes when she rushes carelessly. _______

  1. NO CHANGE
  2. lesser
  3. lots of fewer
  4. fewer

2. We scored much more points during the game this week than we scored last week. ____

  1. NO CHANGE
  2. lots of more
  3. many more
  4. more better

3. Our Booster Club sold less boxes of cookies at the school fair than they sold at the mall. ______

  1. NO CHANGE
  2. fewer boxes
  3. lesser boxes
  4. fewest boxes

SAT Verbal - Answers to Questions from Fall 2021, Week 1 (September 18, 2021)

  1. A
  2. C
  3. B

August Lessons


SAT Math - Questions from Summer 2021, Week 7 (August 28, 2021)

Percent Change

Percent change problems can be somewhat confusing because there may be a question in the minds of some students as to which value to use as a denominator when trying to determine the percent increase or decrease of an amount. We should note that for finding percent changes, the denominator is always the original value or beginning value.

Actually percentage change problems can involve finding:
  1. the percent change,
  2. the amount of change,
  3. the final amount after a percent increase, and
  4. the final amount after a percent decrease.
  1. The percent change = the amount of change/the beginning amount
    Example: A department store was selling a shirt for $20. To cover increased costs, the price was increased by $3. What was the percent change? 3/20 = 0.15 = 15%

  2. There are two ways to determine the amount of change:
    1. The amount of change = The final amount - the beginning amount
    2. The amount of change = The beginning amount x the percent change

      Example:
      • The price of KYZ stock closed at $75 per share today.  The closing price yesterday was $70.  What was the amount of change?     $75  -  $70  =  $5a) The price of KYZ stock closed at $75 per share today.  The closing price yesterday was $70.  What was the amount of change?     $75  -  $70  =  $5

      • Yesterday the closing price of MYQ stock was $35 per share.  The price today increased by 12%.  What was the amount of change?    $35  x  0.12  =  $4.20

  3. There are two ways to determine the final amount after a percent increase.
    1. The final amount   =  The beginning amount + the amount of increase

    2. The final amount  =  The beginning amount(1 + percent increase)  where the percent                                    increase is expressed as a decimal
      Example:
      1. There were 40 members in the Honor Society in 2020, and the number of members increased by 10% in 2021.  How many members were there in 2021?a) There were 40 members in the Honor Society in 2020, and the number of members increased by 10% in 2021.  How many members were there in 2021?
        Amount of increase = 40 x 0.10  =  4
        Final amount  =  40  +  4  =  44

      2. There were 40 members in the Honor Society in 2020, and the number of members increased by 10% in 2021.  How many members were there in 2021?
        Final amount  =  40(1 + 0.10)  =  40(1.10)  =  44

  4. There are two ways to determine the final amount after a percent decrease.
    1. The final amount   =  The beginning amount - the amount of decrease

    2. The final amount  =  The beginning amount(1 - percent decrease)  where the percent                              decrease is expressed as a decimal
        Example:
        1. The budget for the recreation department was $120,000 in 2019. Because of the pandemic, it was decreased by 30% in 2020. What was the budget amount for 2020?
          Amount of decrease = $120,000 x 0.30 = $36,000
          Final amount = $120,000 - $36,000 = $84,000

        2. The budget for the recreation department was $120,000 in 2019. Because of the pandemic, it was decreased by 30% in 2020. What was the budget amount for 2020?
          Final amount = $120,000 (1 - 0.3) = $120,000(0.7) = $84,000

Successive percent increases and percent decreases can be multiplied together.

Note: they are not added together; they are multiplied together.

Example: A clothing store has a suit that has attracted Martin’s attention. The price of the suit has been $250 but the store just announced a discount of 20%. Martin has a coupon for an additional 15% discount to be applied to the already discounted price of the suit. If Martin buys the suit, how much will be pay after both discounts are applied and a 7% sales tax is added to the final price?
- Price after 20% discount = $250(1 – 0.2) = $250(0.8) = $200
- Price after 15% coupon = $200(1 – 0.15) = $200(0.85) = $170
- Final price after 7% tax is added = $170(1 + 0.07) = $170(1.07) = $181.90

We could also write: final price = $250(0.8)(0.85)(1.07) = $181.90.

When successive percent increases or decreases are the same for each successive period, we can use an exponent.

Example: David deposits $1,000 into a bank account that earns an annual interest rate of 3%. How much would he have in the account at the end of one year?
- Final amount = $1,000(1.04) = $1,040

If he does not make any additional deposits and makes no withdrawals, how much would he have in the account at the end of two years?
- Final amount = $1,040(1.04) = $1,081.60 or
- Final amount = $1,000(1.04)(1.04) = $1,081.60 or
- Final amount = $1,000(1.04)2 = $1,000(1.0816) = $1,081.60

If he does not make any additional deposits and makes no withdrawals, how much would he have in the account at the end of three years?
- Final amount = $1,081.60(1.04) = $1,124.86 or
- Final amount = $1,000(1.04)(1.04)(1.04) = $1,124.86 or
- Final amount = $1,000(1.04)3 = $1,000(1.124864) = $1,124.86

This pattern can be used for any number of periods, as long as the interest rate (or percent increase) remains constant and there are no more deposits and no withdrawals.

How much would he have in the account at the end of 12 years?
- Final amount = $1,000(1.04)12 = $1,000(1.60103) = $1,601.03


Keeping in mind the information above, answer the following questions.

  1. The Mason Company increased the salary of its treasurer from $380,000 to $400,000. What was the
    percent increase?

  2. Lloyd received a hot tip about the Zylon Company, a new company in the computer industry with
    great prospects for the future. He was persuaded to invest $10,000 in the stock of the company. Shortly after making the investment the value of the stock dropped by 30%. Several months later the value of the stock increased by 30%. Was this increase sufficient to bring Lloyd’s value back to its original value?

  3. Jackie is a sales clerk at Dade Clothing Store where a 25% clearance sale is currently in effect. Jackie is planning to buy a pair of shoes that originally cost $40, and she will apply her 10% employee discount in addition to the 25% clearance sale discount. How much will she pay for the shoes when an 8% sales tax is added to the final price?

  4. In the current school year 400 students are enrolled in a new major in cyber security at Goldsboro College. The college administration expects an increase of 5% enrollment in the major each year for the next three academic years. If no students drop out of the major, approximately how many students will be enrolled in the major at the end of 3 academic years?

SAT Math - Answers to Questions from Summer 2021, Week 7 (August 28, 2021)

  1. The Mason Company increased the salary of its treasurer from $380,000 to $400,000. What was the
    percent increase?
    Percent increase = ($400,000 - $380,000)/$380,000 = $20,000/$380,000 = 0.0526 = 5.26%

  2. Lloyd received a hot tip about the Zylon Company, a new company in the computer industry with
    great prospects for the future. He was persuaded to invest $10,000 in the stock of the company. Shortly after making the investment the value of the stock dropped by 30%. Several months later the value of the stock increased by 30%. Was this increase sufficient to bring Lloyd’s value back to its original value?

    - Value after 30% drop = $10,000(0.70) = $7,000
    - Value after 30% increase = $7,000 = $7,000(1.30) = $9,100

    The answer is no; a 30% increase is not sufficient to bring the value of the investment back to
    $10,000. This problem illustrates an important fact: percent decrease and percent increase are
    not mirror images of each other.

  3. Jackie is a sales clerk at Dade Clothing Store where a 25% clearance sale is currently in effect. Jackie is planning to buy a pair of shoes that originally cost $40, and she will apply her 10% employee discount in addition to the 25% clearance sale discount. How much will she pay for the shoes when an 8% sales tax is added to the final price?

    Price after 25% discount = $40(1 – 0.25) = $40(0.75) = $30
    Price after 10% employee discount = $30(1 – 0.10) = $30(0.9) = $27
    Final price after 8% tax is added = $27(1 + 0.08) = $27(1.08) = $29.16

    We could also write: final price = $40(0.75)(0.9)(1.08) = $29.16

  4. In the current school year 400 students are enrolled in a new major in cyber security at Goldsboro College. The college administration expects an increase of 5% enrollment in the major each year for the next three academic years. If no students drop out of the major, approximately how many students will be enrolled in the major at the end of 3 academic years?

    Number projected to be enrolled = 400(1.05)3 = 400(1.157625) = 463.05
    Therefore, approximately 463 students will be enrolled in the major.

SAT Verbal - Questions from Summer 2021, Week 7 (August 28, 2021)

SAT Quick Challenge U21C - Parallelism


Infinitives and Gerunds. An infinitive is often described as a word (1) that looks like a verb with the word "to" in front of it, but (2) does not function as a verb. Although an infinitive can function as different parts of speech, it often functions as a noun, as in Sentence A, which follows.
(A) To play the drums in the band has always been Lynn's dream. Since the subject of Sentence A is the infinitive "To play," it is functioning as a noun. A gerund looks like a verb that ends in "ing," but it functions as a noun, Note Sentence B, which follows.
(B) Singing in the choir is Maggie's favorite activity at church. In this sentence, the subject is the gerund "Singing."

Although both infinitives and gerunds can function as nouns, they must not be mixed within the same sentence. Note Sentences C, D, and E, which follow. (C) Ella loves hiking in the woods and to swim at the beach. This sentence is not parallel because it contains both a gerund (hiking) and an infinitive (to swim). Now, note Sentences D and E. (D) Ella loves hiking in the woods and swimming at the beach. Note that the infinitive "to swim" (from Sentence C) has been replaced with the gerund "swimming." Hence, the sentence is now parallel. (E) Ella loves to hike in the woods and to swim at the beach. The gerund hiking (from Sentence C) has been replaced with the infinitive to hike, and this sentence is now parallel.

Do Not Mix Phrases and Clauses in the Same Sentence. Parallelism requires that similar items in a sentence (a list, a series, etc.) be expressed in the same way-- whether those items are individual words, phrases, or clauses. Note Sentence F, which follows. (F) Ernie says that he would enjoy playing football, running track, and he wants to play in the band. This sentence is not parallel because the two phrases in it ("playing football" and "running track") are followed by a clause ("he wants to play in the band"). All three elements in the series must follow the same format, as illustrated in Sentences G and H. Sentence G. Ernie says that he would enjoy playing football, running track, and playing in the band. This sentence is correct because all three elements are expressed as gerund phrases. Sentence H. Ernie says that he would like to play football, run track, and play in the band. This sentence is correct because all three items in the series are expressed as infinitive phrases.

Now, complete the Quick Challenge Exercise below.

QUICK CHALLENGE Practice Exercise U21C - Parallelism

DirectionsReplace the underlined word(s) in each question below with the answer choice that corrects the error in the sentence. If there is no error, select choice A -- NO CHANGE.

1. Whether you will be serving food, cleaning up, or when you operate the cash register, we thank you for helping with our annual spaghetti dinner.

  1. NO CHANGE
  2. near the porch, the mailbox, and along
  3. near the porch, the mailbox, and
  4. near the porch, mailbox, and

2. The Browns want to visit New York to see interesting places, to sample the food, and they would love going to the theaters there.  

  1. NO CHANGE
  2. shows in the theater will be great
  3. they want to see the theater productions
  4. to enjoy the shows

3. Ernie says that he would like playing basketball, running track, and to race sports cars.

  1. NO CHANGE
  2. racing sports cars
  3. he'd enjoy racing sports cars
  4. he loves sports car races

4. People form opinions of you based on what you say and your words.

  1. NO CHANGE
  2. promises made by you
  3. your promises
  4. what you do

SAT Verbal - Answers to Questions from Summer 2021, Week 7 (August 28, 2021)

  1. C
  2. D
  3. B
  4. D

SAT Math - Questions from Summer 2021, Week 6 (August 21, 2021)

Percent Word Problems

In recent administrations of the SAT, there has been an increase in the number of word problems included. The following steps can help you solve word problems:
1. Read the problem carefully an avoid misreading anything important.
2. Identify the key values and determine how they are related mathematically.
3. Solve the problem.
4. Make sure that the answer is reasonable. Ask yourself: does it make sense?

Now examine examples 1 and 2.

Example 1:
All of the students in Mr. Taylor’s math took Exam 1, and each student either passed or failed the exam. 80% of the students passed and 5 students failed. How many students passed the exam?

To answer this question, we need to determine the total number of students in the class. We know that 5 students failed the test and these 5 students represent 20% of the total number of students in the class (100% total - 80% that passed = 20% that failed).
- Thus 20% of the total = 5, or 0.2(Total) = 5;
- Thus the total = 5/0.2 = 25
- There are 25 students in the class. We then multiply 80% times 25 to get the number that passed: 25 x 0.8 = 20.

Example 2:
The Girl Scouts take 500 palm leaves to pin on church members on Palm Sunday. They pinned 80% of the palm leaves on the adult members. They then pinned 40% of the remaining leaves on teen age members. Finally, they pinned 30% on the remaining leaves on the younger members. How many leaves did they have left?

First, they pinned 80% of the leaves on the adult members.
We compute 80% of 500: 0.8 x 500 = 400.
400 are pinned on the adult members.
Thus, they had 500 – 400 = 100 remaining.

Next, they pinned 40% of their remaining leaves on the teenagers.
We compute 40% of 100: 0.4 x 100 = 40.
40 were pinned on the teenagers.
Thus, they had 100 – 40 = 60 remaining.

Finally, they pinned 30% of their remaining leaves on the younger members.
We compute 30% of 60: 0.3 x 60 = 18.
18 are pinned on the younger children.
Thus, they had 60 – 18 = 42 left.

Here is an alternative calculation:
500 x 0.2 = 100 remaining after pinning on the adult members (100% - 80% = 20%) 100 x 0.6 = 60 remaining after pinning on the teenagers.
Finally, 60 x 0.7 = 42 were left after pinning on the younger children. 

Keeping in mind the information above, answer the following questions.
  1. John and Sam played chess against each other many times during the pandemic. Sam won 15% of the time and John won the other 51 games. There were no draws or ties. How many games did Sam win?

  2. In a tryout for the football team, 20 students trying out were seniors and 80 were underclassmen. What percent of the students trying out were seniors?

  3. A large 200 quart container of juice consisted of 60% orange juice and 40% lemonade. 30% of the juice was used, and then 20 quarts of lemonade were added to the container. What percentage of the mixture was lemonade after the new addition?

  4. 800 students applying to Atlantic University took the entrance exam. 80% of the students passed the exam and 20% failed it. Suppose instead only 5% of the students had failed the exam. How many more students would have passed the exam?

SAT Math - Answers to Questions from Summer 2021, Week 6 (August 21, 2021)

  1. John and Sam played chess against each other many times during the pandemic. Sam won 15% of the time and John won the other 51 games. There were no draws or ties. How many games did Sam win?
    - Sam won 15% of the total number of games played, and John won the other 85%.
    - John’s 51 games represent 85% of the total number of games, or 85% of the total = 51, or 0.85 (total) = 51.
    - Thus, the total = 51/0.85 = 60 games played. Sam won 0.15 x 60 = 9 games.

  2. In a tryout for the football team, 20 students trying out were seniors and 80 were underclassmen. What percent of the students trying out were seniors?
    - Some students will incorrectly divide 20 by 80 and get 20/80 = 0.25 = 25%.
    - Actually, we should divide 20 by the total, which is 100:
       20/100 = 0.2 = 20%.

  3. A large 200 quart container of juice consisted of 60% orange juice and 40% lemonade. 30% of the juice was used, and then 20 quarts of lemonade were added to the container. What percentage of the mixture was lemonade after the new addition?

    Amount of orange juice = 0.6 x 200 = 120 quarts
    Amount of lemonade = 0.4 x 200 = 80 quarts

    Amount of each remaining after 30% was used:
    - Amount of orange juice = 0.7 x 120 = 84 quarts
    - Amount of lemonade = 0.7 x 80 = 56 quarts

    Now add 20 quarts of lemonade:
    - Amount of orange juice = 84 quarts
    - Amount of lemonade = 56 quarts + 20 quarts = 76 quarts

    Total juice = 84 + 76 = 160
    Percentage of mixture that is lemonade = 76/160 = 0.475 = 47.5%

  4. 800 students applying to Atlantic University took the entrance exam. 80% of the students passed the exam and 20% failed it. Suppose instead only 5% of the students had failed the exam. How many more students would have passed the exam?

    800 x 0.8 = 640 passed initially.
    If only 5% failed, 800 x .05 = 40 would have failed.

    Total number that would have passed = 800 – 40 = 760
    Total number that initially passed =                           640
                                                                                          ----
                                                                                          120
    Thus, 120 more students would have passed the exam if only 5% of the students had failed the exam.


    Alternative solution:
    If only 5% of the students had failed the exam, 95% would have passed:
    800 x 0.95 =                                                                 760
    Total number that initially passed =                          640
                                                                                          ----
                                                                                         120
    Thus, 120 more students would have passed the exam if only 5% of the students had failed the exam.

SAT Verbal - Questions from Summer 2021, Week 6 (August 21, 2021)

SAT Quick Challenge T21B - Parallelism



Which Word(s) Can You Change To Correct the Error in the Question?  SAT grammar and writing questions ask you to replace the error in the underlined part of a sentence with the answer choice with corrects that error. However, parallelism questions like Question A lbelow can be confusing if you are not familiar with that type of question. Take a look at Question A.

Question A: Compared to a personal vehicle, public transportation is cheaper, safer, and environmentally friendly.
Discussion: The underlined words in this statement (cheaper and safer) are parallel to each other because they are both comparative adjectives (which are used specifically to compare two things), and they compare public transportation to personal vehicles.
Environmentally friendly is not parallel to the underlined words because it is not a comparative adjective; it is an adjective phrase with no limit to the number of things that it can describe. Since the correct answer to Question A requires all the descriptions  of public transportation to be parallel with another, something must be changed. Because only the underlined words in SAT writing and grammar questions can be changed, we must leave environmentally friendly as it is and select an answer choice that is parallel to it.  Now, keeping the information about in mind, answer Question A.

Question A. Compared to a personal vehicle, public transportation is cheaper, safer, and environmentally friendly.   
A. NO CHANGE
B. cheaper, safe
C. cheap, safe
D. cheap, safer

Answer: The correct answer is Choice C because the two words in that choice are adjectives that do the same thing that "environmentally friendly" does. They all describe public transportation, and there is no limit to the number of thing each adjective/adjective phrase can describe.

Prepositions Repeated in a Series of Prepositional Phrases.
When each phrase in a series of prepositional phrases begins with the same preposition, place that preposition in front of the first prepositional phrase only, as in the following: The children left paint smudges on the walls, the sink, and the windows. However, if the same preposition is needed at the beginning of one or more of the phrases, but not all of them, place the preposition needed in front of each phrase, as follows: The children left paint smudges on the walls, on the sink, and under the windows.

Now, keeping in mind the information above, complete the exercise below. Then, use the dropdown in the next section to check your work.

QUICK CHALLENGE Practice Exercise T21B - Parallelism

DirectionsReplace the underlined word(s) in each question below with the answer choice that corrects the error in the sentence. If there is no error, select choice A -- NO CHANGE.

1. Grandma planted flowers near the porch, near the mailbox, and along the driveway.

  1. NO CHANGE
  2. near the porch, the mailbox, and along
  3. near the porch, the mailbox, and
  4. near the porch, mailbox, and

2. Unlike the Afghan army, the Taliban army is toughest, strongest, and extremely vicious.

  1. NO CHANGE
  2. violence, harsher
  3. tough, strong
  4. crueler, meaner

3. My neighbor received a scam call from someone extending her a(n) clever, new offer to make $500,000 in one year.

  1. NO CHANGE
  2. clever, safe
  3. unwise, untrue
  4. sensible, profitable

4. For lunch, we will have vegetable gumbo made with tomatoes, with okra, with corn, and with rice.

  1. NO CHANGE
  2. with tomatoes, okra, corn, and rice
  3. with tomatoes, okra with corn, and rice
  4. with tomatoes, okra, and corn with rice

SAT Verbal - Answers to Questions from Summer 2021, Week 6 (August 21, 2021)

  1. A
  2. C
  3. C
  4. B

SAT Math - Questions from Summer 2021, Week 5 (August 14, 2021)

Decimals and Percent

A percent is a fraction whose denominator is 100:  25%  =  25/100
Percent means “per 100”
If there are 100 questions on your math test and you answer 80 of them correctly, you have answered 80 of 100 correctly, or 80/100, or 80%.

Think of it this way:  part/whole  =  percent
80/100 = 80%

Converting percentages to fractions: sometimes it will be useful to express a percentage as a fraction: put the percentage over a denominator of 100 and reduce.
60% = 60/100 = 6/10 = 3/5
150% = 150/100 = 15/10 = 3/2
25% = 25/100 = 1/4

Converting fractions to percentages:
Divide the numerator by the denominator
Move the decimal point in the result two places to the right
¾ = 0.75 = 75%
1/5 = 0.20 = 20%

Converting percentages to decimals: move the decimal point two places to the left.
16% = .16
2% = .02
.5% = .005

Converting decimals to percentages: move the decimal point two places to the right.
0.85 = 85%
0.4 = 40%
1.7 = 170%
.003 = .3%

Remember that a percent relates part to a whole: 20 is 50% of 40 Thus, there are three problem types involving percent:
  1. Find the percent: 20 is what percent of 40? 20/40 = 0.5 = 50%
  2. Find the part: What number is 50% of 40? 40 x 0.5 = 20 (change the percent to a decimal and multiply)
  3. Find the whole: 20 is 50% of what amount? 20/0.5 = 200/5 = 40 (change the percent to a decimal and divide)

Keeping in mind the information above, answer the following questions without using your calculator.

  1. Find the percent:
    1. 4 is what percent of 8?
    2. 2 is what percent of 5?
    3. 6 is what percent of 4?

  2. Find the percentage amount (or part.)
    1. What number is 50% of 8?
    2. What is 30% of 50?
    3. What is 15% of 60?
    4. What number is 250% of 2?

  3. Find the base (or whole.)
    1. 4 is 25% of what number?
    2. 70 is 40% of what amount?
    3. 12 is 30% of what amount?
    4. 3 is 0.2% of what number?

  4. A summer beach volleyball league has 750 players in it. At the start of the season, 150 of the players are randomly chosen and polled on whether games should be played while it is raining, of if the games should be cancelled. The results of the poll show that 42 of the polled players would prefer to play in the rain. The margin of error is ±4%. What is the range of players in the entire league that would be expected to prefer to play volleyball in the rain rather than cancel the game? (You may use your calculator for this one.)
    1. 24-32
    2. 38-46
    3. 146-154
    4. 180-240

SAT Math - Answers to Questions from Summer 2021, Week 5 (August 14, 2021)

  1. Find the percent:
    1. 4 is what percent of 8?
      4/8 = 0.5 = 50%
    2. 2 is what percent of 5?
      2/5 = 0.4 = 40%
    3. 6 is what percent of 4?
      6/4 = 1.5 = 150%

  2. Find the percentage amount (or part.)
    1. What number is 50% of 8?
      8 x 0.5 = 4
    2. What is 30% of 50?
      50 x 0.3 = 15
    3. What is 15% of 60?
      60 x 0.15 = 9
    4. What number is 250% of 2?
      2 x 2.5 = 5

  3. Find the base (or whole.)
    1. 4 is 25% of what number?
      4/0.25  =  400/25  =  80/5  =  16
    2. 70 is 40% of what amount?
      70/0.4 = 700/4 = 175
    3. 12 is 30% of what amount?
      12/0.3 = 120/3 = 40
    4. 3 is 0.2% of what number?
      3/0.002 = 3000/2 = 1500

  4. A summer beach volleyball league has 750 players in it. At the start of the season, 150 of the players are randomly chosen and polled on whether games should be played while it is raining, of if the games should be cancelled. The results of the poll show that 42 of the polled players would prefer to play in the rain. The margin of error is ±4%. What is the range of players in the entire league that would be expected to prefer to play volleyball in the rain rather than cancel the game? (You may use your calculator for this one.)
      1. 24-32
      2. 38-46
      3. 146-154
      4. 180-240

        First, determine the percent of polled players who wanted to play in the rain: 42/150 = 0.28 = 28%

        Next, apply this percent to the entire population of the league: 750 x 0.28 = 210. The only range that contains this value is D; thus, this is the answer.


        To calculate the actual range, we add and subtract 4% to the 28% to get a range of 24% - 32% of the total:

        750 x 0.24 = 180

        750 x 0.32 = 240

      Thus, the actual range is 180 to 240.

SAT Verbal - Questions from Summer 2021, Week 5 (August 14, 2021)

SAT Quick Challenge T21 - Parallelism



What is Parallelism?  Parallelism involves using the same structure or format to link two or more related words, phrases, or clauses in a sentence. This strategy makes the writing clearer and easier to understand. Note Sentences A and B, which follow. (A) Pam loves cooking food, to clean, and sewing. (B) Pam loves cooking, cleaning, and sewing. The two sentences say similar things, but not in the same way. The verbs in Sentence A are not listed the same way (cooking food, to clean, and sewing), so that sentence is not parallel. The verbs in Sentence B are listed the same way (cooking, cleaning, sewing), so that sentence is parallel.

Phrases or Independent Clauses. You will recall that parallelism requires using the same structure or format to list two or more related words or word groups in a sentence. They can be phrases only or independent clauses only, but not a mixture of both. Note Sentences C and D, which follow. (C) Poor eating habits can cause serious health problems, from high blood pressure to you could get an increased risk of diabetes. Explanation: Sentence C is not parallel because it combines a prepositional phrase (from high blood pressure) with an independent clause (you could get an increased risk of diabetes). Sentence D is parallel because it contains two prepositional phrases (from high blood pressure) and (to an increased risk of diabetes).

Avoiding Unnecessary Prepositions. When prepositional phrases which all begin with the same preposition appear one after another in the same sentence, place that preposition in front of the first prepositional phrase only, and the preposition will apply to all the linked prepositional phrases in the sentence. This strategy avoids the wordiness and redundancy that would result from repeating the preposition for each prepositional phrase. Note Sentences E and F, which follow. (E) On her vacation, Rev. Jones went to London, to Paris, and to Washington, DC. (F) On her vacation, Rev. Jones went to London, Paris, and Washington, DC. Explanation: Sentence E is not correct because it repeats the preposition "to" in front of each prepositional phrase. Sentence F is correct because the common preposition "to" is placed only in front of the first prepositional phrase.



Now, complete QUICK CHALLENGE Practice Exercise T21 below. Then, use the dropdown in the next section to check your work.

QUICK CHALLENGE Exercise T21
Parallelism

DirectionsReplace the underlined word(s) in each question below with the answer choice that corrects the error in the sentence. If there is no error, select choice A -- NO CHANGE.

1. We searched carefully in the house, the garage, and in the car, but we couldn't find the keys.

  1. NO CHANGE
  2. in the house, the garage, and the car
  3. in the house, in the garage, and in the car
  4. in the house, in the garage, and the car

2. Earl's game show prizes are fantastic, especially his new car and he got a trip to Paris.

  1. NO CHANGE
  2. Consequently
  3. Otherwise
  4. Nevertheless

3. My brother likes waffles with butter, syrup, and with cinnamon.

  1. NO CHANGE
  2. with butter, syrup, and cinnamon
  3. with butter, with syrup, and with cinnamon
  4. with butter, with syrup and cinnamon

4. Good study habits can lead to excellent grades and you could get a college scholarship.

  1. NO CHANGE
  2. you might get money for college
  3. college scholarships
  4. colleges could give you scholarships

SAT Verbal - Answers to Questions from Summer 2021, Week 5 (August 14, 2021)

  1. B
  2. D
  3. B
  4. C

SAT Math - Questions from Summer 2021, Week 4 (August 7, 2021)

Decimals

A lesson on decimals is basic and may seem unnecessary. However, many individual problems on the SAT contain more than one math concept, and understanding decimals will be necessary for solving some of them. It is also important to understand the relation between decimals and fractions, and between decimals and percents.

A decimal is just another way of expressing a fraction.
1/5 = 1 ÷ 5 = 0.2 3/4 = 3 ÷ 4 = 0.75

Not all decimals have visible decimal points.
2 is a decimal: 2 = 2.0 (or 2.00 or 2.000); 3 = 3.0 (or 3.00 or 3.000)

Adding and subtracting decimals:
Line up the decimal points and add or subtract (or use your calculator when appropriate - if you are in the calculator section of the test)
Add: 3.7, 14.23, and 9
  3.7
14.23
   9.00
------
 26.93

Multiplying Decimals:
  • Multiply exactly as you would integers
  • Count the total number of digits located to the right of the decimal points in the numbers you are multiplying
  • Place the decimal point in your answer so that there are the same number of digits to the right of it
(or use your calculator when appropriate)
       0.4              2.3              1.23     
     x0.4               x 3              x0.4                  
     ----               ---               ----
     0.16              6.9            0.492

Multiply a decimal by 10: Move the decimal point 1 place to the right
1.06 x 10 = 10.6 0.0076 x 10 = 0.076

Multiply a decimal by 100: Move the decimal point 2 places to the right
62.4 x 100 = 6240 0.0017 x 100 = 0.17

Dividing decimals:

  1. Convert the denominator to a whole number by moving the decimal point in the denominator a sufficient number of places to the right to make it a whole number
  2. The decimal point in the numerator must be moved the same number of places
  3. Then divide

(or use your calculator when appropriate)

Divide 6 by 0.48 6/0.48 = 600/48 reduce (when possible) then divide
                                           600/48 = 100/8 = 25/2 = 12.5

Divide a decimal by 10:  Move the decimal point 1 place to the left
12.635 ÷ 10  =  1.2635                   
0.0076 ÷ 10  =  0.00076

Divide a decimal by 100:  Move the decimal point 2 places to the left   
62.4 ÷ 100  =  .624                     
397 ÷ 100  =  3.97

Convert a fraction to a decimal:  just divide       
2/5  =  0.4                   
7/4  =  1.75

Convert a decimal to a fraction:

  1. Count the number of digits to the right of the decimal point and call this “n”
  2. numerator will be the decimal number without the decimal point
  3. The denominator will be a “1” followed by “n” zeros

Reduce to lowest terms
0.75 = ?
n = 2
75/100 = ¾

0.2 = ?
n = 1
2/10 = 1/5

Stay out of trouble by converting decimals to fractions or by using a calculator when appropriate.

  1. Confusion about decimal points causes more errors on the SAT than confusion about fractions
  2. Therefore, whenever you can conveniently convert a decimal to a fraction, you should do so, especially when the answer choices are fractions.


Keeping in mind the information above, answer the following questions without using your calculator.

  1. Multiply the following decimals.
    1. 0.6 x 84.2
    2. 75 x 0.3
    3. 0.14 x 0.009

  2. Divide the following decimals.
    1. 8/0.32
    2. 48/2.5
    3. 0.9/0.04
    4. 14.4/0.12

  3. Convert the following decimals to fractions.
    1. 0.8
    2. 2.62
    3. .004

  4. Convert the following fractions to decimals.
    1. 3/15
    2. 15/24
    3. 24/32

  5. A coastal geologist estimates that a certain country’s beaches are eroding at a rate of 1.5 feet per year. According to the geologist’s estimate, how long will it take, in years, for the country’s beaches to erode by 21 feet?
    1. 7
    2. 14
    3. 22.5
    4. 31.5

  6. If 0.6w = 4/3, what is the value of w?
    1. 9/20
    2. 4/5
    3. 5/4
    4. 20/9

SAT Math - Answers to Questions from Summer 2021, Week 4 (August 7, 2021)

  1. Multiply the following decimals.
    1. 0.6 x 84.2 = 50.52
    2. 75 x 0.3 = 22.5
    3. 0.14 x 0.009 = 0.00126

  2. Divide the following decimals.
    1. 8/0.32 = 800/32 = 100/4 = 25
    2. 48/2.5 = 480/25 = 96/5 = 19.2
    3. 0.9/0.04 = 90/4 = 45/2 = 22.5
    4. 14.4/0.12 = 1440/12 = 360/3 = 120

  3. Convert the following decimals to fractions.
    1. 0.8                  n = 1             8/10 = 4/5
    2. 2.62                n = 2             262/100 = 131/50
    3. .004                n = 3             4/1000 = 1/250

  4. Convert the following fractions to decimals.
    1. 3/15 = 1/5 = 0.2
    2. 15/24 = 5/8 = 0.625
    3. 24/32 = 3/4 = 0.75

  5. A coastal geologist estimates that a certain country’s beaches are eroding at a rate of 1.5 feet per year. According to the geologist’s estimate, how long will it take, in years, for the country’s beaches to erode by 21 feet?
    1. 7
    2. 14
    3. 22.5
    4. 31.5

    21/1.5 = 210/15 = 70/5 = 14 --------> B

  6. If 0.6w = 4/3, what is the value of w?
    1. 9/20
    2. 4/5
    3. 5/4
    4. 20/9

    Multiply both sides by 3:
    1.8w = 4
    Then w = 4/1.8 = 40/18 = 20/9 ----->D

    An alternative approach: since the answer choices are in fractions, convert the decimal (0.6) to a fraction and then solve.
    0.6 = 3/5
    (3/5)w = 4/3
    3w /5= 4/3
    Cross multiply: 9w = 20
    w = 20/9 ----->D

SAT Verbal - Questions from Summer 2021, Week 4 (August 7, 2021)

SAT Quick Challenge S21
Transition Words and Phrases



What is a Transition?  Sometimes, two sentences that are written one right after the other in a paragraph have a very specific relationship. For instance, the two might contrast with each other, or the second sentence might be an explanation or example of what is stated in the first sentence. In such situations, a word or phrase may be placed in the second sentence to show how the two sentences are related. That word or phrase is called a transition. In some cases, a transition may come between two independent clauses that are in the same sentence. Even so, the transition still shows how the two clauses are related to each other.

How To Determine Which Transition To Use. To decide which transition to use for an SAT question, draw a line through the underlined transition in the question so that it will not distract you. Then read the two sentences and determine their relationship. Finally, select the kind of transition which is needed for that relationship. Note the chart below.

TRANSITION RELATIONSHIPS FREQUENTLY TESTED ON THE SAT

Type of Transition and Job it Does Example Sample Sentences (and How to Select Their Transitions)
Reason/cause effect:
Shows why something did or did not happen
consequently, as
a result, hence, since, therefore,
accordingly
The children were exhausted. Therefore, they went to sleep at the dinner table. (Because the first sentence gives the reason for what happened in the second sentence, a reason or cause effect transition is needed.)
Addition/example:
Points out added information that explains further a point already made
moreover, for example, furthermore, indeed, for instance Dr. Taylor always has huge science classes because the students say that she is an excellent professor. Moreover, her math classes are also large because students also love the way she teaches math. (Since the second sentence further explains how well Dr. Taylor teaches, an addition/example transition is needed.)
Contrast:
Points out conflicting ideas
although, even so, however Frank loves cakes, pies, and other tasty desserts. Nevertheless, he avoids those sweet treats because he has diabetes. (Since the two sentences are in conflict, a contrast transition is needed. )



Keeping in mind the information above, complete QUICK CHALLENGE Exercise S21 below. Then use the dropdown in the next section to check your work.

QUICK CHALLENGE Exercise S21

DirectionsReplace the underlined words in each question below with the answer choice that corrects the error in the sentence. If there is no error, select choice A -- NO CHANGE.

1. Some people disagree with the doctor's thoughts about the cause of this disease. Even so, his theory has never been proven wrong.

  1. NO CHANGE
  2. Therefore
  3. Accordingly
  4. As a result

2. There was a terrible accident on the highway this morning. On the other hand, I did not get to work today until lunch time. 

  1. NO CHANGE
  2. Consequently
  3. Otherwise
  4. Nevertheless

3. Dina is a highly talented writer who does excellent work. However, she has written award-winning commercials for numerous companies.

  1. NO CHANGE
  2. Nevertheless
  3. Since
  4. For instance

SAT Verbal - Answers to Questions from Summer 2021, Week 4 (August 7, 2021)

  1. A
  2. B
  3. D


July Lessons


SAT Math - Questions from Summer 2021, Week 3 (July 24, 2021)

The Average (Mean) and Standard Deviation

Questions about the standard deviation are also asked on the SAT. The standard deviation is a measure of how closely clustered a data set is about the mean of the data set (how spread out the values are).

The standard deviation is low if most of the values are near the mean and close together (narrow spread).

The standard deviation is high if most of the values are spread out over the range of values (wide spread).

The SAT will not require you to calculate the standard deviation, but you need to understand the concept.

Consider the following example:

Arnold, Ronald, and David are the top three scorers on the Eastern High School basketball team. During the first five games of the season, they scored the following points:

Game 1 Game 2 Game 3 Game 4 Game 5
Arnold 18 21 15 24 22
Ronald 14 10 26 32 18
David 20 20 20 20 20

In each case the mean is 20:
Arnold: (18 + 21 + 15 + 24 + 22)/ 5 = 100/5 = 20
Ronald: (14 + 10 + 26 + 32 + 18)/5 = 100/5 = 20
David: (20+ 20 + 20 + 20 + 20)/5 = 100/5 = 20

Let’s compare Arnold and Ronald: The points scored by Arnold are close to the mean, while the points scored by Ronald are more spread out. The standard deviation of Arnold is lower than the standard deviation of Ronald. At the extreme, the points scored by David are all the same; there is no variation in the value of the points. Thus the standard deviation for David is zero.

Keeping in mind the information above, answer the following questions.

    1. Set A 37 42 58 19 66 97 22
      Set B 48 63 55 59 42 51 65
      The table above shows two sets of data. Which of the following statements is true about the standard deviation of the two sets?

      1. The standard deviation of Set A is lower than the standard deviation of Set B.
      2. The standard deviation of Set A is greater than the standard deviation of Set B.
      3. The standard deviation of Set A is equal to the standard deviation of Set B.
      4. The standard deviation of Set A and the standard deviation Set B cannot be compared.

    2. The tables below give the distribution of speeding tickets given by patrol officers in Greensboro and Winston-Salem for the 30 days in June.

      Greensboro   Winston-Salem
      Number of Speeding Tickets Given Frequency   Number of Speeding Tickets Given Frequency
      13 1   13 6
      14 5   14 7
      15 19   15 7
      16 3   16 6
      17 2   17 4
      Which of the following is true about the data shown for these 30 days?

      1. The standard deviation of the number of speeding tickets given in Greensboro is smaller.
      2. The standard deviation of the number of speeding tickets given in Winston-Salem is smaller.
      3. The standard deviation of the number of speeding tickets given in Greensboro is the same as that in Winston-Salem.
      4. The standard deviation shows that the number of speeding tickets given in Winston-Salem is too large.

    3. 13, 22, 29, 17, 9, 15, 27, 21
      If the number 43 is added to the above set of numbers, how will the standard deviation of the set change?

      1. The standard deviation of the set will be higher.
      2. The standard deviation of the set will be lower.
      3. The standard deviation of the set will be unchanged.
      4. The standard deviation of the set cannot be compared.

    4. The weights, in pounds, for 45 players on the football team were reported, and the mean, median, range, and standard deviation were calculated for the data. The player for the lowest reported weight was found to actually weigh 15 pounds less than his reported weight. What value remains unchanged if the four values are reported using the corrected weight?

      1. Mean
      2. Median
      3. Range
      4. Standard deviation

SAT Math - Answers to Questions from Summer 2021, Week 3 (July 24, 2021)

      1. Set A 37 42 58 19 66 97 22
        Set B 48 63 55 59 42 51 65
        The table above shows two sets of data. Which of the following statements is true about the standard deviation of the two sets?

        1. The standard deviation of Set A is lower than the standard deviation of Set B.
        2. The standard deviation of Set A is greater than the standard deviation of Set B.
        3. The standard deviation of Set A is equal to the standard deviation of Set B.
        4. The standard deviation of Set A and the standard deviation Set B cannot be compared.

        The numbers in Set A are spread out farther than the numbers in Set B. Thus Set A has a greater standard deviation than Set B. The answer is B.

      2. The tables below give the distribution of speeding tickets given by patrol officers in Greensboro and Winston-Salem for the 30 days in June.

        Greensboro   Winston-Salem
        Number of Speeding Tickets Given Frequency   Number of Speeding Tickets Given Frequency
        13 1   13 6
        14 5   14 7
        15 19   15 7
        16 3   16 6
        17 2   17 4
        Which of the following is true about the data shown for these 30 days?

        1. The standard deviation of the number of speeding tickets given in Greensboro is smaller.
        2. The standard deviation of the number of speeding tickets given in Winston-Salem is smaller.
        3. The standard deviation of the number of speeding tickets given in Greensboro is the same as that in Winston-Salem.
        4. The standard deviation shows that the number of speeding tickets given in Winston-Salem is too large.

        The number of tickets given out in Greensboro are much more tightly concentrated than those in Winston-Salem. Since most of the values in Greensboro are very closely clustered around 15 tickets, the number of tickets given in Greensboro has a lower standard deviation than that of Winston-Salem. The answer is A.

      3. 13, 22, 29, 17, 9, 15, 27, 21
        If the number 43 is added to the above set of numbers, how will the standard deviation of the set change?

        1. The standard deviation of the set will be higher.
        2. The standard deviation of the set will be lower.
        3. The standard deviation of the set will be unchanged.
        4. The standard deviation of the set cannot be compared.

        Adding the number 43 to the set of numbers will increase the spread of the numbers. Thus the standard deviation will be higher. The answer is A.

      4. The weights, in pounds, for 45 players on the football team were reported, and the mean, median, range, and standard deviation were calculated for the data. The player for the lowest reported weight was found to actually weigh 15 pounds less than his reported weight. What value remains unchanged if the four values are reported using the corrected weight?

        1. Mean
        2. Median
        3. Range
        4. Standard deviation


        The answer is B.

    SAT Verbal - Questions from Summer 2021, Week 3 (July 24, 2021)

    SAT Quick Challenge R21C
    The Semicolon



    Using the Semicolon Correctly. You will recall that an independent clause consists of a subject and verb that work together to express a complete thought that stands alone as a complete sentence. SAT questions about the semicolon often focus on using that punctuation mark to create a new sentence by connecting two closely related independent clauses, as shown in sentences A, B, and C, which follow. (A) Elaine enjoys cooking. (B) She especially loves to create tasty desserts. (C) Elaine enjoys cooking; she especially loves to create tasty desserts. Since sentence B provides further insight into what was said in sentence A, the two clauses are closely related, and the semicolon in sentence C creates a new sentence by connecting A and B properly.

    The Comma Splice Error. A common mistake tested on the SAT is the comma splice, an error created when a writer connects two independent clauses with just a comma. You may recall that a comma can help create a single sentence from two independent clauses if the comma comes just before an appropriate FANBOYS conjunction (for, and, nor, but, or, yet, and so). However, the comma cannot do the job without the FANBOYS conjunction. Note Sentences D-H, which follow. (D) Elaine enjoys cooking. (E) She especially loves to create tasty desserts. (F) Elaine enjoys cooking, she especially loves to create tasty desserts. (G) Elaine enjoys cooking, and she especially loves to create tasty desserts. (H) Elaine enjoys cooking; she especially loves to create tasty desserts. Note the comma splice in sentence F and the corrections in sentences G and H.

    List of Paired Items Connected with Commas. When the items listed in a sentence consist of paired words that must be connected to each other with a comma (such as a city and state), a semicolon instead of a comma must be placed between the listed pairs to make the writing easier to read, as in Sentence J. (J) During Lisa's vacation, she plans to visit Cape Town, South Africa; Atlanta, Georgia; Albany, New York; and Las Vegas, Nevada. As shown in Sentence K, which follows (and which is almost completely identical to sentence J), the semicolon just before the word "and" at the end of the listing in sentence J is optional. Therefore, Sentences J and K are both correct. (K) During Lisa's vacation, she plans to visit Cape Town, South Africa; Atlanta, Georgia; Albany, New York and Las Vegas, Nevada. Now, keeping in mind the information above, complete exercise R21C below.

    QUICK CHALLENGE R21C: The Semicolon

    DirectionsReplace the underlined words in each question below with the answer choice that corrects the error in the sentence. If there is no error, select choice A -- NO CHANGE.

    1. Next summer, students from our College International Fellowship will have summer internships in New Delhi, India, London, England, and Paris, France.

    1. NO CHANGE
    2. India, London; England,
    3. India; London, England;
    4. India, London England

    2. Donnie Gray does not plan to get a COVID 19 shot, he is extremely afraid of needles. 

    1. NO CHANGE
    2. COVID, 19 shot he is;
    3. COVID 19 shot; he is
    4. COVID 19 shot, he is;

    3. We slept too late this morning; we need Mom to take us to school so we won't be late.

    1. NO CHANGE
    2. late this morning; we need Mom,
    3. late this morning, we need Mom
    4. late this morning; we need Mom;

    SAT Verbal - Answers to Questions from Summer 2021, Week 3 (July 24, 2021)

    1. C
    2. C
    3. A

    SAT Math - Questions from Summer 2021, Week 2 (July 17, 2021)

    The Average (Mean) Revisited

    It was noted in the last lesson that on the SAT the use of the word average usually refers to the mean and is indicated by “average (arithmetic mean).”

    It was also noted that the key to solving any problem involving the average (mean) is to find the total of the items before you do anything else. There are two ways to find the total: (1) add the numbers and (2) multiply the average (mean) by the total number of items. The second method is frequently used on the SAT.

    In some problems it will be necessary to calculate two or three totals. Some problems require maximizing or minimizing values. In order to minimize one value, you will need to maximize some other values. For example, if you have an upcoming test in a class and you are trying to determine the minimum score you can get on that test to give you a specific average for all of the tests in that class, you will need to assume that you will get the maximum score of 100 on the remaining tests.

    Consider the following example:

    An online store receives customer satisfaction ratings between 0 and 100, inclusive. In the first 10 ratings the store received, the average (arithmetic mean) of the ratings was 75. What is the least value the store can receive for the 11th rating and still be able to have an average of at least 85 for the first 20 ratings?

    We begin by getting the total for the first 10 ratings. Total = 10 x 75 = 750.

    Since we want an average of 85 for 20 ratings, we get the total for 20 ratings: Total = 20 x 85 = 1700.

    We can now determine the total for the last 10 ratings: 1700 - 750 = 950.

    Now if one rating, the 11th, is as small as possible, the other 9 ratings must be as large as possible. The highest possible rating is 100. Thus the maximum total for the other 9 ratings = 9 x 100 = 900.

    Thus the least possible rating for the 11th rating = 950 - 900 = 50.


    Keeping in mind the information above, answer the following questions.

    1. In a set of 15 integers, three of the integers are 12, 19, and 23. The mean of the 15 integers is 44. If 12, 19, and 23 are removed from the set, what is the mean of the remaining 12 numbers in the set?

    2. A new computer game receives critical reviews between 1 and 50, inclusive. In the first 6 ratings, the average (mean) of the ratings was 40. What is the least value the game can receive for the 8th rating and still be able to have an average of at least 42 for the first 10 ratings?

    3. Jackie took 6 tests in the fall semester of school. The mean score of the 6 tests was 84. If the mean score of the first four tests is 80, what is the mean score of the last 2 tests?

    SAT Math - Answers to Questions from Summer 2021, Week 2 (July 17, 2021)

    1. In a set of 15 integers, three of the integers are 12, 19, and 23. The mean of the 15 integers is 44. If 12, 19, and 23 are removed from the set, what is the mean of the remaining 12 numbers in the set?

      We begin by getting the total for the 15 integers. Total = 15 x 44 = 660
      We then subtract 12, 19, and 23 from 660 to get the new total: 660 – 12 – 19 – 23 = 606

      This new total divided by 12 gives us the mean of the remaining 12 numbers:
      606/12 = 50.5


    2. A new computer game receives critical reviews between 1 and 50, inclusive. In the first 6 ratings, the average (mean) of the ratings was 40. What is the least value the game can receive for the 8th rating and still be able to have an average of at least 42 for the first 10 ratings?

      We begin by getting the total for the first 10 ratings. Total = 6 x 40 = 240.

      Since we want an average of 42 for 10 ratings, we get the total for 10 ratings:
      Total = 10 x 42 = 420

      We can now determine the total for the last 4 ratings: 420 – 240 = 180.

      Now if one rating, the 8th, is as small as possible, the other 3 ratings must be as large as possible. The highest possible rating is 50. Thus the maximum total for the other 3 ratings = 3 x 50 = 150.

      Thus the least possible rating for the 8th rating = 180 - 150 = 30.


    3. Jackie took 6 tests in the fall semester of school. The mean score of the 6 tests was 84. If the mean score of the first four tests is 80, what is the mean score of the last 2 tests?

      The total for the 6 tests: total = 6 x 84 = 504 The total for the first 4 tests: total = 4 x 80 = 320
      Total for the last 2 tests = 504 - 320 = 184
      Thus the mean of the last 2 tests = 184/2 = 92

    SAT Verbal - Questions from Summer 2021, Week 2 (July 17, 2021)

    SAT Quick Challenge R21B
    Routine Uses of the Comma, Part II



    The Versatile Comma.  As noted in previous lessons, the comma does different kinds of jobs. For instance, when two separate independent clauses (also sentences by definition) are combined to make a single sentence, a FANBOYS conjunction (for, and, nor, but, or, yet, and so) can be used to create that new sentence. Note Sentences A, B, and C, which follow. (A) We expected our cousins to arrive this morning. (B) They missed their flight and won't get here until tonight. (C) We expected our cousins to arrive this morning, but they missed their flight and won't get here until tonight. As Sentence C shows, a comma must be placed in front of "but," the FANBOYS conjunction that turns the two independent clauses into a single sentence. As you will note below, today's lesson identifies additional ways in which commas are used in writing.

    Separating Items in a List. When a list of items is written as part of a sentence, a comma is often placed just before the word "and" to show that the last item is about to be named. However, that comma is optional. Note Sentences D and E, which follow. (D) People are advised to keep on hand masks, gloves, hand sanitizer, first aid kits, non-perishable foods, water, and other essential emergency supplies. (E) People are advised to keep on hand masks, gloves, hand sanitizer, first aid kits, non-perishable foods, water and other essential emergency supplies. Since the comma is optional, sentences D and E are both correct, so the SAT will not ask you to choose between the two; you will just need to remember that both are correct.

    Introductory Words and Phrases. An introductory word/phrase comes at the beginning of a sentence and sets the tone for what will be said. As illustrated in Sentence F, which follows, a comma must be used after an introductory word/phrase (in fact, initially, nevertheless, however, etc.) in a sentence. (F) Initially, we were going to the beach, but we decided to cook out at home, instead.

    The "Self" Words Exception. "Self" pronouns are used to emphasize the fact that a particular person or thing is being referred to. Every "object" pronoun (both direct and indirect objects) has a "self" counterpart, as in the following: me/myself, you/yourself, he/himself, she/herself, it/itself, etc. As illustrated in Sentence G, which follows, generally, a writer must not place a comma in front of or behind a self word. (G) Britney Spears herself will sing "America, the Beautiful" during the halftime show at the game on Friday. However, if a grammar rule requires a comma where a self word is used, you must follow that rule. (That is why commas follow the bold, italicized self words listed above in this paragraph.) Another example of the exception is the required comma for combining two independent clauses with a FANBOYS conjunction, as in Sentence H, which follows: (H) I will take Eric to school myself, but Mom will pick him up after school.

    Keeping in mind the information above, complete QUICK CHALLENGE R21B below. Then use the dropdown in the next section to check your work.

    QUICK CHALLENGE R21B: Routine Uses of the Comma, Part II

    DirectionsReplace the underlined words in each question below with the answer choice that corrects the error in the sentence. If there is no error, select choice A -- NO CHANGE.

    1. Mom said that you, yourself, must was the dishes after lunch today.

    1. NO CHANGE
    2. you, yourself must,
    3. you yourself, must
    4. you yourself must

    2. Little Andy wants to have cupcakes, pie ice cream: and candy for his birthday dinner. 

    1. NO CHANGE
    2. cupcakes, pie. ice cream: and
    3. cupcakes, pie, ice cream and
    4. cupcakes, pie ice cream, and

    3. The hotel overbooked its rooms; so our guests will be staying at our home during their visit.

    1. NO CHANGE
    2. its rooms; so our guests,
    3. its rooms, so our guests
    4. its rooms so our guests;

    4. Finally, Mom said, that we could go to the beach, but bad weather spoiled our plans. 

    1. NO CHANGE
    2. Finally, Mom said
    3. Finally Mom said
    4. Finally, Mom, said

    SAT Verbal - Answers to Questions from Summer 2021, Week 2 (July 17, 2021)

    1. D
    2. C
    3. C
    4. B

    SAT Math - Questions from Summer 2021, Week 1 (July 10, 2021)

    Averages and Range

    There are three averages that are tested on the SAT: mean, median, and mode.
    • Mean: the total of the items divided by the number of items
    • Median: the number that is exactly in the middle of a group of numbers when the numbers are arranged from smallest to largest; the median is always the middle value in a data set
    • Mode: the number that appears most often
    Range: the largest number – the smallest number

    Find the averages and range of these numbers: 6, 18, 12, 6, 8
    • Mean = (6 + 18 + 12 + 6 + 8)/5 = 50/5 = 10
    • Median; arrange the items in order: 6, 6, 8, 12, 18
    • Median = 8
    • Mode = 6
    • Range = 18 – 6 = 12
    Find the median of these numbers: 7, 4, 15, 20, 8, 15
    • Arrange in order: 4, 7, 8, 15, 15, 20
    • In this case, the median is midway between the two middle numbers:
      Median = (8 + 15)/2 = 23/2 = 11.5
    On the SAT the use of the word average usually refers to the mean and is indicated by “average (arithmetic mean).” Questions involving the median and mode will have those terms stated as part of the question’s text.

    The key to solving any problem involving the average (mean) is to find the total of the items before you do anything else. There are two ways to find the total: (1) add the numbers and (2) multiply the average (mean) by the total number of items. The second method is frequently used on the SAT.

    The average of 4 numbers is 5. If three of the four numbers are 3, 4, and 5, what is the fourth number? The first thing to do is to get the total. The total = 4 x 5 = 20.
    Thus the sum of the four numbers must total 20.
    3 + 4 + 5 = 12; to make the total = 20, the fourth number must be 8.

    Keeping in mind the information above, answer the following questions.

    1. 1, 6, 4, 10, 16, 4, 10, 25, 4, 20
      Calculate the mean, median, mode, and range for the above set of numbers.

    2. Weight (Pounds) Number of Dumbbells
      5 6
      10 10
      20 4

      David bought dumbbells in three different weights, in pounds. The table above shows the weight of the dumbbells, in pounds, and the number of dumbbells for each weight David bought. What is the mean weight of the dumbbells, in pounds?
      1. 11.67
      2. 10.50
      3. 8.50
      4. 6.67


    3. Player Height Player Height
      Alice 77 Florence 73
      Barbara 69 Geraldine 76
      Carolyn 71 Helen 68
      Denise 72 Ivey 70
      Edith 67 Jane 74

      The table above shows the heights of 10 players on the Greensboro High School women’s basketball team. If the coach takes Alice out of the game and substitutes Geraldine in her place, and makes no other substitutions, which of the following must be true? (In basketball, exactly five players from a team are allowed on the court at a time.)

      1. The median height of players on the court from Greensboro High School will not change.
      2. The median height of players on the court from Greensboro High School will increase.
      3. The median height of players on the court from Greensboro High School will decrease.
      4. A change in the median height of players on the court from Greensboro High School cannot be determined from the information given.

    SAT Math - Answers to Questions from Summer 2021, Week 1 (July 10, 2021)

      1. 1, 6, 4, 10, 16, 4, 10, 25, 4, 20
        Calculate the mean, median, mode, and range for the above set of numbers.

        • Mean = (1+6+4+10+16+4+10+25+4+20)10 = 100/10 = 10
        • Median: arrange the numbers in order from lowest to highest:
          1, 4, 4, 4, 6, 10, 10, 16, 20, 25 The two middle numbers are 6 and 10.
          Thus the median = (6 + 10)/2 = 16/2 = 8
        • Mode = 4 (the number that occurs most often)
        • Range = highest - lowest = 25 – 1 = 24

      2. Weight (Pounds) Number of Dumbbells
        5 6
        10 10
        20 4

        David bought dumbbells in three different weights, in pounds. The table above shows the weight of the dumbbells, in pounds, and the number of dumbbells for each weight David bought. What is the mean weight of the dumbbells, in pounds?
        1. 11.67
        2. 10.50
        3. 8.50
        4. 6.67

        Remember in an average problem, the first thing to do is to find the total.

        • The total weight of 6 dumbbells that weigh 5 pounds each is 6 x 5 = 30.
        • The total weight of 10 dumbbells that weigh 10 pounds each is 10 x 10 = 100.
        • The total weight of 4 dumbbells that weigh 20 pounds each is 4 x 20 = 80.
        • There are 20 dumbbells (6 + 10 + 4 = 20).
        • The total weight of the 20 dumbbells = 30 + 100 + 80 = 210.
        • Thus, the mean = 210/20 = 10.50 ------------------> B



      3. Player Height Player Height
        Alice 77 Florence 73
        Barbara 69 Geraldine 76
        Carolyn 71 Helen 68
        Denise 72 Ivey 70
        Edith 67 Jane 74

        The table above shows the heights of 10 players on the Greensboro High School women’s basketball team. If the coach takes Alice out of the game and substitutes Geraldine in her place, and makes no other substitutions, which of the following must be true? (In basketball, exactly five players from a team are allowed on the court at a time.)

        1. The median height of players on the court from Greensboro High School will not change.
        2. The median height of players on the court from Greensboro High School will increase.
        3. The median height of players on the court from Greensboro High School will decrease.
        4. A change in the median height of players on the court from Greensboro High School cannot be determined from the information given.


        1. This problem does not require any calculation or rearranging of data; it requires only an understanding of the concept of the median.
        2. If Alice is playing and Geraldine is not, it does not matter who the other four players on the team are playing with her; Alice is the tallest player.
        3. If Alice is taken out of the game and is replaced by Geraldine, then Geraldine is the tallest player on the court. All that has happened is that the tallest player on the court from Greensboro High School has changed.
        4. Since the median is the middle value in a data set, changing only the greatest value has no impact on the median. Thus the answer is A.

    SAT Verbal - Questions from Summer 2021, Week 1 (July 10, 2021)

    SAT Quick Challenge R21-A
    Routine Uses of the Comma, Part I



    Punctuating Nonessential Information (NESI).  A nonessential word or word group provides extra information (NOT the main idea) about another word or word group in a sentence -- generally the word that the nesi follows. If the nesi is removed, we will still have the main idea of the sentence, and the sentence will still make sense. Therefore, the nesi should be enclosed in commas. However, if the word or word group is needed to express the main idea clearly, do not use commas.

    When deciding whether to place commas around a word group, and where to place them if they are needed, draw a line under the entire possible nesi. The underlining can help you remember (1) to omit the nesi when you are reading to see if the sentence makes sense without it and (2) to read the entire nesi as you decide whether or not to use commas when an SAT question underlines only a small part of a long nesi.

    A relative pronoun (such as who, which, or that) is sometimes used to introduce a relative clause that modifies the word the clause follows. If the clause provides nonessential information (nesi) that does not help the reader understand clearly the main idea the writer is trying to state, use commas to separate the nesi from the rest of the sentence. If the clause is needed to help convey the main idea, do not separate it.

    Comparing NESI Punctuation. The three sentences which follow show how relative pronouns are sometimes used to introduce nesi. (A) Nan Epps, who just graduated from Northwest High School, has accepted a scholarship to Salem College. (B) Nan Epps has accepted a scholarship to Salem College. (C) The student who has accepted a scholarship to Salem College is Nan Epps.

    Commas separate the underlined relative clause in sentence A from the main idea because the reader does not need the information in the clause in order to understand the main idea. Sentence B simply states its main idea; it has no relative clause. In sentence C, commas do not separate the underlined relative clause from the rest of the sentence because the reader needs that clause in order to understand who Nan Epps is.

    The Interrupter. An interrupter comes within a sentence and creates emphasis or shows emotion by temporarily breaking the flow of thought in that sentence. Expressions routinely used as introductory words/phrases (in fact, however, initially, etc.) are also used as interrupters. Other common interrupters include expressions such as as you know and for example. Even a person's name can be used as an interrupter.

    You can use two commas, two dashes, or two parentheses to separate an interrupter from the rest of the sentence, as in sentences C and D, which follow. (C) What, Kenny, did you think would happen after you broke that window? (D) The heroic teenagers -- as we had expected -- received awards for their bravery.

    Keeping in mind the information above, complete QUICK CHALLENGE R21-A below. Then use the dropdown in the next section to check your work.

    QUICK CHALLENGE R21-A: Routine Comma Uses, Part I

    DirectionsReplace the underlined words in each question below with the answer choice that corrects the error in the sentence. If there is no error, select choice A -- NO CHANGE.

    1. Our Business Honor Society speaker is a lady who, we were told --  ran a business when she was 11 years old.

    1. NO CHANGE
    2. who we were told,
    3. who -- we were told
    4. who, we were told,

    2. A musical instrument, that, has been mishandled frequently, may never work properly again. 

    1. NO CHANGE
    2. instrument -- that has been mishandled frequently,
    3. instrument, that has been mishandled frequently
    4. instrument that has been mishandled frequently

    3. Crystal Beach, which attracts huge crowds to its annual Summer Fest events have great restaurants.

    1. NO CHANGE
    2. events; have
    3. events, has
    4. events, having

    SAT Verbal - Answers to Questions from Summer 2021, Week 1 (July 10, 2021)

    1. D
    2. D
    3. C


    June Lessons


    SAT Math - Questions from Week 18 (June 26, 2021)

    Systems of Linear Equations, Part 6:  Word Problems

    In recent years, word problems have become more numerous on the SAT. Some word problems will require that you create a system of equations from the information given. Sometimes you will be asked to solve the system of equations, but often you will just be asked to develop the two equations. It is essential, then, that you be able to find and translate the information in the problem into the equations. The following procedure is useful in determining the required information.

    Step 1: Read the problem carefully and identify these important items for each equation:

    1. The variables
    2. The coefficients of the variables
    3. The total

    Remember that each of the two equations is in the format: Ax +By = C where x and y are variables,
    A and B are coefficients, and C is the total.

    The coefficients can be negative or positive;
    the coefficients can equal 1;
    each equation has a total on the right side of the equation;
    and the total can be negative or positive

    Step 2: Write the appropriate coefficients and total for each equation.

    Step 3. Review the problem to ensure that your equations are consistent with the information in the problem.


    Let’s look at the following example:

    Albert, Brad, and Charles are selling hamburgers and hot dogs. The hamburgers are $2.50 each and the hot dogs are $1.75 each. On yesterday they had 58 customers each buying only one item and they made $128.50. Which of the following equations will let them know how many hamburgers and how many hot dogs they sold?

    1. x + y = 58
      1.75x + 2.50y = 128.50

    2. x + y = 58
      250x + 175y = 128.50

    3. x + y = 58
      2.50x = 1.75y = 128.50

    4. x + y = 128.5
      2.50x + 1.75y = 58

    Let’s use the steps outlined above:

    Step 1:   Read the problem carefully and identify these important items for each equation:

    1. The variables: The variables are the number of hamburgers and the number of hot dogs. Let x = the number of hamburgers and y = the number of hot dogs.
    2. The coefficients of the variables: in one equation the coefficient of x is 1 and the coefficient of y is 1; in the other equation the coefficient of x is 2.50 and the coefficient of y is 1.75.
    3. The total: the total for one equation is 58, and the total for the other equation is 128.50.

    Step 2:  Write the appropriate coefficients and total for each equation.
         x + y = 58
        2.50x + 1.75y = 128.50

    Step 3:   Review the problem to ensure that your equations are consistent with the information in the problem.
         The equations are consistent. The answer is C.


    Here is another example:

    Last week Charlie worked 11 more hours than Sam. If they worked a combined total of 59 hours, which system of equations could be used to determine the number of hours Sam worked last week?

    Let’s use the steps outlined above:

    Step 1: Read the problem carefully and identify these important items for each equation:

    1. The variables: The variables are the number of hours Charlie worked and the number of hours Sam worked. Let x = the number of hours Charlie worked and y = the number of hours Sam worked.
    2. The coefficients of the variables: in one equation the coefficient of x is 1 and the coefficient of y is -1; in the other equation the coefficient of x is 1 and the coefficient of y is 1.
    3. The total: the total for one equation is 11 and the total for the other equation is 59

    Step 2.  Write the appropriate coefficients and total for each equation.
          x  -  y = 11
          x +  y = 59

    Step 3. Review the problem to ensure that your equations are consistent with the information in the problem.
          The equations are consistent.

    Keeping in mind the information above, answer the following system of equations questions.


    1. Delta State University has separate tuition rates for in-state students and out-of-state students. In-state students are charged $421 per semester and out-of-state students are charged $879 per semester. The university’s junior class of 1,980 students paid a total of $1,170,210 in tuition fees for the most recent semester. Which of the following systems of equations represents the number of in-state (x) and out-of-state (y) juniors and the amount of tuition fees the two groups paid?
      1. x + y = 1,170,210
        421x + 879y = 1,980

      2. x + y = 1,980
        879x+ 421y= 1,170,210

      3. x + y = 1,980
        421x+ 879y= 1,170,210

      4. x + y = 1,170,210
        879x + 421y = 1,980

    2. An online bookstore sells novels and magazines. Each novel sells for $4, and each magazine sells for $1. If Jackie purchased a total of 11 novels and magazines that have a combined selling price of $20, how many novels did she purchase?
      1. 2
      2. 3
      3. 4
      4. 5

    3. A bead shop sells wooden beads for $0.20 each and crystal beads for $0.50 each. If a jewelry artist buys 127 beads total and pays $41 for them, how much more did she spend on crystal beads than wooden beads?
      1. $11
      2. $15
      3. $23
      4. $26

    SAT Math - Answers to Questions from Week 18 (June 26, 2021)

        1. Delta State University has separate tuition rates for in-state students and out-of-state students. In-state students are charged $421 per semester and out-of-state students are charged $879 per semester. The university’s junior class of 1,980 students paid a total of $1,170,210 in tuition fees for the most recent semester. Which of the following systems of equations represents the number of in-state (x) and out-of-state (y) juniors and the amount of tuition fees the two groups paid?

          1. x + y = 1,170,210
            421x + 879y = 1,980

          2. x + y = 1,980
            879x + 421y = 1,170,210

          3. x + y = 1,980
            421x + 879y = 1,170,210

          4. x + y = 1,170,210
            879x + 421y = 1,980

          In this case we are not asked to solve the system of equations; just determine what they are. Use the process described earlier:

          Step 1: Read the problem carefully and identify these important items for each equation:

          1. The variables: The variables are the number of in-state juniors and the number of out-of-state juniors. Let x = the number of in-state juniors and y = the number of out-of-state juniors.
          2. The coefficients of the variables: in one equation the coefficient of x is 1 and the coefficient of y is 1; in the other equation the coefficient of x is 421 and the coefficient of y is 879.
          3. The total: the total for one equation is 1,980 and the total for the other equation is 1,170,210.

          Step 2: Write the appropriate coefficients and total for each equation.

          x + y = 1.980
          421x + 879y = 1,170,210 The answer is C.

        2. An online bookstore sells novels and magazines. Each novel sells for $4, and each magazine sells for $1. If Jackie purchased a total of 11 novels and magazines that have a combined selling price of $20, how many novels did she purchase?
            1. 2
            2. 3
            3. 4
            4. 5

          In this case we are asked to determine the system of equations and then solve the problem. Use the process described earlier to determine the system of equations.

          Step 1: Read the problem carefully and identify these important items for each equation:
          1. The variables: The variables are the number of novels and the number of magazines. Let x = the number of novels and y = the number of magazines.
          2. The coefficients of the variables: in one equation the coefficient of x is $4 and the coefficient of y is $1; in the other equation the coefficient of x is 1 and the coefficient of y is 1.
          3. The total: the total for one equation is $20 and the total for the other equation is 11.

          Step 2:  Write the appropriate coefficients and total for each equation.

          4x + y = 20
            x + y = 11

          Now we can solve the system of equations; let’s use the elimination approach.
          Multiply the second equation by -1:

          -x - y = -11. Then
          4x + y = 20
          -x - y = -11
          3x = 9

          x = 3
          The answer is B.

        3. A bead shop sells wooden beads for $0.20 each and crystal beads for $0.50 each. If a jewelry artist buys 127 beads total and pays $41 for them, how much more did she spend on crystal beads than wooden beads?
          1. $11
          2. $15
          3. $23
          4. $26

          In this case we are asked to determine the system of equations and then solve the problem. Use the process described earlier to determine the system of equations.

          Step 1: Read the problem carefully and identify these important items for each equation:
          1. The variables: The variables are the number of wooden beads and the number of crystal beads.
            Let x = the number of wooden beads and y = the number of crystal beads.
          2. The coefficients of the variables: in one equation, the coefficient of x is $0.20, and the coefficient of y is $0.50; in the other equation, the coefficient of x is 1. and the coefficient of y is 1.
          3. The total: the total for one equation is $41 and the total for the other equation is 127.

          Step 2:  Write the appropriate coefficients and total for each equation.

          0.2x + 0.5y = 41
            x + y = 127

          Now we can solve the system of equations; let’s use the elimination approach.
          Multiply the second equation by -0.5:

          -0.5xx - 0.5y = -63.5.  Then

           0.2x  + 0.5y =   41
          -0.5x - 0.5y = -63.5
          -------------------
          .3x = -22.5
          x = 75 = number of wooden beans

              x + y = 127
          75 + y = 127
          y = 127 - 75 = 52 = number of crystal beads

          Cost of crystal beads = $0.5 (52) =   $26
          Cost of wooden beads = $0.2 (75) = $15

          $26
          $15
          ----
          $11            The answer is A.

    SAT Verbal - Questions from Week 18 (June 26, 2021)

    SAT Quick Challenge Q21B
    Common SAT Fragment Errors -- Part 2



    The Sentence and a Fragment.  You may recall that a sentence is a word group that has three essential characteristics: (1) It contains a subject and its verb, (2) it expresses a complete thought, and (3) it makes sense all by itself. (This description also applies to an independent clause, since it is also a complete sentence.) A fragment is a word group that is written as though it were a sentence, but does not have all three essential sentence characteristics.

    The Unnecessary Relative Pronoun Fragment. A relative pronoun (such as "where," "who," or "which") begins a dependent clause that is referred to as a relative clause because of that relative pronoun. Like all other dependent clauses, the relative clause (1) contains a subject and verb, and (2) gives extra information about the noun that the clause modifies, as in the following sentence, which has both an independent and a dependent clause: "Mr. Long, who won last year's cooking contest, is a fantastic chef." The independent clause tells us that Mr. Long is a fantastic chef, and the bold, underlined, italicized relative clause gives us extra information about Mr. Long -- that he won last year's cooking contest. If written as a complete statement all by itself, the relative clause (Who won last year's cooking contest.) would be a fragment because it does not contain all three essential sentence characteristics.

    Sometimes an SAT question will require you to determine whether a word group is a fragment. If it is, you will need to determine what is missing -- perhaps a subject or a verb. Then, you must select the answer choice that provides the missing element(s). Note the fragments with missing elements and the corrections in the chart below.


    Fragment to Analyze Missing Element Correction
    Will take a cruise to Hawaii. Subject Our class will take a cruise to Hawaii.
    While waiting for the bus. Subject and Verb While waiting for the bus, Terri found $10.

    Keeping the information above in mind, complete Quick Challenge Exercise Q21B below.

    .

    SAT Quick Challenge Exercise Q21B - Turning Fragments into Complete Sentences

    Directions. Replace the underlined words in each fragment below with the answer choice that turns that fragment into a complete sentence. If the word group is already a complete sentence, select choice A -- NO CHANGE.

    1. Mr. Long, waiting for class to begin, always ready and eager to help his students.

    1. NO CHANGE
    2. sharpened extra pencils just in case they would be needed
    3. the new band instruments that just arrived 
    4. hoping to become a principal

    2. The new kindergarteners, who were very excited about school. enjoyed the entire day.

    1. NO CHANGE
    2. you may go right in
    3. the dog barking loudly for attention
    4. parents felt a little uneasy

    3. Proud and happy, that will be presented to the winner is made of bronze.

    1. NO CHANGE
    2. Waiting eagerly for the race to get started.
    3. The shiny new trophy
    4. Felt good about his likely victory, 

    SAT Verbal - Answers to Questions from Week 18 (June 26, 2021)

    1. B
    2. A
    3. C

    SAT Math - Questions from Week 17 (June 12, 2021)

    Systems of Linear Equations, A Short Cut – Part 4: : One Solution, No Solution, An Infinite Number of Solutions

    Most of the system of linear equations problems on the SAT will have one solution. Occasionally, however, there will be a problem where there is no solution or there is an infinite number of solutions. For test takers it is important to be able to determine which situation exists. A standard approach to make this determination is to express each equation in slope-intercept form: y = mx + b, where “m” is the slope and “b” is the y-intercept. In other words, solve each equation for y.

    You will also recognize that “y = mx + b” is the formula for a line, and thus there is a line for each of the two equations in a system of linear equations. The slopes and y-intercepts of the two lines determine whether there is one solution, no solution, or an infinite number of solutions to the system of linear equations, as follows:

    One solution. If the two lines have different slopes, there will be one solution to the system of equations. Geometrically, the two lines will cross, and the point of intersection (x,y) will be the solution to the system of equations.

    No Solution. If the two lines have the same slope but different y-intercepts, there will be no solution to the system of equations. Geometrically, the two lines are parallel and will never cross; thus there is no point of intersection, and there is no solution to the system of equations.

    An Infinite Number of Solutions: If the two lines have the same slope and the same y-intercept, there will be an infinite number of solutions to the system of equations. Geometrically, the two lines are really the same line, and one is a multiple of the other. Here is an example:

      y = 2x + 7
    2y = 4x + 14     The second equation is a multiple of the first one.

    The coefficients of the variables and y-intercept of one equation are multiples of the coefficients in the other equation. In this case:

    Variable y: 2 in the second equation is a multiple of the 1in the first equation
    Variable x: 4 in the second equation is a multiple of the 2 in the first equation
    y-intercept: 14 is a multiple of 7.         The common multiple is 2.


    Keeping in mind the information above, answer the following system of equations questions.
    1. 6x + 2y = 3
       3x + y = 2
      How many solutions (x,y) are there to the system of equations above?
      1. Zero
      2. One
      3. Two
      4. More than two


    2. 8x - 2y = 12
      6y = kx - 42
      In the system of equations above, k represents a constant. If the system of equations has no solution, what is the value of 2k?
      1. 6
      2. 12
      3. 24
      4. 48

    3. 3x - 2y = a
      -15x + by = 10

      In the system of equations above, a and b are constants.  If the system of equations has infinitely many solutions, what is the value of a?

    SAT Math - Answers to Questions from Week 17 (June 12, 2021)

      1. 6x + 2y = 3
        3x + y = 2
        How many solutions (x,y) are there to the system of equations above?
        1. Zero
        2. One
        3. Two
        4. More than two


        Express each equation in slope-intercept form (solve for y) and compare the slopes and y-intercepts.
        6x + 2y = 3                                                                             3x + y = 2
        2y = -6x + 3                                                                                   y = -3x + 2
        y = -3x + 1.5

        The two equations have the same slope (-3) but different y-intercepts (1.5 and 2). Thus there is no solution to the system of equations. The answer is A.

        Let’s see what happens if we solve the equations by substitution. Solve the second equation for y: y = -3x + 2. Now substitute this value of y in the first equation and solve for x:

        6x + 2(-3x + 2) = 3
        6x – 6x + 4 = 3
        4 = 3
        Of course this is not true; thus there is no solution.

      2. 8x - 2y = 12
        6y = kx - 42
        In the system of equations above, k represents a constant. If the system of equations has no solution, what is the value of 2k?
        1. 6
        2. 12
        3. 24
        4. 48

        There is no solution if the slopes are equal and the y-intercepts are unequal. Let’s express each equation in slope-intercept form.
        8x – 2y = 12                                                6y = kx – 42
              –2y = -8x + 12                                        y = k/6 - 7
                  y = 4x - 6

        The y-intercepts are unequal; now we need to determine whether the slopes are equal. They are equal if 4 = k/6. Thus we need to solve for k.
        k/6 = 4
        k = 24
        If the system of equations has no solution, then k must - 24. The question asks us for 2k.
        Thus, the answer is 2(24) - 48 -----------> D

      3. 3x - 2y = a
        -15x + by = 10

        In the system of equations above, a and b are constants.  If the system of equations has infinitely many solutions, what is the value of a?

        If a system of equations has infinitely many solutions, the two lines have the same slope and the same y-intercept. The two lines are really the same line, and one is a multiple of the other.

        There is a common multiple that is applied to each of the two variables (x and y) and to the y-intercept.

        In this case the second equation is a multiple of the first equation. What is the common multiple?

        Look at the coefficients of x, the coefficients of y, and the y-intercepts. Is there a multiple in either of these?
        Yes, there is a multiple in the x coefficients; -15 is a multiple of 3.

        The multiple is -5 since 3(-5) = -15. -5 is the common multiple, and we will use it to determine b and a.

        – 2(-5) = b                                                                   a(– 5) = 10
                 b = 10                                                                        a = 10/(– 5) = – 2

        We can check our work by calculating the slope and y-intercept for each equation using the values of a and b that we calculated. The slopes and y-intercepts should be the same for both equations.

        First equation:                 3x – 2y = a
                                                 3x – 2y = – 2
                                                –2y = – 3x – 2
                                                    y = 1.5x + 1

        Second equation:         –15x + by = 10
                                               –15x + 10y = 10
                                                 10y = 15x + 10
                                                      y = 1.5x + 1

        The slopes and y-intercepts are the same.

    SAT Verbal - Questions from Week 17 (June 12, 2021)

    SAT Quick Challenge Q21A
    Common SAT Fragment Errors -- Part 1




    What is a Sentence? You may recall that a sentence is a word group that has three essential characteristics: (1) contains a subject and its verb, (2) expresses a complete thought, and (3) makes sense all by itself. (This description also applies to an independent clause, since it is also a complete sentence.)
    What Is a Fragment? (Participial Phrase Fragment -- Which one?) A fragment is a word group that is written as though it were a sentence, but does not have all three essential sentence characteristics identified above. Note the italicized phrase in the following: sentence: "The gift hidden in the laundry room is Carlotta's birthday present." Some students think that the italicized phrase is a complete sentence all by itself, with the word "gift" as the subject and the word "hidden" as the verb. But the word "gift" is the subject of the complete sentence, and the verb for that sentence is the word "is." "Hidden" is part of the phrase that describes the noun "gift" by indicating which gift we are talking about -- the gift "hidden in the laundry room." Since the phrase describes a noun, the phrase functions as an adjective. Because "hidden" is the past participle of the verb "hide," the phrase is called a participial phrase, and "hidden" is referred to as a participle. Written as a complete sentence, that phrase would be a fragment because it does not have all three essential sentence elements identified above.

    Subordinating Conjunction Fragment -- Reason. A subordinating conjunction links the dependent clause in a sentence to the independent clause in that sentence. The dependent clause gives extra information about what is stated in the independent clause. For example, a dependent clause that begins with a word such as "because" or "since" (when "since" means "because") explains the reason for the action in the independent clause, as follows: "The children fell asleep right away because they were exhausted." The underlined independent clause tells the main idea of the sentence: "The children fell asleep right away," and the italicized dependent clause gives the reason that they fell asleep right away -- "because they were exhausted." Written all by itself, the dependent clause would be a fragment.

    Subordinating Conjunction Fragment -- Time. Subordinating conjunctions such as "after" or "when" tell the time of the action in the independent clause, as in the following: "The children fell asleep before we got home from the party." The underlined independent clause (which is also the main idea) tells that the children fell asleep, and the italicized dependent clause tells when they fell asleep (before we got home from the party). Again, written as a complete sentence all by itself, that clause would be a fragment.
    .
    Now, keeping in mind the information above, complete SAT Quick Challenge Exercise Q21A below.

    SAT Quick Challenge Exercise Q21A - Fragments

    Directions. On the line that follows each statement below, place the letter of the answer choice which corrects the underlined part of that statement. If you believe that the underlined part is already correct, select Choice A -- NO CHANGE. After completing this activity, use the answer key to check your work.

    1. Mom will meet us, at the mall after she leaves work.                

    1. NO CHANGE
    2. us at the mall.  After
    3. us at the mall after
    4. us.  At the mall after

    2. The puppies playing with the children.  Are getting plenty of exercise.               

    1. NO CHANGE
    2. the children.  are
    3. The children are
    4. the children are

    3. We laughed. When Mrs. Lee told her jokes.                

    1. NO CHANGE
    2. laughed when Mrs. Lee
    3. laughed When Mrs. Lee
    4. laughed. when Mrs. Lee

    SAT Verbal - Answers to Questions from Week 17 (June 12, 2021)

    1. C
    2. D
    3. B

    SAT Math - Questions from Week 16 (June 5, 2021)

    Systems of Linear Equations, A Short Cut – Part 4

    The traditional method of determining the answer in a system of linear equations problem involves calculating the values of each of the two variables by substitution, by elimination, or by using the answer choices. In some cases it is possible to find the answer without determining the values of each of the two variables. Consider the following example:

    3x + 4y  =  8
      x - 4y  =  12

    Based on the system of equations above, what is the value of 4x - 2y?
    Let’s solve this problem by substitution.


    Step 1.  Solve for x in the second equation.

    x - 4y = 12
            x = 12 + 4y

    Step 2. Substitute this expression of x in the first equation, and solve for y.

                 3x + 2y = 8
    3(12 + 4y) + 2y = 8
      36 + 12y + 2y = 8
               14y = 8 - 36
               14y = -28
                   y = -2

    Step 3.  Substitute this value of y in the second equation, and solve for x.
    x - 4y = 12
    x - 4(-2) = 12
    x + 8 = 12
          x = 4

    Now that we have values for x and y, we can find the value of 4x – 2y:
    4x – 2y = 4(4) – 2(-2) = 16 + 4 = 20


    In this case an alternative for finding our answer is simply to add the two equations:
    3x + 2y = 8
      x - 4y = 12
    ------------
    4x – 2y = 20

    We get 4x – 2y = 20, the same answer we got using the longer traditional approach. Whenever you have a problem similar to this one (where you are asked questions such as find the value of 3x + 5y or 8x + 10y or 3x + 7y), try adding the two equations or subtracting one equation from the other one.

    Keeping in mind the information above, answer the following systems of equations questions.
    1. 2x + 3y = 2
        x - 4y = 12
        Based on the system of equations above, what is the value of 3x – y?

    2. 4x + y = 14
      3x + 2y = 13
      Based on the system of equations above, what is the value of x - y?

    3. 2x + y = 6
      7x + 2y = 27
      Based on the system of equations above, what is the value of 3x + y?

    SAT Math - Answers to Questions from Week 16 (June 5, 2021)

    1. 2x + 3y = 2
        x - 4y = 12
        Based on the system of equations above, what is the value of 3x – y?

        Solving this system of equations be substitution or by elimination, we find that x = 4 and y =  -2;      
        thus 3x – y = 3(4) – (-2) = 12 + 2 = 14

        The short cut, which will work in this case, is to add the two equations together:

        2x + 3y = 2
          x – 4y = 12
        ------------
         3x  – y = 14         By adding, we get 3x – y = 14, which is our answer; we did not need to determine the individual values of x and y.  

    2. 4x + y = 14
      3x + 2y = 13
      Based on the system of equations above, what is the value of x - y?

      Try adding the equations:
      4x +   y = 14
      3x + 2y = 13
      ------------
      7x + 3y = 27

      Adding did not work. Try subtracting one equation from the other one. Subtract the second equation from the first equation.

        4x +    y = 14
      -(3x + 2y = 13)
      --------------

        4x +   y =  14
      -3x - 2y = -13
      --------------
          x – y =      1                This is our answer, and again we did not need to determine the individual values
                                             of x and y. (For your information, if you choose to calculate the values of x
                                             and y, x = 3, y = 2.)

    3. 2x + y = 6
      7x + 2y = 27
      Based on the system of equations above, what is the value of 3x + y?

      Try adding the equations:
      2x  +  y =   6
      7x + 2y = 27
      ------------
      9x + 3y = 33

      This is not our answer, but we observe that 9x + 3y is a multiple of what we are trying to find, 3x + y.

      We can divide the 9x + 3y = 33 by 3, and we get 3x + y = 11. Thus 11 is our answer.

      (For your information, if you choose to calculate the values of x and y, x = 5, y = -4.)

    SAT Verbal - Questions from Week 16 (June 5, 2021)

    SAT Quick Challenge P21
    Preciseness

    Preciseness refers to selecting the right word for the situation. For example, if a statement focuses on something negative, the correct answer will maintain that negative focus. Similarly, the answer for a statement with a positive or neutral focus will maintain that positive or neutral focus.

    Some students believe that the correct answer for a preciseness question is a "hard" word, and sometimes it is. Often, however, the correct answer is not so hard. To select the correct answer, just look for the choice that is most consistent with (1) the tone of the question and (2) the point being made. However, if you believe that all the easy words are incorrect, the hard word may actually be the correct answer. Note the Precision Sample Question below.

    Precision Sample Question. Select the correct answer for the question which follows.
    To advance good health, we must eat properly, exercise appropriately, and get the right amount of sleep.

    1. NO CHANGE
    2. popularize
    3. urge
    4. endanger

    Explanation and Answer: The sentence tells how to have good health, so we need a word that tells what a person must do in order to have good health. Hence, we must examine each possibility and then select the choice that does the best job of providing that information. "Popularize" means to make something popular, "urge" means to encourage people to do something, "endanger" means to cause danger to, and "advance" means to develop something or make it happen -- in this case, to make good health happen. Therefore, choice A is correct.

    Now, keeping in mind the information above, complete Quick Challenge Exercise P21 below.

    SAT Quick Challenge Exercise P21 - Preciseness

    Directions. On the line that follows each statement below, place the letter of the answer choice which corrects the underlined part of that statement. If you believe that the underlined part is already correct, select Choice A -- NO CHANGE. After completing this activity, use the answer key to check your work.

    1. Doctors say that since obese children generally become significantly heavier as they get older, their excess weight increases the chances that they will develop diabetes.               

    1. NO CHANGE
    2. inadequate
    3. friendly
    4. shy

    2. Wanda is always reading because of her inexcusable thirst for books and knowledge.               

    1. NO CHANGE
    2. unreliable
    3. unfavorable
    4. unquenchable

    3. Swerving skillfully, the alert driver avoided the vulnerable old lady, who could have been injured badly when she stumbled into the path of
    his car .               

    1. NO CHANGE
    2. athletic
    3. flexible
    4. insignificant

    SAT Verbal - Answers to Questions from Week 16 (June 5, 2021)

    1. A
    2. D
    3. A


    May Lessons


    SAT Math - Questions from Week 15 (May 22, 2021)

    Systems of Linear Equations – Part 3

    A system of equations can be solved in three ways: by substitution, by elimination, and by using the answer choices. When elimination is used, we find a way to eliminate one of the variables from the calculation. The first step is to multiply one of the equations by a number so that when the two equations are added together, one of the variables falls out. (For example, we might get 6x – 6x = 0 or 8x – 8x = 0.) We then solve for the remaining variable, then use the value of that variable to find the value of the variable that was eliminated.

    Consider the following example:
    3x + 4y  =  -23
     -x + 2y  =  -19

    Which ordered pair (x,y) satisfies the system of equations above?
    1. (-5, -2)
    2. (3, -8)
    3. (4, -6)
    4. (9, -6)

    Step 1.  .To eliminate one of the variables, we could multiply the second equation by 3 so that the x variable would be eliminated (we would get 3x – 3x = 0). Or we could multiply the second equation by -2 so that the y variable could be eliminated (we would get 4y – 4y = 0).
    In this case, multiply the second equation by 3.

    -x + 2y  =  -19 -------------> -3x + 6y = -57


    Step 2. Add the two equations together, and solve for y. 

                                       3x + 4y = -23
                                     -3x + 6y = -57
                                    ----------------
                                               10y = -80
                                                    y = -8


    Step 3.  Substitute this value of y into either of the two equations and solve for x. 
    First equation:         3x + 4y = -23
                                     3x + 4(-8) = -23
                                     3x - 32 = -23
                                     3x = 9
                                       x = 3

    Second equation:    -x + 2y = -19
                                     -x + 2(-8) = -19
                                     -x - 16 = 19
                                     -x = -3
                                       x = 3

    Thus the answer choice is B:  x = 3 and y = -8.


    Keeping in mind the information above, solve the following systems of equations by elimination.

    1. x + y = 0
      3x - 2y = 10

      Which of the following ordered pairs (x,y) satisfies the system of equations above?
      1. (3, -2)
      2. (2, -2)
      3. (-2, 2)
      4. (-2, -2)


    2. x + 2y = 7
      2x + 3y = 11

      Which of the following ordered pairs (x,y) satisfies the system of equations above?
      1. (-1, -3)
      2. (-1, 3)
      3. (1, 3)
      4. (1, -3)

    3. -2x = 4y + 6
      2(2y + 3) = 3x - 5

      Which of the following ordered pairs (x,y) satisfies the system of equations above?

      1. (1, 2)
      2. (1, -2)
      3. (-1, -1)
      4. (-1, 1)

    SAT Math - Answers to Questions from Week 15 (May 22, 2021)

    1. x + y = 0
      3x - 2y = 10

      Which of the following ordered pairs (x,y) satisfies the system of equations above?
      1. (3, -2)
      2. (2, -2)
      3. (-2, 2)
      4. (-2, -2)


      Step 1. Multiply the first equation by 2 so that the y variable can be eliminated.

      x + y = 0                 ------------->                 2x + 2y = 0

      Step 2. Add the two equations together, and solve for x.

      2x + 2y = 0
      3x  - 2y = 10
      -------------
      5x         = 10
      x = 2

      Step 3. Substitute this value of x into either of the two equations, and solve for y. 

      First equation:                          x + y = 0
                                                        2 + y = 0
                                                              y = -2

      Second equation:                    3x - 2y = 10
                                                       3(2) - 2y = 10
                                                         6 - 2y - 10
                                                          -2y = 4
                                                             y = -2

      Thus our answer choice is B:  x = 2 and y = -2.

    2. x + 2y = 7
      2x + 3y = 11

      Which of the following ordered pairs (x, y) satisfies the system of equations above?
      1. (-1, -3)
      2. (-1, 3)
      3. (1, 3)
      4. (1, -3)

      Step 1:
        Multiply the first equation by -2 so that the x variable can be eliminated.
      x + 2y = 7                 ------------->                -2x - 4y = -14

      Step 2. Add the two equations together, and solve for x.
      -2x - 4y = -14
        2x + 3y = 11
      ---------------
      -y = -3
      y = 3

      Step 3. Substitute this value of y into either of the two equations, and solve for x.
                                          
      First equation:                            x + 2y = 7
                                                          x + 2(3) = 7
                                                          x + 6 = 7
                                                          x = 1

      Second equation:                    2x + 3y = 11
                                                       2x + 3(3) = 11
                                                       2x + 9 = 11
                                                       2x = 2
                                                       x = 1

      Thus, our answer choice is C:  x = 1 and y = 3.

    3. -2x = 4y + 6
      2(2y + 3) = 3x - 5

      Which of the following ordered pairs (x,y) satisfies the system of equations above?
      1. (1, 2)
      2. (1, -2)
      3. (-1, -1)
      4. (-1, 1)

      We first should rewrite both equations in standard form: ax + by = c
      -2x = 4y + 6                                           2(2y + 3) = 3x - 5
      -2x -4y = 6                                            4y + 6 = 3x – 5
                                                                    -3x + 4y = -11


      Now we have this system of equations:
      -2x -4y = 6
      -3x + 4y = -11

      Step 1.
      In this case we do not need to multiply either equation by a number since the two equations can be added and the y variable can be eliminated.

      Step 2. Add the two equations together and solve for x.

      -2x - 4y = 6
      -3x + 4y = -11
      ---------------
      -5x          = -5
      x = 1

      Step 3. Substitute this value of x into either of the two equations and solve for y.

      First equation:                                       -2x -4y = 6
                                                                     -2(1) -4y = 6
                                                                     -2 - 4y = 6
                                                                     -4y = 8
                                                                         y = -2

      Second equation:                                 -3x + 4y = -11
                                                                    -3(1) + 4y = -11
                                                                     -3 + 4y = -11
                                                                     4y = -8
                                                                       y = -2

      Thus our answer choice is B: x = 1 and y = -2.

    SAT Verbal - Questions from Week 15 (May 22, 2021)

    SAT Quick Challenge O21
    Conciseness B

    Conciseness is the ability to write clearly without repeating ideas unnecessarily. For instance, if two words mean the same thing, you must not use them both in the same sentence because the unnecessary repetition destroys conciseness and makes the sentence redundant. Some SAT questions test your ability to write concisely by assessing how well you handle (1) synonymous adverb errors and (2) implied description errors. Both types are explained below.

    Synonymous Adverb Errors. When two or more adverbs mean the same thing, they are referred to as synonymous adverbs. However, using them in the same sentence creates unnecessary repetition, as in the following: "Hastily, Gloria ran to the bus stop quickly, hoping that she would not miss the bus." The sentence can say that Gloria ran "hastily" or that she ran "quickly," but not both, since the two words mean the same thing.

    Implied Description Errors. An implied description is a word that does two grammatical jobs: (1) it names something, and (2) it gives a "definition-related" description of the thing it names, as in the following sentence: "Carla needs the tan, four-sided square." This sentence tells us what Carla needs -- a figure called a square. However, the word "square" itself implies a figure that has four right angles and four straight sides that are the same length. Since the word "square" clearly indicates the figure that is being talked about, a writer does not need to mention the square's sides, and doing so makes the sentence redundant.

    Note that since the color of the square does not help define the figure, mentioning the color is fine. However, if the color had been needed in order to identify which square Carla needs, pointing that color out is not simply fine, but actually necessary. In this case, then, stating the color is important, and the sentence should say, "Carla needs the tan square."

    Keeping in mind the information above, complete SAT Quick Challenge Exercise O21 below.

    SAT Quick Challenge Exercise O21
    Conciseness B

    Directions. On the line that follows each statement below, place the letter of the answer choice which corrects the underlined part of that statement. If you believe that the underlined part is already correct, select Choice A -- NO CHANGE. After completing this activity, use the answer key to check your work.

    1. Rocky happily and joyfully barked greetings to his owners when they picked him up from the doggie
    sitter after their two-week vacation.                 

    1. NO CHANGE
    2. joyfully
    3. happily and cheerfully
    4. joyfully and gladly

    2. To score points, the children had to run through the four-sided rectangular door without being tagged by the catcher.                 

    1. NO CHANGE
    2. rectangular in form
    3. rectangular
    4. rectangular shaped

    3. The girls spoke quietly so that they would not wake the sleeping baby up.                 

    1. NO CHANGE
    2. wake the baby up
    3. wake up the slumbering baby
    4. awaken the sleeping baby

    SAT Verbal - Answers to Questions from Week 15 (May 22, 2021)

    1. B
    2. C
    3. B

    SAT Math - Questions from Week 14 (May 15, 2021)

    Systems of Linear Equations – Part 2

    As noted last week a system of equations can be solved in three ways: by substitution, by elimination, and by using the answer choices. The alternative of using the answer choices can be used only when the answer choices include values for both variables. Remember answer choices are not given for 5 of the problems in Section 3 of the SAT and answer choices are not given for 8 of the problems in Section 4. Also in some Problems the question might be “What is the value of x?” or “What is the value of x + y?” In these cases we would have to solve the problem by substitution or by elimination.

    Let's look at the substitution process.

    Consider the following example:
    3x + 4y  =  -23
     -x + 2y  =  -19

    Which ordered pair (x,y) satisfies the system of equations above?
    1. (-5, -2)
    2. (3, -8)
    3. (4, -6)
    4. (9, -6)

    These are the steps for solving by substitution.

    Step 1.  In one of the equations solve for one of the variables in terms of the other variable. Let’s solve for x in the second equation.
    3x + 4y = -23
    -x = -19 - 2y
    x = 19 + 2y

    Step 2. Substitute this expression of x in the first equation, and solve for y. 
    3x + 4y = -23
    3(19 + 2y) + 4y = -23
    57 + 6y + 4y = -23
    10y = -23 - 57
    10y = -80
    y = -8

    Step 3.  Substitute this value of y in the second equation, and solve for x. 
    -x + 2y = -19
    -x + 2(-8) = -19
    -x - 16 = 19
    -x = -3
    x = 3

    Thus the answer choice is B, x = 3 and y = -8.


    Keeping in mind the information above, solve the following systems of equations by substitution.
    1. x + y = 0
      3x - 2y = 10

      Which of the following ordered pairs (x,y) satisfies the system of equations above?
      1. (3, -2)
      2. (2, -2)
      3. (-2, 2)
      4. (-2, -2)

    2. x + 2y = 7
      2x + 3y = 11

      Based on the system of equations above, what is the value of x? (No answer choices are given here.)

    3. -2x = 4y + 6
      2(2y + 3) = 3x - 5

      If the ordered pair (x,y) that satisfies the system of equations above is (x,y), what is the value of x + y?

      Which of the following ordered pairs (x,y) satisfies the system of equations above?

      1. -1
      2. 0
      3. 1
      4. 2

    SAT Math - Answers to Questions from Week 14 (May 15, 2021)

    1. x + y = 0
      3x - 2y = 10

      Which of the following ordered pairs (x,y) satisfies the system of equations above?
      1. (3, -2)
      2. (2, -2)
      3. (-2, 2)
      4. (-2, -2)

      These are the steps for solving by substitution.

      Step 1. In one of the equations solve for one of the variables in terms of the other variable. Let’s solve for x in the first equation.

      x = -y

      Step 2. Substitute this expression of x in the second equation, and solve for y. 

      3x - 2y = 10
      3(-y) -2y = 10
      -3y - 2y = 10
      -5y = 10
      y = -2

      Step 3. Substitute this value of y in the first equation, and solve for x. 

      x + y = 0
      x - 2 = 0
      x = 2

      Thus our answer choice is B, x = 2 and y = -2.

    2. x + 2y = 7
      2x + 3y = 11

      Based on the system of equations above, what is the value of x? (No answer choices are given here.)

      These are the steps for solving by substitution.

      Step 1:
       In one of the equations solve for one of the variables in terms of the other variable. Let’s solve for x in the first equation.
      x = -2y + 7

      Step 2. Substitute this expression of x in the second equation, and solve for y.

      2x + 3y = 11
      2(-2y + 7) + 3y = 11
      -4y + 3y = 11 - 14
      -y = -3
      y = 3

      Step 3. Substitute this value of y in the first equation, and solve for x.
                                          
      x +  2y  =  7
      x + 2(3) = 7
      x + 6 = 7
      x = 1

      Thus, our answer is: x = 1.

    3. -2x = 4y + 6
      2(2y + 3) = 3x - 5

      If the ordered pair (x,y) that satisfies the system of equations above is (x,y), what is the value of x + y?

      1. -1
      2. 0
      3. 1
      4. 2

      We first should rewrite both equations in standard form: ax + by = c
      -2x = 4y + 6                                           2(2y + 3) = 3x - 5
      -2x -4y = 6                                            4y + 6 = 3x – 5
                                                                    -3x + 4y = -11


      Now we have this system of equations:
      -2x -4y = 6
      -3x + 4y = -11


      These are the steps for solving by substitution.

      Step 1.
      In one of the equations solve for one of the variables in terms of the other variable. Let’s solve for x in the first equation.

      -2x - 4y = 6
      -2x = 4y + 6
      x = -2y - 3

      Step 2. Substitute this expression of x in the second equation, and solve for y.

      -3x + 4y = 6
      -3(-2y -3) + 4y = -11
      6y +9 + 4y = -11
      10y = -20
      y = -2

      Step 3. Substitute this value of y in the first equation, and solve for x.

      -2x -4y = 6
      -2x -4(-2) = 6
      -2x +8 = 6
      -2x = -2
      x = 1

      Now we have values for x and y: x = 1 and y = -2. Thus x + y = 1 + (-2) = -1, and our answer choice is A.

    SAT Verbal - Questions from Week 14 (May 15, 2021)

    SAT Quick Challenge N21
    Conciseness A

    Conciseness is the ability to express ideas clearly without using unnecessary repetition. Some SAT questions test your ability to avoid unneeded words that destroy conciseness. For instance, if two words mean the same thing, you must not use them both in the same sentence. Note example A below.

    Example A: The customer was very angry and furious about the manager's refusal to give her a refund for the spoiled fruit she had just purchased. Explanation: Because very angry and furious mean the same thing, you must use only one of the two terms in your sentence, as shown in Examples B and C below. Each option corrects the redundancy in Example A by using just one of the two synonyms in that sentence.

    Example B: The customer was very angry about the manager's refusal to give her a refund for the spoiled fruit she had just purchased. 

    Example C:
    The customer was furious about the manager's refusal to give her a refund for the spoiled fruit she had just purchased. 

    Common SAT conciseness errors that you must avoid include (1) using compound verb phrases and (2) using synonymous adjectives. Both are explained below.

    Compound Verb Phrase Errors. A compound verb phrase is a phrase that contains two verbs that mean the same thing, as in the following sentence: Jessica screamed and shouted for the intruder to leave. The sentence is redundant because the two verbs, "screamed" and "shouted," mean the same thing. Using either one of the verbs will make the sentence correct, as in the following: Jessica screamed for the intruder to leave. OR Jessica shouted for the intruder to leave.

    Synonymous Adjectives Errors. When two or more adjectives mean the same thing, they are referred to as synonymous adjectives. Using those types of modifiers in the same sentence is incorrect because doing so makes the sentence redundant, as in the following: The Hill family's mansion is huge, "gorgeous," and "beautiful." Since "gorgeous" and "beautiful" mean the same thing, the sentence is redundant. Your sentence can say that the mansion is "beautiful" or that it is "gorgeous," but not both.


    Keeping in mind the information above, complete SAT Quick Challenge Exercise N21 below.

    SAT Quick Challenge Exercise N21
    Conciseness A

    Directions. On the line that follows each statement below, place the letter of the answer choice which corrects the underlined part of that statement. If you believe that the underlined part is already correct, select Choice A -- NO CHANGE. After completing this activity, use the answer key to check your work.

    1. Last night's storm was so scary and frightening that the children all huddled together in the same bed.                 

    1. NO CHANGE
    2. terrifying and scary
    3. frightening and terrifying
    4. frightening

    2. After working in the field all day, the boys felt really hungry and starving.                 

    1. NO CHANGE
    2. really hungry
    3. starving and famished
    4. starving hungry

    3. The driver sped and flew along the road the way racers zip around a race track.                  

    1. NO CHANGE
    2. zipped and flew
    3. zoomed
    4. zoomed and sped

    SAT Verbal - Answers to Questions from Week 14 (May 15, 2021)

    1. D
    2. B
    3. C

    SAT Math - Questions from Week 13 (May 1, 2021)

    Systems of Linear Equations – Part 1

    One of the types of questions you will see on the SAT is solving systems of linear equations, sometimes called simultaneous equations or two equations in two unknowns. A system of equations can be solved in three ways: by substitution, by elimination, and by using the answer choices (when answer choices are given). When the alternative of using the answer choices is used, we substitute an answer choice in each of the two equations and determine which one works. The correct answer choice must work in both equations. Consider the following example:

    3x + 4y  =  -23
     -x + 2y  =  -19

    Which ordered pair (x,y) satisfies the system of equations above?
    1. (-5, -2)
    2. (3, -8)
    3. (4, -6)
    4. (9, -6)
    Try choice A): x = -5 and y = -2

    First equation:
    3x + 4y = -23
    3(-5) + 4(-2) = -23
    -15 -8 = -23
    YES

    Second equation:
    -x + 2y = -19
    -(-5) + 2(-2) = -19
    5 -4 = 1, not -19
    NO; choice A does not work.

    Now try choice B):  x = 3 and y = -8

    First equation:
    3x +  4y  =  -23
    (3) + 4(-8) = -23
    9  - 32  =  -23
    YES            

    Second equation:   
    -x +  2y  =  -19
    -(3) + 2(-8)  =  -19
    -3  -16  =  -19       
    YES           

    Thus our answer is B; these numbers work for both equations. Answer choices C and D do not work.

    Keeping in mind the information above, solve the following systems of equations by using the answer choices.
    1. x + y = 0
      3x - 2y = 10

      Which of the following ordered pairs (x,y) satisfies the system of equations above?
      1. (3, -2)
      2. (2, -2)
      3. (-2, 2)
      4. (-2, -2)

    2. x + 2y = 7
      2x + 3y = 11

      Which of the following ordered pairs (x,y) satisfies the system of equations above?
      1. (-1, -3)
      2. (-1, 3)
      3. (1, 3)
      4. (1, -3)

    3. -2x = 4y + 6
      2(2y + 3) = 3x - 5

      Which of the following ordered pairs (x,y) satisfies the system of equations above?
      1. (1, 2)
      2. (1, -2)
      3. (-1, -1)
      4. (-1, 1)

    SAT Math - Answers to Questions from Week 13 (May 1, 2021)

    1. x + y = 0
      3x - 2y = 10

      Which of the following ordered pairs (x,y) satisfies the system of equations above?
      1. (3, -2)
      2. (2, -2)
      3. (-2, 2)
      4. (-2, -2)

      Try choice A): x = 3 and y = -2
      First equation:
      x + y = 0
      3 -2 = 1, not 0
      NO; choice A does not work.

      Now try choice B): x = 2 and y = -2
      First equation:
      x + y = 0
      2 -2 = 0
      YES

      Second equation:
      3x - 2y = 10
      3(2) -2(-2) = 6 + 4 = 10
      YES
      Thus our answer is B; answer choices C and D do not work.

    2. x + 2y = 7
      2x + 3y = 11

      Which of the following ordered pairs (x,y) satisfies the system of equations above?
      1. (-1, -3)
      2. (-1, 3)
      3. (1, 3)
      4. (1, -3)

      Try choice A):  x = -1 and y = -3
      First equation:    x +  2y  =  7
      -1 +2(-3) = -1 -6  =  -7, not 7         
      NO; choice A does not work                                                                                                                                                                                                 

      Try choice B):   x = -1 and y = 3
      First equation:    x +  2y  =  7
      -1 +2(3) = -1 +6  =  5, not 7
      NO; choice B does not work

      Try choice C):   x = 1 and y = 3
      First equation:    x +  2y  =  7
      1 +2(3) = 1 +6  =  7    YES

      Second equation:   2x +  3y  =  11
      2(1)  +  3(3)  =  2 + 9 = 11   YES
      Thus our answer is C; answer choice D does not work.

    3. -2x = 4y + 6
      2(2y + 3) = 3x - 5

      Which of the following ordered pairs (x,y) satisfies the system of equations above?
      1. (1, 2)
      2. (1, -2)
      3. (-1, -1)
      4. (-1, 1)

      Try choice A): x = 1 and y = 2
      First equation:
      -2x = 4y + 6
      -2x = -2(1) = -2 and
      4y + 6 = 4(2) +6 = 8 + 6 = 14, not -2
      NO; choice A does not work

      Now try choice B): x = 1 and y = -2
      First equation:
      -2x = 4y + 6
      -2x = -2(1) = -2 and
      4y + 6 =
      4(-2) + 6 =
      -8 + 6 = -2
      YES

      Second equation:
      2(2y + 3) = 3x -5
      2(2y + 3) = 2(2(-2) + 3) =
      2(-4 + 3) = 2(-1) = -2 and
      3x – 5 = 3(1) – 5 = 3 – 5 = -2
      YES

      Thus our answer is B; answer choices C and D do not work.

    SAT Verbal - Questions from Week 13 (May 1, 2021)

    SAT Quick Challenge M21
    Placing Modifiers Correctly

    A modifier is a word or word group that describes someone or something. If the modifier is not placed next to (or as close as possible to) the word/word group it is describing; it will be misplaced, and the sentence will not convey the meaning intended. The possessive noun error and the relative clause error are misplaced modifier errors often tested on the SAT. Both are explained below.

    The Possessive Noun Error. Whether a noun is possessive or not, the regular modification rule still applies: place the modifier next to (or as close as possible to) the word/word group that it describes. Note the following sentence: "Speaking with authority, the teacher's reassuring smile and her expert rescue experience helped us remain calm and cooperative during her heroic, lifesaving efforts." Note that the closest nouns/noun phrases to the introductory modifier (Speaking with authority) are "the teacher's reassuring smile" and "her expert rescue experience." Accordingly, the sentence says that the teacher's smile and rescue experience were speaking with authority, but those things cannot speak. Hence, the sentence has a possessive noun error. Note the following correction: "Speaking with authority, the teacher beamed her reassuring smile and used her rescue experience to help us remain calm and cooperative during her heroic, lifesaving efforts." Now, the correct word, "teacher," is being modified.

    The Relative Clause Error. You may recall that a clause is a word group that contains a subject and a verb. (An independent clause expresses a complete thought and can stand alone as a complete sentence; a dependent clause does not express a complete thought and cannot stand alone as a complete sentence.) A relative clause, which begins with words such as "when," "who," "where," "which," or "that," is a dependent clause that functions as a modifier and must be placed right after the noun it describes. Note that the italicized relative clause in the sentence which follows is misplaced because it is not right after the noun it modifies. "The school is in Atlanta where Mr. Johnson will be coaching next year." The clause should modify the noun "school," but it is not placed right after that word." Note the following corrected sentence. Correction: "The school where Mr. Johnson will be coaching next year is in Atlanta." Now, the correct word, "school," is being modified. Keeping in mind the information above, complete Exercise M21 below.

    SAT Quick Challenge Exercise M21
    Placing Modifiers Correctly

    Directions. On the line that follows each statement below, place the letter of the answer choice which corrects the underlined part of that statement. If you believe that the underlined part is already correct, select Choice A -- NO CHANGE. After completing this activity, use the answer key in the "SAT Verbal - Answers to Questions from Week 13 (May 1, 2021)" dropdown below to check your work.

    1. Filled with excitement, Jessica's joy was uncontrollable as she showed us the letter offering her a full college scholarship.                 

    1. NO CHANGE
    2. Jessica could not keep her hands from shaking
    3. the shaking of Jessica's hands would not stop
    4. Jessica's shaking could not be controlled

    2. The guest speaker's many awards is a professor at Harvard.                 

    1. NO CHANGE
    2. will be our main speaker
    3. speaker's respected and honored awards
    4. who will be our main speaker

    3. The day the most miserable weather ever was one of our most unforgettable days ever.                 

    1. NO CHANGE
    2. stranded vehicles all along the highway
    3. when we got the unexpected blizzard
    4. sudden surprisingly miserable weather

    SAT Verbal - Answers to Questions from Week 13 (May 1, 2021)

    1. B
    2. D
    3. C


    April Lessons


    SAT Math - Questions from Week 12 (April 24, 2021)

    Statistical Sampling: Cause and Effect

    Some sampling questions on the SAT present a cause and effect situation. A treatment of some type is applied to the participants in a randomly chosen sample, and the effect on the participants is noted. The question is whether the results can be generalized to the entire population. Remember that we use statistical sampling to make generalizations about large populations based on observations we observe from randomly selected small groups from those populations. Consider the following example:

    A notice was placed in the A&T student newspaper indicating that assistance would be provided to students in preparing their cover letters and resumes, important documents for students seeking employment after college graduation. The time and place for the meeting were indicated, and 300 students attended. The meeting organizer randomly assigned 100 of the students to Group A and the other 200 students to Group B. All 300 students prepared cover letters and resumes before the assistance was provided and after assistance was provided; the cover letters and resumes were reviewed by corporate recruiters. Students in Group A were provided detailed instruction with examples and personal assistance, and students in Group B were provided no assistance or instruction. The corporate recruiters reported that the students in Group A improved their cover letters and resumes significantly compared to those in Group B. What generalization can be made from this analysis? What conclusions can be made?

    It should be noted that the population is not all students; it is not all A&T students; it is not all A&T students who read the student newspaper. The population is the 300 students who attended the meeting, and a random sample was taken from that population. Therefore, we can conclude that it is likely that the assistance improved the cover letters and resumes of students in Group A and that the result can be generalized to the other students in the population of the 300 students. We cannot, however, make any generalization about other students or individuals who were not in that population of 300 students. Sample results can be generalized only to the specific population from which the random sample was taken.


    Keeping in mind the information above, answer the following questions.

    1. Mr. Mason, a math teacher, offers extra help every Wednesday after school. He keeps careful attendance records at these sessions. At the end of the school year, he compares the average score of the students who attended extra help sessions regularly to the average score of the students who didn’t attend, and finds that students who attended extra help sessions regularly did better in the class. Which of the following is an appropriate conclusion?
      1. Attending extra help sessions regularly will cause an improvement in performance for any student in any subject.
      2. Attending extra help sessions regularly will cause an improvement in performance for any student taking a math course.
      3. Attending extra help sessions regularly with Mr. Mason was the cause of the improvement for the students in Mr. Mason’s class who did so.
      4. No conclusion about cause and effect can be made regarding students in Mr. Mason’s class who attended extra help sessions regularly and their performance in the course.

    2. In order to determine if treatment X is successful in improving eyesight, a research study was conducted. From a large population of people with poor eyesight, 400 participants were selected at random. Half of the participants were randomly assigned to receive treatment X, and the other half did not receive treatment X. The resulting data showed that participants who received treatment X had significantly improved eyesight as compared to those who did not receive treatment X. Based on the design and results of the study, which of the following is an appropriate conclusion?
      1. Treatment X is likely to improve the eyesight of people who have poor eyesight.
      2. Treatment X improves eyesight better than all other available treatments.
      3. Treatment X will improve the eyesight of anyone who takes it.
      4. Treatment X will cause a substantial improvement in eyesight.

    3. In order to determine if 30 minutes of intense cardiovascular exercise before bed can help people with insomnia sleep better, a research study was conducted. From a large population of people with insomnia, 500 participants were selected at random. Half of those participants were randomly assigned an intense cardiovascular exercise regimen to complete each night before bed. The other half were not assigned an exercise regimen. The resulting data showed that throughout the study, participants who were assigned an exercise regimen slept better than participants who were not assigned an exercise regimen. Based on the design and results of the study, which of the following is an appropriate conclusion?
      1. Intense cardiovascular exercise is likely to help anyone who tries it to sleep better.
      2. Intense cardiovascular exercise is likely to help people with insomnia sleep better.
      3. Any kind of exercise is likely to help people with insomnia sleep better.
      4. No conclusion about cause and effect can be made regarding intense cardiovascular exercise and people with insomnia sleeping better.

    SAT Math - Answers to Questions  from Week 12 (April 24, 2021)

    1. D is correct.
      Since the students who received the extra help were not randomly selected, we cannot conclude that it was the extra help that resulted in the improvement. It may be that those students who sought the extra help were more motivated than others, and were more willing to study harder on their own.

    2. A is correct since the participants were selected at random and the participants were randomly assigned to treatment groups.
      B is incorrect since all available treatments were not included in the study.
      C is incorrect since the population consisted only of persons with poor eyesight, not everyone.
      D is incorrect since the population consisted only of persons with poor eyesight, not everyone. Also the treatment is likely to improve the eyesight of people who have poor eyesight, not that it definitely will cause improvement in eyesight for everyone.

    3. B is correct since the participants were randomly selected from people with insomnia and the intense cardiovascular exercise regimen was assigned randomly to half of the participants.
      A is not correct because the sample was not selected from all people.
      C is not correct because all kinds of exercise were not included in the study.

    SAT Verbal - Questions  from Week 12 (April 24, 2021)

    SAT Quick Challenge L21
    Placing Modifiers Correctly

    A modifier is a word or word group that describes someone or something. The modifier must be placed next to (or as close as possible to) the word/word group that it is describing; otherwise, the modifier will be misplaced, and the sentence will not convey the meaning intended. Two common types of misplaced modifier errors found on the SAT are the misplaced introductory modifying phrase and the misplaced "extra information" description. Both are explained below.

    The Misplaced Introductory Modifying Phrase. In the sentence "Speeding through the air, the catcher was injured when the ball hit his shoulder," the introductory modifying phrase Speeding through the air is next to the underlined noun phrase the catcher. Therefore, the sentence says that the catcher was speeding through the air, but that idea makes no sense. The ball was speeding through the air, and the modifier should be next to the ball. Correction: Speeding through the air, the ball injured the catcher when it hit his shoulder. Now, the modifier is next to the phrase it modifies, and the sentence makes sense.

    The Misplaced "Extra Information" Description. Sometimes, extra information that is not part of the main idea is added to a sentence, as follows: "The student playing Miss Jane Pittman used special makeup, a 17-year-old senior, to make herself appear to be over 100 years old." Enclosed in commas because it is extra, the description a 17-year-old senior has been placed next to the noun phrase special makeup. In that location the modifier says that the special makeup is a 17-year-old senior. Yet, the extra information description should modify the noun phrase The student playing Miss Jane Pittman, and it should be placed next to that phrase. Correction. The student playing Miss Jane Pittman, a 17-year-old senior, used special makeup to make herself appear to be over 100 years old. Now, the sentence makes sense. Keeping in mind the information above, complete Exercise L21 below. After you complete this exercise, use the answer key in the "SAT Verbal - Answers to Questions from Week 11" dropdown below to check your work.

    SAT Quick Challenge Exercise L21
    Placing Modifiers Correctly

    Directions. On the line that follows each statement below, place the letter of the answer choice which corrects the underlined part of that statement. If you believe that the underlined part is already correct, select Choice A -- NO CHANGE. 

    1. Wincing from the insect bite, Marissa's tears made her eyes red and swollen.             

    A.  NO CHANGE                                                                      
    B.  Marissa's tears would not stop                                   
    C. Marissa could not hold back her tears
    D. the sting brought tears to Marissa's eyes

    2. Just before dinner time, Mom spooned her special cheese dip into individual serving dishes, a long-time family favorite.             

    A.  NO CHANGE                                                                      
    B.  her special cheese dip, a long-time family favorite, into individual serving dishes
    C. a long-time family favorite into individual serving dishes her special cheese dip
    D.  her special cheese dip a long-time family favorite into individual serving dishes

    3. Disposable diapers are the top choice everywhere of families, especially because of their timesavers. They don't have to be washed, dried, or folded; you just throw them away..             

    A.  NO CHANGE                                                                      
    B.  because of families everywhere in which they save time
    C. of families everywhere, this is because of the time they save
    D. of families everywhere because they save time

    SAT Verbal - Answers to Questions  from Week 12 (April 24, 2021)

    1. C
    2. B
    3. D

    SAT  Math - Questions  from Week 11 (April 17, 2021)

    Statistical Sampling: Margin of Error

    Some sampling questions on the SAT require the use of a “margin of error.” Consider the following situation: The Guilford County School Board has directed the superintendent of schools to prepare an obesity report on children in the Guilford County public schools. Among other information in the data section of the report, the average weight of 8th graders will be indicated. It will be expensive and time consuming to weigh every 8th grader, since there are over 6,000 8th graders in Guilford County public schools. Instead, the average weight of a random sample of Guilford County 8th graders will be determined.

    Suppose this average is 94 pounds. Based on this result, what conclusion can be made about the average weight of the entire population of 8th graders? Since the sample was a random sample, it is plausible that the average weight of the population of 8th grades is 94 pounds. The estimate is reasonable, but it is unlikely to be exactly correct; the actual average for the population may be somewhat less than 94 pounds or somewhat more than 94 pounds. Statisticians use a “margin of error” to describe the precision of an estimate.

    If this example were an SAT question, you might be given results indicating that, for a random sample of 100 8th graders, the estimated average weight was 94 pounds with a margin of error of 3 pounds. A logical interpretation of this result is that it is plausible that the average weight of all 8th graders in the population is greater than 91 pounds and less than 97 pounds.

    There are two factors that affect the value of the margin of error: the variability (standard deviation) in the data and the sample size.

    1. Variability: the larger the standard deviation, the larger the margin of error; the smaller the standard deviation, the smaller the margin of error.
    2. Sample size: the larger the sample size, the smaller the margin of error.

    Answer the following questions. 

    1. A researcher surveyed a random sample of students from a large university about how often they see movies. Using the sample data, the researcher estimated that 23% of the students in the population saw a movie at least once each month, with a margin of error of 4%. Which of the following is the most appropriate conclusion about all students at the university, based on the given estimate and margin of error?
      1. It is unlikely that less than 23% of the students see a movie at least once per month
      2. At least 23%, but no more than 25%, of the students see a movie at least once per month
      3. The researcher is between 19% and 27% sure that most students see a movie at least once per month
      4. It is plausible that the percentage of students who see a movie at least once per month is between 19% and 27%.

    2. Jason conducted a survey of a randomly selected group of 18-24-year old American males to determine how many basketball games young men of that age watch in a month. He excluded tournament games, “March madness,” and other playoff games. After reviewing his results, he decided that his margin of error was too high and that he would conduct the survey again. Which of the following changes could Jason make to decrease his margin of error?
      1. Increase his sample size.
      2. Decrease his sample size.
      3. Increase the age range in his sample.
      4. Include 18-24 year old American females in his sample.

    3. A group of researchers conducted a phone survey of 200 randomly selected people in Durham in an attempt to determine the average amount spent by people in the city every month on groceries. They calculate that the average is $197 with a margin of error of 6%. Rounded to whole dollars, which of the following represents the confidence interval the researchers should report?
      1. $102 - $192
      2. $185 - $209
      3. $187 - $207
      4. $191 - $203

    SAT  Math  -  Answers to Questions  from Week 11 (April 17, 2021)

    1. D is correct.
      A is incorrect; it could be less than 23%.
      B is incorrect: it could be less than 23%, and it could be more than 25%.
      C is incorrect: this is an incorrect interpretation of the margin of error

    2. A is correct - Increase the sample size.
      B and C would increase the margin of error.
      D is incorrect since the research is about young men.

    3. This is one of those SAT trap questions:  if you do not read the question carefully, you will be trapped into choosing the wrong answer.   The trap is the  “6%”  margin of error and  the average of $197.  The trap is adding 6 to 197 and getting 203, subtracting 6 from 197 and getting 191, and choosing  D - $191 - $203.  This calculation is incorrect because we should add and subtract dollars, not a percentage point.  We must express the margin of error in dollars.
      6% of $197  =  $197 x .06  =  $11.82     This is the margin of error.
      $197  -  $11.82  =  $185.18   and  $197  +  $11.82  =  $208.82
      These round to $185  and  $209  
      Thus, the correct answer is B:  $185 - $209            

    SAT  Verbal - Questions  from Week 11 (April 17, 2021)

    SAT Quick Challenge K21
    Formal or Casual Language

    Formal speech is reserved for people who understand, and may have special training in, what is being discussed (science, technology, politics, etc.). Therefore, such communication can involve long, complex sentences that use elaborate, unusual words and phrases related specifically to the topic being discussed. Casual speech, however, is used for everyday language. It is noted for shorter, conversational sentences which may include friendly chatter, figures of speech, or even slang. Some SAT questions ask you (1) to determine whether a passage uses formal or casual language and then (2) to complete a sentence with a word or word group which matches the language used in the passage.

    Practice classifying language as formal or casual by reading the sample passage below. Then select the choice beneath the passage which best replaces the underlined part of the last sentence in the paragraph. However, if you think that the underlined part is already correct, select choice A -- NO CHANGE.
    Tip: A formal passage must have a formal answer, and a casual passage must have a casual answer.

    Sample Passage: Kinds of Locksets. Locksets routinely fall into two main categories: mortise and cylindrical. The former has a rectangular body that will glide into a similarly shaped pocket. The latter has a rotund body that slips into a bore hole and connects with the latch bolt. Mortise locksets are generally used for warehouses and industrial sites, while cylindrical locksets are routinely used for a wide range of residential spaces. Locksets are generally sold (1) in major chain stores such as Lowe's and Home Depot, and (2) in department stores such as Walmart and Sears, but you can often find larger selections and the best prices online. Hence, locksets are likely to be less expensive at Lowe's.

    A. NO CHANGE         B. cheaper at Sears         C. higher at Walmart         D. more cost-effective online

    Answer: Since you can often find larger selections and the best prices online, locksets sold online are likely to cost less than the others. Also, cost-effective is a formal term that matches the language used in the passage. Hence, D is the correct answer. NOW, COMPLETE EXERCISE K21 BELOW.

    Exercise K21 -- Formal or Casual Language

    Directions. On the blank line after each passage, write the answer choice which best replaces the underlined part
    of the last sentence in the passage. If you think the underlined part is already correct, select choice A -- NO
    CHANGE. Then use the answer key in the "SAT Verbal - Answers to Questions from Week 11" dropdown below to check your work.

    1. Sisu, a clever little guy, would wait outside near the door, quietly go into the store as people would leave, head straight for the toys.  Then he would grab the little purple stuffed unicorn and head back outside. To his dismay, however, he would always get caught before making it out the door.  After he was caught for the fifth time, an employee called the authorities, who promptly came and confronted Sisu.  Finding out that he was homeless, the officer felt sorry for him and bought him the little toy.  Also, thanks to publicity about Sisu, dog lovers are calling to find out how to provide Sisu with information about employment opportunities.            

      A.  NO CHANGE          B. demand Sisu's incarceration          C. adopt Sisu          D. inform Sisu of local soup kitchens
    1. Mr. Al Jacobs was notorious for his infractions against the law -- all carried out, along with his accomplices, under the the guise of legitimate, upstanding citizens such as bankers, lawyers, and realtors. Through online and telephone scams, they bilked innocent victims, many of them senior citizens, out of millions of dollars. After escaping arrest multiple times, they all were finally captured, thanks to ingenious FBI schemes. Ultimately, the scammers were all sent to the big house for decades.           

      A.  NO CHANGE       B. tossed in the slammer     C. imprisoned         D. sentenced to free food, clothing, and shelter

    SAT Verbal - Answers to Questions  from Week 11 (April 17, 2021)

    1. C
    2. C

    SAT Verbal - Questions  from Week 10 (April 10, 2021)

    SAT QUICK CHALLENGE J21
    Using the Correct Word


    Deciding which of two commonly confused words should be used may not be as difficult as one might think initially. The tips below explain how to use some words in that category correctly.

    Commonly Confused Word Pair A"less" vs. "fewer" 

    RULE:  If the commonly confused word is followed by a singular noun and refers to something that cannot  be counted, use the word "less."  If the commonly confused word is followed by a plural noun and refers to something that can be counted, use the word fewer.  

    Sentence 1: 
    During meals, Mrs. Parker usually has              food on her plate than her son has on his. 
    Explanation 1: 
    A singular noun (food) follows the word in question and names something  that cannot be counted.
      (NOTE:  You can count individual food items, but not the concept of  "food."Therefore, the missing word must be "less." 
    Correct Sentence: 
    Mrs. Parks usually  has less food on her plate than her son has on his.

    Sentence 2: 
    When I take my time, I make             mistakes than I do when I rush. 
    Explanation 2: 
    A  plural noun (mistakes) follows the commonly confused word and refers to something that can be  counted.  Therefore, the missing word is "fewer." 
    Correct Sentence:   
    When I take my time, I make fewer mistakes than I do when I rush.

    Commonly Confused Word Pair B"much" vs. "many" 

    RULE:  If the commonly confused word is followed by a singular noun and refers to something that cannot be counted, use the word "much."  If the commonly confused word is followed by a plural noun and refers to something that can be counted, use the word many.  

    Sentence 3: 
    During meals, Mrs. Parker's son usually has             more food on his plate than she  has on hers. 
    Explanation 3: 
    A singular noun (food) follows the commonly confused word and names  something that cannot be counted.  (NOTE
    :  You can count individual food items, but not the  concept of "food.")  Therefore, the missing word is "much." 
    Correct Sentence
    Mrs. Hall's  son usually has much more food on his plate than she has on hers.

    Sentence 4: 
    When I rush, I make             more mistakes than I do when I take my time.
    Explanation 4: 
    A plural noun (mistakes) follows the commonly confused word and refers to something  that can be counted.  Therefore, the missing word is "many." 
    Correct Sentence:   
    When I  rush, I make many more mistakes than I do when I take my time.

    Keeping in mind the information above, complete Exercise J21 below.

    Directions. On the line that follows each statement below, place the letter of the answer choice that corrects any error in the statement. If there is no error, mark choice "A" as your answer. When you have finished the exercise, use the answer key in the "SAT Verbal -  Answers to Questions from Week 10" dropdown below to check your work.

    1. The debate team students say that they have much more fun at the amusement park than at the zoo.                   

    A.  NO CHANGE                B. more better                            C. lots of more                       D. many more fun

    2. We scored much more points in the game this week than we scored last week.                                    

    A.  NO CHANGE                B. lots of more                            C. many more                       D. more better

    3.  Our Girl Scouts sold less boxes of cookies at the school fair than they sold at the mall.                   

    A.  NO CHANGE                B. fewer boxes                             C. lesser boxes                     D. fewest boxes

    SAT Verbal - Answers to Questions  from Week 10 (April 10, 2021)

    1. A
    2. C
    3. B

    SAT Math - Questions  from Week 10 (April 10, 2021)

    Statistical Sampling: Sample vs Population

    The SAT will usually include two or three questions where you are asked to interpret the results of a sample based on a research study. For example, you might want to know what percent of the registered voters in Greensboro prefer a specific candidate. It would be expensive and time consuming to contact every registered voter in Greensboro to get their preference. Instead, pollsters would contact a sample of the voters and determine their preferences, and then use this figure as an approximation for the entire Greensboro population. However, to be able to draw conclusions about the population from the sample results, the sample must be randomly selected. To repeat, in order to generalize your findings to the entire population, the sample must be randomly selected from that population.

    1. The members of the Greensboro City Council wanted to assess the opinions of all city residents about converting an open field into a dog park. The council surveyed a sample of 500 city residents who owned dogs. The survey showed that the majority of those surveyed were in favor of the dog park. Which of the following is true about the city council’s survey?

      1. It shows that the majority of city residents are in favor of the dog park.
      2. The survey sample should have included more residents who are dog owners.
      3. The survey sample should have consisted entirely of residents who do not own dogs.
      4. The survey sample is biased because it is not representative of all city residents.

    2. A study was done on the weights of different types of fish in a pond. A random sample of fish were caught and marked in order to ensure that none were weighed more than once. The sample contained 150 largemouth bass, of which 30% weighed more than 2 pounds. Which of the following conclusions is best supported by the sample data?

      1. The majority of all fish in the pond weigh less than 2 pounds.
      2. The average weight of all fish in the pond is approximately 2 pounds.
      3. Approximately 30% of all fish in the pond weigh more than 2 pounds.
      4. Approximately 30% of all largemouth bass in the pond weigh more than 2 pounds.

    3. A market researcher selected 200 people at random from a group of people who indicated that they liked a certain book. The 200 people were shown a movie based on the book and then asked whether they liked or disliked the movie. Of those surveyed, 95% said they disliked the movie. Which of the following inferences can appropriately be drawn from this survey result?
      1. At least 95% of people who go to see movies will dislike this movie.
      2. At least 95% of people who read books will dislike this movie.
      3. Most people who like this book will dislike this movie.
      4. Most people who dislike this book will like this movie.

    SAT Math - Answers to Questions  from Week 10 (April 10, 2021)

    1. The sample was not randomly selected; it consisted only of residents who owned dogs. Thus the sample is biased and the results cannot be generalized to the entire population of Greensboro. The changes suggested in B and C would not make the sample a random sample. The correct choice is D.

    2. What is the population here? It is not all of the fish in the pond. The population in this problem consists of all of the largemouth bass in the pond, and the sample consists of 150 of the largemouth bass. Thus, any conclusions drawn from the sample relate only to the population of largemouth bass, not to the population of all of the fish in the pond. The answer is D.

    3. What is the population here? It is the group of people who indicated that they liked a certain book, and a random sample of 200 people was taken from that population. The population in this problem was not people who go to see movies, and it is not people who read books. Thus A and B are incorrect. There is no information about people who dislike the book; thus D is incorrect. Since the sample was randomly selected from the group of people who indicated that they liked the book, we can conclude that most people who like this book will dislike this movie. Thus answer choice C is correct.


    March Lessons


    SAT Verbal - Questions  from Week 9 (March 27, 2021)

    SAT QUICK CHALLENGE
    Exercise I21
    Gender Specific and Gender Neutral Pronouns

    Pronouns for "Gender Specific" and "Gender Neutral" Nouns. A noun that identifies a male (for example, "boy") can be replaced with a corresponding gender specific pronoun such as "he" or "him." Likewise, a gender specific noun that identifies a female (for example, "girl") can be replaced with a corresponding gender specific pronoun such as "she" or "her." Be sure to replace a singular noun with a singular pronoun or pronoun phrase. Also, remember that if the noun that is being replaced has the word "each" or "every" in front of it, that noun is singular and must be replaced with a singular pronoun or pronoun phrase.

    To replace a noun which does not indicate gender, you must use a gender neutral pronoun or pronoun phrase. Hence, you can replace the noun "leader" with a pronoun phrase such as "he or she" or "him or her." Plural pronouns do not indicate gender, so you must replace plural nouns, regardless to the gender they indicate, with gender neutral pronouns such as "they" or "them."

    Using "One" and "You" Correctly. Whether a noun being replaced names a male or female, you can use the gender neutral pronoun "one" or "you" to replace that noun. However, do not mix or match those two pronouns (1) with each other or (2) with any other pronouns within either the same sentence or the same paragraph. Instead, maintain consistency by using the same pronoun you began with. Note the ERROR below, its explanation, and the two ways shown to correct it.

    ERROR:  Doctors say that to avoid catching COVID-19, you should wear a mask, wash your hands at least 20 seconds, and wait at least six feet away from other people.  One should also get one's COVID-19 vaccination as soon as possible.  
    EXPLANATION:  You must not use in the same sentence or paragraph both "you" and "one."  Note the following two ways of correcting that error.
    POSSIBLE CORRECTION 1:  Doctors say that to avoid catching COVID-19, one should wear a mask, wash one's hands at least 20 seconds, wait at least six feet away from other people, and get the COVID-19 vaccination as soon as possible. 
    POSSIBLE CORRECTION 2:  Doctors say that to avoid catching COVID-19, you should wear a mask, wash your hands at least 20 seconds, wait at least six feet away from other people, and get the COVID-19 vaccination as soon as possible.  

    Keeping in mind the information above, complete Exercise I21 below.

    Exercise I21 -- Using Gender Specific and Gender Neutral Pronouns Correctly

    Directions. On the line that follows each statement below, place the letter of the answer choice that corrects any error in the statement. If there is no error, mark choice "A" as your answer. When you have finished the exercise, use the answer key in the "SAT Verbal -  Answers to Questions from Week 9" dropdown below to check your work.

    1. Each coach can get Activity Kits for his or her team from the Special Activities Desk in the lobby.             

    A.  NO CHANGE                B. their                                       C. its                          D. his

    2.  They can get a 15% discount on the entire cost of the event if you register before the "Early Bird" deadline of May 27..            

    A.  NO CHANGE                B. Him or her                            C. One                       D. You

    3.  A guest who would like to arrive early and get settled before the 10 am practice time will be glad that you can check in at 8 am.           

    A.  NO CHANGE                B. they                                      C. he or she                D. it

    SAT Verbal - Answers to Questions  from Week 9 (March 27, 2021)

    1. A
    2. D
    3. C

    SAT Math - Questions  from Week 9 (March 27, 2021)

    Standard Deviation

    The standard deviation is a measure of how far the data set values are from the mean.
    The standard deviation is low if most of the values are near the mean and close together.
    The standard deviation is high if most of the values are spread out over the range of values.

    The SAT will not require you to calculate the standard deviation, but you must be familiar with the concept.

    1. The table below gives the distribution of high temperatures in degrees Fahrenheit (°F) for City A and City B over the same 21 days in March.

      City A City B
      Temperature (°F) Frequency   Temperature (°F) Frequency
      80 3   80 6
      79 14   79 3
      78 2   78 2
      77 1   77 4
      76 1   76 6

      Which of the following is true about the data shown for these 21 days? 

      1. The standard deviation of temperatures in City A is larger.
      2. The standard deviation of temperatures in City B is larger.
      3. The standard deviation of temperatures in City A is the same as that of City B.
      4. The standard deviation of temperatures in City A is 0 and the standard deviation of temperatures in City B is negative.

    2. The table below shows two lists of numbers.

      List A 1 2 3 4 5 6
      List B 2 3 3 4 4 5

      Which of the following is a true statement comparing List A and List B? 

      1. The means are the same and the standard deviations are different.
      2. The means are the same and the standard deviations are the same.
      3. The means are different and the standard deviations are different.
      4. The means are different and the standard deviations are the same.

    3. Martin Zimmer, the star basketball player at Spingarn High School, is being evaluated by the University of Maryland for a basketball scholarship. The points he scored per game in the 19 games he played in his senior year ranged from a low of 12 points to a high of 26 points. The mean, median, and standard deviation were calculated for these points. If the lowest number of points scored and the highest number of points scored were removed from the data set, and the mean, median, and standard deviation were recalculated, which of the following is true with regard to the recalculated figures?
      1. The median is the same and the standard deviation is higher.
      2. The median is the same and the standard deviation is smaller,
      3. The median is different and the standard deviation is higher.
      4. The median is different and the standard deviation is smaller.

    SAT Math - Answers to Questions  from Week 9 (March 27, 2021)

    1. The data for City B are more spread out than the data for City A, indicating a higher standard deviation.
      ------------> B

      Note: the standard deviation will never be negative. The standard deviation = 0 only if all of the numbers in the data set are exactly the same.

    2. The mean is the same for List A and List B; it is 3.5. List B contains numbers that are closer to
      the mean than are the numbers is List A. The numbers in List A are more spread out than the
      numbers in List B. Thus the standard deviations are different.
      ---------> A

      (Generally, the wider the range, the greater is the standard deviation.)

    3. The median is the same; removing the smallest and largest values will not change the middle
      number, which is the median. The standard deviation will be lower since numbers that are the
      greatest distance from the mean are removed.
      -----------> B

    SAT Verbal - Questions  from Week 8 (March 20, 2021)

    SAT QUICK CHALLENGE
    Exercise H21
    Unclear Pronoun Use

    Missing, Ambiguous, or Unclear Antecedent. An antecedent is the noun (or noun phrase or noun clause) that a pronoun replaces. When that noun is not provided, a reader may not understand clearly the message the writer is trying to get across. Note ERROR 1 below.

    ERROR 1. The new 4-Star Video System can record and save what is happening in a baby's room much longer than any other system currently being sold. They can watch the video up to six months after it was recorded.
    EXPLANATION. The message is unclear because the antecedent for the pronoun "they" was not given, and we do not know who can watch the video up to six months later. Is it police officials who are trying to find out about an incident that happened in that room? Is it family members who want to find out how the baby got our of her bed the night before? We do not know the answer.
    POSSIBLE CORRECTION: "The new 4-Star Video System can record and save for parents and other observers" what is happening in a baby's room much longer than any other system now being sold. They can watch the video up to six months after it was recorded." The added antecedent lets us know that the underlined pronoun "they" is talking about "parents and other observers."

    Incorrect Use of "This" and "That." The pronouns "this" and "that," as well as their plural counterparts, "these" and "those," should be followed by a noun to indicate what those pronouns are referring to. Otherwise, the message will be ambiguous or unclear. Note ERROR 2 below.


    ERROR 2. Health officials say that we should all fight COVID-19 by wearing masks, washing our hands properly, staying at least six feet away from other people, and getting vaccinated as soon as we can. These can help us stay safe and healthy.
    EXPLANATION. These what? The sentence should say what the pronoun "these" refers to.
    POSSIBLE CORRECTION: "Health officials say that we should all fight COVID-19 by wearing masks, washing our hands properly, and staying at least six feet away from other people. These precautions can help us stay safe and healthy." Placing the noun precautions right after the pronoun "these" tells us that the pronoun refers to the precautions.

    Now complete Exercise H21 below.

    Exercise H21 -- Unclear Pronoun Use

    Directions. On the line that follows each statement below, place the letter of the answer choice that corrects any error in that statement. If there is no error, mark choice "A" as your answer. When you have finished the exercise, use the answer key in the "SAT Verbal -  Answers to Questions from Week 8" dropdown below to check your work.

    1. According to Mrs. Eva Harper, whether it's laying out school clothes, preparing breakfast, checking
    homework, or getting dinner, a mother's work is never finished. Mrs. Harper adds that no matter how
    well moms do those tasks each day, they will just have to do more of them again the next day.             

    A.  NO CHANGE                B. theirs               C. they're tasks               D. it

    2.  Mr. Ed Murray says that he must turn the temperature in the house down to 73 degrees at night, or the baby's
    crying will keep waking people up. This is the only thing that will help her sleep until morning.            

    A.  NO CHANGE                B. This adjustment is               C. Theirs are               D. It is

    3.  Parents must now accompany their children who are under 18 whenever those children are at the mall. They are
    enforced at all times.           

    A.  NO CHANGE                B. That new policy is               C. There it's               D. It is

    SAT Verbal -  Answers to Questions from Week 8 (March 20, 2021)

    SAT QUICK CHALLENGE H21 ANSWER KEY
    Unclear Pronoun Use

    1. A
    2. B
    3. B

    SAT Math - Questions from Week 8 (March 20, 2021 )

    Averages
    On the SAT, the use of the word average usually refers to the mean and is indicated by "average (arithmetic mean)."
    1. The key to solving any problem involving an average (mean) is to find the total of the items before you do anything else.
    2. There are two ways to find the total:
      1) Add all of the items
      2) Multiply the average by the total number of items. For many SAT problems we must use this second method.

    1. On a given day there are 12 trains to City X with an average (arithmetic mean) of 1,400 commuters per train. If the number of trains was cut to 7 and the total number of commuters remained the same, there would be an average of how many more commuters per train?
      A) 800          B) 1,000          C) 1,600          D) 2,400

    2. Peter’s average (arithmetic mean) score on the first three of four tests is 85. If Peter wants to raise his average by 2 points, what score must he earn on the fourth test?

    3. The average (arithmetic mean) of 6 positive numbers is 5. If the average of the least and greatest of these numbers is 7, what is the average of the other four numbers?

      A) 3          B) 4          C) 5          D) 6

    SAT Math - Answers to Questions from Week 8 (March 20, 2021 )

    1. The first thing to do is to get the total. The total number of commuters = 1,400 x 12 = 16,800 . Since the number of trains is cut to 7, we divide this total of 16,800 by 7: 16,800/7 = 2,400.
      Many students make the mistake of choosing D here since we got 2,400. But the question asks: how many more commuters per train?

      The answer is:        2,400
                                    -1,400                               
                                      -----
                                      1,000        The answer is (B)    1,000

    2. Total number of points for the 3 tests = 85 x 3 = 255
      Total number of points for 4 tests if the average is 87 = 87 x 4 = 348
      The difference between the two totals is what he needs on test number 4 = 348 – 255 = 93.

    3. There are three totals in this problem: (1) the sum of the six numbers, (2) the sum of the largest and smallest numbers, and (3) the sum of the other four numbers

      The total of the six numbers = 6 x 5 = 30

      The total of the largest and smallest numbers = 7 x 2 = 14

      The total of the other four numbers = 30 – 14 = 16

      The average of the other four numbers = 16/4 = 4 ---------


    SAT Verbal - Questions  from Week 7 (March 13, 2021)

    SAT QUICK CHALLENGE
    Exercise G21 -- The Pronoun and the Apostrophe

    What is the difference between a/ noun and a pronoun? Well, there are many, but this lesson focuses on one -- the use of the apostrophe. We add an apostrophe and the letter "s" to a noun to show ownership. For example, we write Carla's dog or Ron's pigeon, or even the children's school.

    The Pronoun and the Apostrophe. We never add an apostrophe and the letter "s" to a pronoun to show ownership. Instead, we use special pronoun ownership words. For Carla, we would write "her" dog; for Ron, we would write "his" pigeon; and for the children, we would write "their" school.

    Sometimes, we do use the apostrophe and the letter "s" for pronouns -- but only to make contractions.
    You will recall that a contraction is a ''short cut;" it combines two words to make one word. Examples of contractions include "I'm" (I am), "You're" (You are), etc. Remembering the difference between how apostrophes are used for nouns and how they are used for pronouns will help you earn points each time an SAT question requires you to show that you know the difference.

    Study the possessive pronouns, subject pronouns, and pronoun contractions in the chart below. Then, complete Exercise G21 (beneath the chart), and use the Answer Key in the section below to check your work.

    SUBJECT PRONOUNS POSSESSIVE PRONOUNS PRONOUN CONTRACTIONS
    (with the verb "to be")
    Singular Plural Singular Plural Singular Plural
    I We   My Our I'm (I am) We're (We are)
    You You Your Your You're (You are) You're (You are)
    He, She, It They His, Her, Its Their He's, She's, It's
    (He is, She is, It is)
    They're (They are)



    Now, complete Exercise G21 below.

    Exercise G21 -- The Pronoun and the Apostrophe

    Directions. On the line that follows each statement below, place the letter of the answer choice that corrects any error in that statement. If there is no error, mark choice "A" as your answer. When you have finished the exercise, use the answer key in the section below to check your work.

    1. Your going to be late for school if you don't hurry up.             

    A.  NO CHANGE
    B.  You'll going to be late
    C.  You've going to be late
    D.  You're going to be late

    2.  Its not wise to ignore the CDC guidelines for avoiding COVID-19 infections.           

    A.  NO CHANGE
    B.  It's not wise to ignore
    C.  Its' not wise to ignore
    D.  It not wise to ignore

    3.  Our dance students will win the trophy because they're such excellent dancers.           

    A.  NO CHANGE
    B.  there such excellent dancers
    C.  their such excellent dancers
    D.  theirs such excellent dancers

     

    SAT Verbal -  Answers to Questions from Week 7 (March 13, 2021)

    SAT QUICK CHALLENGE G21 ANSWER KEY
    The Pronoun and the Apostrophe

    1. D
    2. B
    3. A

    SAT Math - Questions from Week 7 (March 13, 2021 )

    Averages and Range
    There are three averages: mean, median, and mode.
    1. Mean; the total of the items divided by the number of items
    2. Median: the number that is exactly in the middle of a group of numbers when the numbers are arranged from smallest to largest,
    3. Mode: the number that appears most often
    4. The range is the difference between the largest number and the smallest number
    5. On the SAT the use of the word average usually refers to the mean and is indicated by “average (arithmetic mean).”
    6. The key to solving any problem involving an average (mean) is to find the total of the items before you do anything else.
    7. There are two ways to find the total:
      1) Add all of the items
      2) Multiply the average by the total number of items. For many SAT problems we must use this second method.

    Now answer the following questions.

    1. The average of four numbers is 5. If three of the four numbers are 3, 4, and 5, what is the fourth number?

    2. In a class of 27 students, the average (arithmetic mean) score of boys on the final exam was 83. If the average score of the 15 girls in the class was 92, what was the average for the whole class?
      A) 85          B) 86          C) 88          D) 90

    3. Mary’s average salary for her first 6 years of work was $30,000; her average salary for the next two years was $32,000. What was her average salary over the entire 8 years?
      A) $30,900          B) $30,500          C) $31,200          D) $31,700

    SAT Math - Answers to Questions from Week 7 (March 13, 2021 )

    1. The first thing to do is to get the total.
      The total = 5 x 4 = 20.
      The four numbers must total 20.
      3 + 4 + 5 = 12; to make 20, the fourth number must be 8.

    2. Girls:   92 x 15  =  1380        total for girls
      Boys:   83 x 12  =    996        total for boys                               
                                      ----
                                    2376        total for the entire class

      Average for the entire class = 2376/27 = 88

    3. 30,000 x 6 = 180,000                total for the first 6 years
      32,000 x 2 =   64,000                total for the next 2 years
                            --------
                            244,000               total for the entire 8 years
      Average salary over the entire 8 years = 244,000/8 = 30,500.

    SAT Verbal - Questions  from Week 6 (March 6, 2021)

    SAT QUICK CHALLENGE
    Exercise F21 -- Noun Pronoun Agreement Errors

    A pronoun is a general word that takes the place of a noun -- often, its antecedent. The antecedent is the specific noun that the pronoun is replacing, The pronoun and its antecedent will be somewhere near each other -- frequently, in the same sentence. Often, the antecedent comes first.

    The antecedent and the pronoun replacing it must match. Therefore, a female antecedent (Loretta or girl) requires a female replacement pronoun (she or her), and a male antecedent (Matt or boy) requires a male replacement pronoun (he or him). A singular antecedent requires a singular replacement pronoun (as in Matt or boy), and a plural antecedent (as in Loretta and Matt or the students) requires a plural replacement pronoun (as in they or them). Did you notice that the plural replacement pronouns they and them do not indicate gender? Now, note Errors 1 and 2 below.

    ERROR 1:           Human resources workers interview potential employees, asking him or her questions that can indicate whether those job candidates are likely to help the business achieve important company goals.
    PROBLEM:          The plural antecedent "potential employees" and the plural, synonymous follow-up noun phrase "job candidates" indicate that the missing pronoun (or pronoun phrase) must also be plural. However, the pronoun phrase "him or her" is singular.
    CORRECTION:     Human resources workers interview potential employees, asking them questions that can indicate whether those job candidates are likely to help the business achieve important company goals.
    Now, the antecedent ("potential employees"), the needed pronoun ("them"), and the synonymous follow-up noun phrase ("job candidates") are all plural as they should be.

    ERROR 2:        Because of their keen sense of smell, a dog can be trained to locate bombs and other dangerous objects.
    PROBLEM:       Dog, the subject, is singular, but the pronoun antecedent, their, is plural. Since both the subject and its antecedent must match, both must be the same -- both singular OR both plural.
    CORRECTION:  Because of its keen sense of smell, a dog can be trained to locate bombs and other dangerous objects.  
    Now, the the subject (dog) and the pronoun antecedent (its) are both singular.


    Now, complete Exercise F21 below.

    Exercise F21 -- Noun / Pronoun Agreement Errors

    Directions. Place on the line that follows each statement below the letter of the answer choice that corrects the error in the statement. If there is no error, mark choice "A" as your answer. Use the answer key in the section below to check your work.

    1. While cucumbers are generally thought of as a vegetable, it is actually a fruit.             

    A.  NO CHANGE
    B.  they are really fruit.
    C.  fruit is how it is classified.
    D.  actually, it is fruit.

    2.  When an unknown singer suddenly comes out with a big hit, they can get popular very fast.           

    A.  NO CHANGE
    B.  its song can rise to number one quickly.
    C.  one's song can rapidly become number one.
    D.  he or she can quickly become very popular.

    3.  Thanks to his team's superior practices, they threw three Super Bowl touchdown passes          

    A.  NO CHANGE
    B.  one threw three Super Bowl touchdown passes
    C.  Brady threw three Super Bowl touchdown passes
    D.  he or she threw three Super Bowl touchdown passes

     

    SAT Verbal -  Answers to Questions from Week 6 (March 6, 2021)

    SAT QUICK CHALLENGE F21 ANSWER KEY
    Noun Pronoun Agreement Errors

    1. B
    2. D
    3. C

    SAT Math - Questions from Week 6 (March 6, 2021 )

    Solving Equations by Cross Multiplying - Continued

    1. We cross multiply when both sides of an equation are fractions.
    2. Multiply the denominator of the left side of the equation by the numerator of the right side of the equation, and multiply the numerator of the left side of the equation by the denominator of the right side of the equation.
    3. Set the two items equal to each other, and solve the resulting equation.

    Now solve the following equations.

    1.     Assume that the following is true.

                1                     4                    5              1    
               ---  (x)   +      --- (x)     =      ---     -     ---
                3                     9                    9              6

        What is the value of x?       

          

    2.     Assume that the following three items are true.

      1. x ≠ 0
      2. x = y
      3.   3a                 9b
        ------    =    ------
            x                   y

               What is the value of a in terms of b?  
                   

      A)  b / 3
      B) b
      C) 3b
      D) 6b
    3.  Assume that the following two items are true.

      1. x and y are positive integers
      2.                     1            7
             x     -     ---    =   --- 
                            y            2

               What is the value of x?  
                   

      A) 3
      B) 4
      C) 6
      D) 7

    SAT Math - Answers to Questions from Week 6 (March 6, 2021 )

    1.     Assume that the following is true.

                1                     4                    5              1    
               ---  (x)   +      --- (x)     =      ---     -     ---
                3                     9                    9              6

        What is the value of x?     

                1                     4                    5              1    
               ---  (x)   +      --- (x)     =      ---     -     ---
                3                     9                    9              6  

                 The least common denominator for the left side of the equation is 9.



                  3                     4                       10              3    
                ----  (x)   +     ---- (x)     =      ----     -     ----
                  9                     9                       18             18
        
                 The least common denominator for the right side of the equation is 18.
                 

                  7                     7    
                ----  (x)   =     ----
                  9                     18

                63x = 126
                     x = 0.5

    2.     Assume that the following three items are true.

      1. x ≠ 0
      2. x = y
      3.   3a                 9b
        ------    =    ------
            x                   y

               What is the value of a in terms of b?  

               Since x = y, substitute x for y, then solve for a.

               3a           9b
              ----     =    ---
                x               x
             
               3ax = 9bx
               3a = 9b
                  a = 3b
                 
               --------->  

    3.  Assume that the following two items are true.

      1. x and y are positive integers
      2.                     1            7
             x     -     ---    =   --- 
                            y            2

               What is the value of x?  

               This problem might appear to be unsolvable since there are two unknowns and only one equation. But read the question                   carefully and proceed to solve for x.
       
                                1             7
                 x     -     ---     =    ---
                                y              2

                                7             1
                 x     =     ---     +    ---
                                2              y

                                      1               1
                 x     =   (3)  ---     +     ---
                                        2               y                  

                 x is a positive integer; from the answer choices we know that x = 3 or 4 or 6 or 7.         

                                      1              
                 x     =   (3)  ---     +     some other value that will make the total = 3 or 4 or 6 or 7.
                                      2               

                                                                   1      
                 What can we add to (3)  ---    so that we will have an integer?
                                                                   2              

                                                                                   1              
                 The other value that we add is   ---   and y = 2 since y is a positive integer.
                                                                                    2           

                 Thus x = 4 ---------> B


    February Lessons


    SAT Verbal - Questions from Week 5  (February 27, 2021)

    SAT QUICK CHALLENGE
    Exercise E21 -- Noun Pronoun Agreement Errors

    Effective writers use synonymous words or phrases to rename or refer to words that have already been used in a sentence or passage. Even so, making comparisons, a singular noun must be replaced by a singular noun/noun phrase, and a plural noun must be replaced by a plural noun/noun phrase. Note Errors 1 and 2 below.

    ERROR 1:           The little boys in Mrs. Brown's class want to be a basketball player when they grow up.
    PROBLEM:          The plural noun "boys" has been replaced with the singular noun phrase "a basketball player."
    CORRECTION:     The little boys in Mrs. Brown's class want to be basketball players when they grow up.  Now, the noun "boys" and the replacement noun phrase "basketball players" are both plural.

    ERROR 2:         Tina and Kenny will be a star in the school's spring play.
    PROBLEM:        The plural, compound subject names two people -- Tina and Kenny.  Yet, a singular noun phrase -- "a star" -- is used to replace that plural compound subject. 
    CORRECTION:  Tina and Kenny will be stars in the school's spring play.  Now, the plural compound subject, "Tina and Kenny," and the replacement noun, "stars," are both plural.



    Now, complete Exercise E21 below.

    Directions. Place on the line that follows each statement below the letter of the answer choice that corrects any error in the statement. If there is no error, mark choice "A" as your answer. Use the answer key at the bottom of the page to check your work.

    1. New attendance rules for football games say that fans no longer have to be limited to just the family and a few close friends of the players.            

    A.  NO CHANGE
    B.  just the families
    C.  just the family's
    D.  just his family

    2.  People say that the star of the new horror film is crooked thieves who cannot be trusted.           

    A.  NO CHANGE
    B.  crooks and thieves
    C.  crooks and a thief
    D.  a crooked thief

    3.  The honor students received a special pass that lets them get into talent shows free.           

    A.  NO CHANGE
    B.  an honor pass
    C.  a special honor pass
    D.  special passes

     

    SAT Verbal -  Answers to Questions from Week 5 (February 27, 2021)

    SAT QUICK CHALLENGE E21 ANSWER KEY
    Noun Pronoun Agreement Errors

    1. B
    2. D
    3. D

    SAT Math - Questions from Week 5 (February 27, 2021 )

    Solving Equations by Cross Multiplying    

    1. We cross multiply when both sides of an equation are fractions.
    2. Multiply the denominator of the left side of the equation by the numerator of the right side of the equation, and multiply the numerator of the left side of the equation by the denominator of the right side of the equation.
    3. Set the two items equal to each other, and solve the resulting equation.

    Here is an example:

      6                  3
    -----     =     -----
      x                  10

    3x = 60
    x = 20

    Now solve the following equations.

    1.     5                    2x + 7
      -------     =        --------
          2                        3

      A) 0.25
      B) 1.00
      C) 2.5
      D) 3

    2.     Assume that the following is true.
         
          3x + 2y                   17
          ---------     =        --------
                y                         4

      Compute the value of the following:

              x
           ------     
              y       
          

    3.     Assume that the following is true.
         
          p + q + r                p + q
          -----------     =      ----------
                  3                          2

      What is r?
               
      A) q + p
      B) 2p + 2q
      C) (1 / 2) * (p + q)
      D) 1

    SAT Math -  Answers to Questions from Week 5 (February 27, 2021)

    1.     5                    2x + 7
      -------     =        --------
          2                        3

      2(2x + 7) = 15
      4x + 14 = 15
      4x = 1
      x = 1/4 = 0.25
      -----> A
    2.     3x + 2y                   17
          ---------     =        --------
                y                         4

        17y = 4(3x + 2y)
        17y = 12x + 8y
          9y = 12x

              x                 9              3
           ------    =    -----   =   -----     
              y                12             4

    3.      p + q + r                p + q
          -----------     =      ----------
                  3                          2

         2(p + q + r) = 3(p + q)
         2p + 2q + 2r = 3p + 3q
         2r = p + q
               

                        p + q                       
          r  =       --------     =      (1/2) (p + q)
                            2                          

         -----> C

    SAT Math - Questions from Week 3 (February 13, 2021)

    Solving Linear Equations Continued
    1. Solve for x (or for any variable) by isolating x on one side of the equal sign and everything else on the other side.
    2. Always remember to do the same thing to both sides of the equation: add, subtract, multiply, divide, raise to a power, or take the square root

    1. If z <> 0, x = 4 / z , and yz = 8, then x / y =
      1. 0.5
      2. 1
      3. 2
      4. 16
    2. If r <> -s and a = (r-s) / (r+s) , then a + 1 =
      1. (2r) / (r +s )
      2. (r - s + 1) / (r + s)
      3. (2r - 2s) / (r + s)
      4. (2s) / (r + s)
    3. D = (3+k) / 45               Solve for k in terms of D.
      1. k = 45D - 3
      2. k = 45D + 3
      3. k = 45(D – 3)
      4. k = 45(D + 3)
    4. d = tc + 4                       Solve for t in terms of c and d.
      1. (d - 4) / c
      2. (c - 4) / d
      3. (4 - d) / c
      4. (4 - c) / d
    5. E = (L + 4M + P) / 6              Solve for P in terms of E, L, and M.
      1. P = 6E – L – 4M
      2. P = -6E + L + 4M
      3. P = (L + 4M + E) / 6
      4. P = (L + 4M -E) / 6
    6. If 2 / (a - 1) = 4 / y , and y <> 0 where a <> 1, what is y in terms of a?
      1. y = 2a – 2
      2. y = 2a – 4
      3. y = 2a - (1 / 2)
      4. y = (1 / 2)(a) + 1
    7. h = -16t2 + vt + k Solve for v in terms of h, t, and k.
      1. v = h + k – 16t
      2. v = (h - k + 16) / t
      3. v = (h + k) / t – 16t
      4. v = (h - k)/ t + 16t

    SAT Math - Answers to Questions from Week 3 (February 13, 2021)


    1. If z <> 0, x = 4 / z , and yz = 8, then x / y =
      This problem can be solved in one of three ways.

      Alternative one:

      We are given yz; put it in the denominator. Then get xz and put it the numerator; we have yz = 8 and xz = 4: 
      (xz) / (yz) = 4/8. Thus, x/y = 4/8 = 0.5 -----> A

      Alternative two:
      We are given x (x = 4 / z); solve for y and then get (x / y) . (yz) = 8, then y = 8 / z
      Thus (x / y) = (4 / z) / (8 / z) = (4 / z) . (z / 8) = 4 / 8 = 0.5 --------> A

      Alternative three:
      Pick numbers and substitute. This method can be used when there are answer choices. Pick a number for z. Let x = 2; then solve for x and y.
      x = 4 / z                                        yz = 8
      x = 4 / 2 = 2                                  y(2) = 8
                                                           y = 4

      Thus x / y = 2 / 4 = 0.5 --------------> A

    2. Express 1 as (r+s) / (r+s) and combine the terms.

      a + 1 = r-sr+s + r+sr+s = r-s+r+sr+s = 2rr+s -----------> A

      Then c = (1 / x) + (1 / y) = (1 / 6) + (1 / 3) = (1 / 6) + (2 / 6) = (3 / 6) = (1 / 2)
      Then (1 / c) = (2 / 1) = 2
      Now, using our numbers, which answer choice = 2?

      1. x + y = 6 + 3 = 9                                                        NO
      2. x - y = 6 - 3 = 3                                                         NO
      3. (x + y) / xy = (6 + 3) / 6(3) = 9 / 18 = 1 / 2                 NO
      4. (xy) / (x + y) = 6(3) / (6 + 3) = 18 / 9 = 2                    YES  ----------------> D
    3. D = (3+k) / 45               Solve for k in terms of D.
      1. k = 45D - 3
      2. k = 45D + 3
      3. k = 45(D – 3)
      4. k = 45(D + 3)
    4. d = tc + 4                       Solve for t in terms of c and d.
      d - 4 = tc
      (d - 4) / c = t ------------------> A

    5. E = (L + 4M + P) / 6              Solve for P in terms of E, L, and M./
      6E = L + 4M + P
      6E - L - 4M = P   ----------------> A

    6. If 2 / (a - 1) = 4 / y , and y <> 0 where a <> 1, what is y in terms of a?
      2y = 4(a - 1)
      y = 2(a - 1) = 2a - 2 -------------> A

    7. h = -16t2 + vt + k Solve for v in terms of h, t, and k.
      h + 16t2 -k = vt
      v = (h + 16t2 – k) / t = h / t + (16t/ t) - (k / t) = (h / t) + 16t - (k / t) = (h-k) / t + 16t  -------------> D

    SAT Verbal - Questions from Week 3 (February 13, 2021)

    SAT QUICK CHALLENGE C21
    Faulty Comparisons

    Using "That of" and "Those of." Both "that of" and "those of" (1) show ownership and (2) help you avoid being repetitive when making comparisons. For example, to compare the success of two singers, you could say, "Singer A's annual income is higher than the annual income of Singer B." However, you can make that point more concisely by saying, "Singer A's annual income is higher than that of Singer B." When you are talking about plural nouns, use the pronoun "those" in place of "that." Hence, you could say, "Singer A's annual sales are higher than those of Singer B." 

    Comparing the Same Kinds of Things. When making comparisons, you must compare the same kinds of things: living things with living things, cars with cars, sports with sports, etc. Some SAT questions ask you to correct comparisons which do not compare the same kinds of things. Note the following faulty comparison error, along with an explanation of the problem and various ways of correcting it.

    ERROR: William Shakespeare's poetry is much less popular with teenagers than *Amanda Gorman.
    EXPLANATION: This statement compares a thing (William Shakespeare's poetry) with a person (poet Amanda Gorman). Note the possible corrections below.

    Correction 1:
    William Shakespeare's poetry is much less popular with teenagers than the poetry of Amanda Gorman. (Compares poetry with poetry)

    Correction 2:
    William Shakespeare's poetry is much less popular with teenagers than that of Amanda Gorman. (Compares poetry with poetry)

    Correction 3:
    William Shakespeare's poetry
    is much less popular with teenagers than Amanda Gorman's. (Compares poetry with poetry; repeating "poetry" after her name is optional)

    Correction 4:
    William Shakespeare's poems
    are much less popular with teenagers than those of Amanda Gorman. (Compares poems with poems)

    *NOTE:
    Amanda Gorman became the Nation's first Youth Poet Laureate, and the first African- American Youth Poet Laureate, in 2017. At age 22, the Harvard honor graduate became America's youngest inaugural poet when she read her own original poem, "The Hill We Climb," at President Joe Biden's inauguration in January of 2021. On Sunday, February 7, 2021, she became the first person to read a poem at the Super Bowl when she read her poem "Chorus of the Captains."

    Practice Exercise C21 -- Faulty Comparisons

    Directions. Keeping in mind the information above, read the questions below, and place on the line after each question the answer choice that corrects the error in the question. If there is no error, mark choice "A." After completing the exercise, use the answer key at the bottom of the page to check your answers.

    1. Our science teacher says that his parrot has intelligence that is similar to a three-year-old. _____
      1. NO CHANGE
      2. that of a three-year-old
      3. those of a three-year-old
      4. the ones from a three-year-old
    2. Coach Hamilton said that Tom Brady's passes are more accurate than Patrick Mahomes. _____
      1. NO CHANGE
      2. more accurate than of Patrick Mahomes
      3. more accurate than from Patrick Mahomes
      4. more accurate than those of Patrick Mahomes

    SAT Verbal - Answers to Questions from Week 3 (February 13, 2021)

    1. B
    2. D

    SAT Math - Questions from Week 2 (February 6, 2021)

    Solving Linear Equations Continued
    Simple equations
    1. Solve for x (or for any variable) by isolating x on one side of the equal sign and everything else on the other side.
    2. Always remember to do the same thing to both sides of the equation: add, subtract, multiply, divide, raise to a power, or take the square root
    1. If c = (1/x + 1/y) and x > y > 0, then which of the following is equal to 1/c ?
      1.  x + y
      2. x - y
      3. (x + y) / xy
      4. xy / (x + y)

    2. If c = 1/x + 1/y , solve for x in terms of c and y.

    3. b = 2.35 + 0.25x
      c = 1.75 + 0.40x

      In the equations above, b and c represent the price per pound, in dollars, of beef and chicken, respectively, x weeks after July 1 during last summer. What was the price per pound of beef when it was equal to the price per pound of chicken?

      1. $2.60
      2. $2.85
      3. $2.95
      4. $3.35
    4. 3(1/2 – y) = 3/5 + 15y
      What is the solution to the equation above?

    5. If (1/2)x + (1/3)y = 4, what is the value of 3x + 2y?

    SAT Math - Answers to Questions from Week 2 (February 6, 2021)

    1. If c = (1/x + 1/y) and x > y > 0, then which of the following is equal to 1/c ?
      1.  x + y
      2. x - y
      3. (x + y) / xy
      4. xy / (x + y)




      This problem can be solved in one of two ways.

      Alternative one:

      Find the least common denominator and combine the two terms. The least common denominator is xy.

      c = (1 / x )+ (1 / y)
         = (1 / x) . [ (xy) / (xy) ] + 1y . [ (xy) / (xy) ]
         = [ y / (xy) ] + [ x / (xy) ]
         = [ (y + x) ] / xy

      Then 1 / c = xy / [x + y]
      ------> D


      Alternative two:
      Pick numbers and substitute. This method can b used when there are answer choices. Let x = 6 and y = 3.

      Then c = (1 / x) + (1 / y)
                  = (1 / 6) + (1 / 3)
                  = (1 / 6) + (2 / 6)
                  = (3 / 6) = (1 / 2)

        Then 1 / c = 2 / 1 = 2.
        Now, using our numbers, which answer choice = 2?
      1.  x + y
        = 6 + 3 = 9 NO

      2. x - y
        = 6 - 3 = 3 NO

      3. (x + y) / xy
        = (6 + 3) / (6 * 3) 
        = 9 / 18
        = 1 / 2  NO

      4. xy / (x + y)
        = (6 * 3) / (6 + 3)
        = 18 / 9
        = 2      YES

        ------> D




    2. If c = 1/x + 1/y , solve for x in terms of c and y.
      c - (1 / y) = 1 / x
      (1 / x) = c - (1 / y) = (cy / y) - 1 / y = (cy - 1)  / y
      Then x = y / (cy - 1)

    3. b = 2.35 + 0.25x
      c = 1.75 + 0.40x

      In the equations above, b and c represent the price per pound, in dollars, of beef and chicken, respectively, x weeks after July 1 during last summer. What was the price per pound of beef when it was equal to the price per pound of chicken?

      1. $2.60
      2. $2.85
      3. $2.95
      4. $3.35

      Set the two expressions equal to each other, solve for x, then substitute the value of x into either expression.

      1.75 + 0.40x = 2.35 + 0.25x
      Now solve for x.
      0.15x = 0.6
      x = 0.6 / 0.15 = 4

      Now substitute into either expression.
      b = 2.35 + 0.25(4) = 2.35 + 1.00 = 3.35
      ------> D

      Also c = 1.75 + 0.4(4) = 1.75 + 1.60 = 3.35
      ------> D

    4. 3(1/2 – y) = 3/5 + 15y
      What is the solution to the equation above?

      Solve for y. First, remove the parentheses.
      3/2 – 3y = 3/5 + 15y

      Get rid of the denominators. Multiply both sides by 2.
      3 – 6y = 6/5 + 30y

      Now multiply both sides by 5.
      15 – 30y = 6 + 150y
      (Note: instead of multiplying both sides by 2 and then by 5, we could have initially multiplied both sides by 10.)

      Now, solve for y:
      15 – 30y = 6 + 150y
      9 = 180y
      y = 9/180 = 1/20 = .05

    5. If (1/2)x + (1/3)y = 4, what is the value of 3x + 2y?
      This type of problem frustrates many students taking the SAT. They would like to get a value for x and multiply it by 3, get a value for y and multiply it by 2, and then add the two terms together. From the information given, however, we cannot get a value for x and we cannot get a value for y.

      We can try multiplying both sides by a constant or dividing both sides by a constant. In this case, multiply both sides by 6.
      (6)  * [ (1/2)(x) ] + (6) * [ (1/3)( y) ] = (6) (4)
      3x + 2y - 24     This is our answer. We got it without getting the individual values of x and y.


    SAT Verbal - Questions from Week 2 (February 6, 2021)

    SAT QUICK CHALLENGE B21
    Parallelism and Faulty Word Pairs

    What Is Parallelism? When you list two or more things that have the same level of importance in a sentence, maintain parallelism by using the same format for each of the items. That is, if you use a noun to name one of them, use a noun to name each of the others. The same is true for phrases, clauses, and sentences. Note Sentences 1A and 1B below, along with their explanations.

    Sentence 1A 
    There was no direct flight from Washington, DC to New York City, but driving to New York actually took less time than to fly there.
    Comment:
    This sentence is not parallel because it uses a gerund ("driving") and an infinitive ("to fly") to name the two ways of traveling.

    Sentence 1B
    Because there was no direct flight from Washington, DC to New York City, driving to  New York actually took less time than flying there.
    Comment:
    In this sentence, the two actions compared are both written in the same format -- gerunds.  Therefore, the sentence is parallel.    

    Word Pairs and Comparisons. You can expect some SAT questions to ask about word pairs noted for pointing out similarities and differences. Always use the pairs together; do not mix or match them with any other words. Also, the words or phrases that follow each part of a word pair must be parallel. Note the examples in the chart below.

    WORD PAIRS AND COMPARISONS
    Word Pair Purpose Sample Sentence
    Either...or Tells that only one of two people or things will be involved in something. Either Lisa or her mother will go to the show.
    Neither...nor Excludes two or more people / things Neither Lisa nor her mother will go to the show.
    Not only...but also Points out two different qualities of a person or thing Beyonce is not only a great singer, but also a fantastic dancer.

    Practice Exercise

    Directions. On the line after each question, write the letter of the answer choice which corrects the underlined part of the sentence. If the underlined part is already correct, select choice A (No change). Use the answer key in the dropdown below to check your work.

     

    1.    Vickie's art classes not only helped her to create useful household gadgets, and also learning to design stylish women's fashions.. ____ 

    1. NO CHANGE
    2. and designing
    3. but also taught her to
    4. but additionally teaching to

    2. Most of the children in the class wanted either pizza and hamburgers for lunch. ____

    1. NO CHANGE
    2. or hamburgers
    3. plus hamburgers
    4. with hamburgers

    3. Scattered paper scraps, India's assorted paint spills, and marking pens that Tara had abandoned were all unmistakable signs that the children had made their mom surprise              birthday gifts.

    1. NO CHANGE
    2. assorted paint spills and abandoned marking pens
    3. spilled paint and marking pens
    4. paint spills and abandoned marking pens

    SAT Verbal - Answers to Questions from Week 2 (February 6, 2021)

    1. C
    2. B
    3. B


    January Lessons

    SAT Math - Questions from Week 1 (January 30, 2021)

    Solving Linear Equations
    Simple Equations

    1. Solve for x (or for any variable) by isolating x on one side of the equal sign and everything else on the other side.
    2. Always remember to do the same thing to both sides of the equation: add, subtract, multiply, divide, raise to a power, or take the square root
    1. If x - 6 = 11, what is the value of x?

    2. if a + 1/2 = 4 + 2/3, what is the value of x?

    3. If 2(x  - 40) = 3(x  - 30), what is the value of x?

    4. If (x-1)/3 = k and k = 3, what is the value of x?
      1. 2
      2. 4
      3. 9
      4. 10

    5. If y = kx, where k is a constant, and y = 24 when x = 6, what is the value of y when x = 5?
      1. 6
      2. 15
      3. 20
      4. 23

    6. 1/4 + 1/5 = ?

    7. 2/7 + 3/y = ?

    SAT Math - Answers to Questions from Week 1 (January 30, 2021)

    1. x - 6 = 11
      x = 17

    2. (a + 1/2) = 4 + 2/3
      6a + 3 = 24 + 4
      6a = 24 + 4 - 3 = 25
      a = 25/6 = 4 1/6

    3. If 2(x  - 40) = 3(x  - 30)
      2x - 80 = 3x - 90
      10 = x

    4. If (x-1)/3 = 3
      x - 1 = 9
      x = 10

    5. If y = kx
      24 = k(6)
      k = 4

      y = kx = 4(5) = 20

    We need the least common denominator for numbers 6 and 7.

    1. LCD = 20
      1/4 + 1/5 = 5/20 + 4/20 = 9/20

    2. LCD = 7y
      2/7 + 3/y = [(2y) / (7y)] + [(7)(3)] / [(7)(y)] = (2y + 21) / 7y

    SAT Verbal - Questions from Week 1 (January 30, 2021)

    SAT QUICK CHALLENGE EXERCISE A21
    Parallelism and Sentence Patterns

    Parallelism Review. When you list two or more of the same types of things in a sentence, (whether single words, phrases, clauses, or complete sentences), keep the sentence parallel by listing all the items in the same way. Note Sentences 1 - 3 below.

    Sentence 1: My brother likes playing basketball, but my sister prefers to play soccer.
    Sentence 2: My brother likes playing basketball, but my sister prefers playing soccer.
    Sentence 3: My brother likes to play basketball, but my sister prefers to play soccer.

    Sentence 1 is not parallel because the phrases telling what the brother and sister like are not written in the same way (playing basketball vs. to play soccer). However, sentence 2 is parallel because it has two gerund phrases: playing basketball and playing soccer. Sentence 3 is also parallel because it has two infinitive phrases: to play basketball and to play soccer.

    Parallel Sentence Patterns. Some SAT questions refer to a section of the passage in which all but one of the sentences are parallel, and you are asked to find the answer choice which corrects that sentence. Keeping the information above in mind, read the sentence in the box below, and then write on the line which follows the sentence the letter of the choice which corrects the underlined part of the sentence. (As always, if the underlined part is already correct, write the letter A (NO CHANGE) as your answer.)

    1. The color of a rose is said to have its own meaning. Yellow roses represent friendship. Pink roses suggest admiration. Love is symbolized by red roses. _____

    A. NO CHANGE
    B. Love is what red stands for.
    C. Red roses symbolize love.
    D. Love is suggested by red.

    If you selected choice C, your answer is correct. Now, complete the exercise below.

    Exercise A21 - Parallelism and Sentence Patterns.

    Directions. On the line which follows each question below, write the letter of the choice which completes the underlined part of the question correctly. If the underlined part is already correct, place the letter A (NO CHANGE) on that line. Use the Answer Key in the dropdown below to check your work.

    1.    Pediatricians treat children. Geriatric specialists treat older adults. Without cardiologists, we would not have experts who focus on what puts us at risk for heart disease and heart
           attacks. ____ 

    1. NO CHANGE
    2. Heart disease and heart attack issues are treated by cardiologists.
    3. Fatal heart ailments are what cardiologists keep from happening
    4. Cardiologists treat patients with heart ailments.

    2. My family looks forward to the feasts at our family reunions. Granny bakes her tangy apple cobbler. Aunt Mary makes her luscious lemon ice cream. Uncle Joe's ribs are
        everybody's favorite.

    1. NO CHANGE
    2. Uncle Joe's barbecued ribs melt in your mouth.
    3. Uncle Joe grills his tender, juicy, "melt-in-your mouth" barbecued ribs.
    4. We all love Uncle Joe's barbecued ribs.

    SAT Verbal - Answers to Questions from Week 1 (January 30, 2021)

    1. D
    2. C


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