section 4 - educational testing service · web viewthe answer space is followed by the word...

GRADUATE RECORD EXAMINATIONS®

Official G R E Quantitative ReasoningPractice Questions, Volume 1

Chapter 4 – AlgebraAnswer Key with Answers and Explanations

Copyright© 2014 by Educational Testing Service. All rights reserved. E T S, the E T S logo, GRADUATE RECORD EXAMINATIONS, and G R E are registered trademarks of Educational Testing Service (E T S) in the United States and other countries.

document.doc

AlgebraThis document begins with the answer key for questions found in the Chapter 4 Algebra Practice Questions document. Following the answer key are the complete explanations for each question. If you wish to work through the questions before consulting the answers and explanations, please use the Chapter 4 Algebra Practice Questions document.

Answer KeyQuestion 1.

Answer: Choice A. Quantity A is greater.

Question 2.

Answer: Choice B. Quantity B is greater.

Question 3.

Answer: Choice D. The relationship cannot be determined from the information given.

Question 4.


Question 5.

Answer: Choice C. The two quantities are equal.

Question 6.

Answer: Choice B. Quantity B is greater.

Question 7.


document.doc

Question 8.

Answer: Choice E. four thirds

Question 9.

Answer: Choice D. x y over 8

Question 10.

Answer: Choice B. 4 raised to the negative y power

Question 11.

Answer: Choice D. Seven

Question 12.

Answer: Choice B. 1,000 minus r

Question 13.

Answer: Choice E. 25 r = negative 16 s

Question 14.

The answer to question 14 consists of two of the answer choices.

Choice B. the point with coordinates 5 comma 12

Choice D. the point with coordinates 12 comma 3

document.doc

Question 15.

The answer to question 15 consists of two of the answer choices.

Choice A. the fraction x over 2, end fraction, minus the fraction 3x over 20, end

fraction, +8

Choice B. the fraction 7x over 20, end fraction, +8

Question 16.

In question 16 you were asked to enter an integer or a decimal. The answer to question 16 is 7.2.

Question 17.

In question 17 you were asked to enter an integer or a decimal. The answer to question 17 is 17.

document.doc

Answers and ExplanationsQuestion 1.

It is given that the fraction with numerator x, times, open parenthesis x minus

2 close parenthesis, and denominator, open parenthesis x + 3 close parenthesis, times, open parenthesis x minus 4 close parenthesis squared, = 0.

Quantity A: x

Quantity B: negative 2

A. Quantity A is greater.

B. Quantity B is greater.

C. The two quantities are equal.

D. The relationship cannot be determined from the information given.

From the answer choices given, select and indicate the one that describes the relationship between quantity A and quantity B.

Explanation for Question 1.

To compare x with you should first solve the equation the fraction with

numerator x, times, open parenthesis x minus 2 close parenthesis, and denominator, open parenthesis x + 3 close parenthesis, times, open parenthesis x minus 4 close parenthesis squared, =

0 for x. To solve the equation, recall that a fraction a over b is equal to 0 only if a = 0 and

b is not equal to 0. So the fraction with numerator x, times, open

parenthesis x minus 2 close parenthesis, and denominator, open parenthesis x + 3 close parenthesis, times, open parenthesis x minus 4 close parenthesis squared, is equal to 0 only if

x times, open parenthesis, x minus 2, close parenthesis, is equal to 0 and

open parenthesis, x +3, close parenthesis, times, open parenthesis, x minus 4, close parenthesis,

squared, is not equal to 0. Note that the only values of x for which x times, open

document.doc

parenthesis, x minus 2, = 0, are x = 0 and x = 2, and for these two values, open

parenthesis, x +3, close parenthesis, times, open parenthesis, x minus 4, close parenthesis, squared,

is not equal to 0. Therefore the only values of x for which the fraction with

numerator x, times, open parenthesis x minus 2 close parenthesis, and denominator, open parenthesis x + 3 close parenthesis, times, open parenthesis x minus 4 close parenthesis squared, is equal to 0 are x = 0 and x = 2. Since both 0 and 2 are greater than Quantity B, negative 2, the correct answer is Choice A.

This explanation uses the following strategy.Strategy 8: Search for a Mathematical Relationship

Question 2.

It is given that x is greater than 0.

Quantity A: the fraction 1 over x.

Quantity B: the fraction with numerator x+1 and denominator x squared.






document.doc


Note that Quantity B, the fraction with numerator x + 1 and denominator x squared, can be

expressed as the fraction with numerator x, and denominator x squared + the fraction

with numerator 1 and denominator x squared, which can be simplified to the fraction with

numerator 1 and denominator x + the fraction with numerator 1 and denominator x squared. Note

that Quantity A is the fraction 1 over x, and for all nonzero values of x, the fraction 1

over x squared is greater than 0. It follows that the fraction with numerator 1, and

denominator x + the fraction with numerator 1 and denominator x squared, is greater than the fraction 1 over x; that is, Quantity B is greater than Quantity A. Thus the correct answer is Choice B.


Question 3.

Quantity A: the absolute value of m + 25, end absolute value

Quantity B: 25 minus m






document.doc

Explanation for Question 3.To compare Quantity A the absolute value of m + 25, end absolute value with Quantity B

25 minus m, first note that because the absolute value of any quantity is greater than or

equal to 0, it must be true that the absolute value of m + 25, end absolute value, is

greater than 0, for all values of m. But you know that 25 minus m is less than 0 when m is greater than 25. So Quantity A is greater than Quantity B when m is greater than 25.

On the other hand, both the absolute value of m + 25, end absolute value and 25

minus m are equal to 25 when m = 0. So in one case Quantity A is greater than Quantity B, and in the other case the two quantities are equal. Thus the correct answer is Choice D.

This explanation uses the following strategies.

Strategy 11: Divide into Cases

Strategy 13: Determine Whether a Conclusion Follows from the Information Given

Question 4.

During an experiment, the pressure of a fixed mass of gas increased from 40 pounds per square inch (p.s.i) to 50 p.s.i. Throughout the experiment, the pressure, P p .s.i, and the volume, V cubic inches, of the gas varied in such a way that the value of the product PV was constant.

Quantity A: The volume of the gas when the pressure was 40 p .s.i

Quantity B: 1.2 times the volume of the gas when the pressure was 50 p .s.i






document.doc

Explanation for Question 4.You are given that the relationship between the pressure and volume of the gas throughout the experiment was PV = C, where C is a positive constant.Quantity A is the volume of the gas when the pressure was 40 psi. At this pressure, the equation

PV = C becomes 40V = C. Solving this equation for V, you get V = C over 40 =

0.025C; that is, the volume of the gas was 0.025C cubic inches.Quantity B is 1.2 times the volume of the gas when the pressure was 50 psi. At this pressure, the

equation PV = C becomes 50V = C or V = C over 50 = 0.02C Therefore

1.2V = 1.2 times 0.02C = 0.024C, that is, Quantity B is 0.024C

cubic inches. Thus the correct answer is Choice A.

This explanation uses the following strategies.Strategy 1: Translate from Words to an Arithmetic or Algebraic RepresentationStrategy 8: Search for a Mathematical Relationship

Question 5.

It is given that x is greater than 0.

Quantity A: x percent of 100 x

Quantity B: x squared






document.doc


Note that Quantity A, x percent of 100x, can be expressed algebraically as x over

100, times, 100x. Simplifying this algebraic expression gives x squared, which is Quantity B.

The correct answer is therefore Choice C.

This explanation uses the following strategy.Strategy 1: Translate from Words to an Arithmetic or Algebraic Representation.

Question 6.

It is given that open parenthesis 4 x minus 2 y close parenthesis, times,

open parenthesis 6 x + 3 y close parenthesis, = 18.

Quantity A: 4 x squared, minus y squared

Quantity B: 6






document.doc

Explanation for Question 6.The given equation open parenthesis 4 x minus 2 y close parenthesis,

times, open parenthesis 6 x + 3 y close parenthesis, = 18 can be simplified as follows.

Step 1: Note that open parenthesis 4 x minus 2 y

close parenthesis, = 2 times open parenthesis, 2x minus y, close parenthesis, and, open parenthesis 6x + 3y close parenthesis, = 3 times, open parenthesis, 2x + y, close parenthesis, so the given

equation can be rewritten as 2 times open parenthesis, 2x minus y,

close parenthesis, times 3 times, open parenthesis 2x + y close parenthesis, = 18.

Step 2: Dividing both sides of the rewritten equation by 6 gives open

parenthesis, 2x minus y, close parenthesis, times, open parenthesis, 2x + y, close parenthesis = 3.

Step 3: Multiplying out the left side of the equation in Step 2 gives 4, x squared,

minus, y squared = 3.

Since Quantity A is 4, x squared, minus, y squared, it follows that Quantity A is equal

to 3. Since Quantity B is 6, the correct answer is Choice B.


Question 7.

Figure for Algebra Question 7

document.doc

Begin skippable part of figure description.The figure shows a line and three points in the first quadrant of the xy-plane. One of the points is on the line, one of the points is above the line, and one of the points is below the line.

The line starts at the origin, slants upward and to the right. The point on the line has coordinates

6 comma 6. The point above the line has coordinates c comma d, and the point

below the line has coordinates w comma z.

End skippable part of figure description.

Quantity A: w + d

Quantity B: c + z






Explanation for Question 7.The line in the xy-plane passes through the origin and the point with coordinates 6 comma

6, so the equation of the line is y =x. Since the point with coordinates c comma d is above

the line, it follows that d is greater than c, and since the point with coordinates w

comma z is below the line, it follows that w is greater than z. From the fact that d is greater than c and w is greater than z, you get w + d is greater

than c + z; that is, Quantity A is greater than Quantity B. Thus the correct answer is Choice A.


document.doc

Question 8.This question has five answer choices, labeled A through E. Select the best one of the answer choices given.

If x, y squared = 12 and xy = 4, then x =

A. 1

B. 2

C. the positive square root of 3

D. 2 thirds

E. 4 thirds

Select and indicate the best one of the answer choices given.


From the given equations x, y squared = 12 and x y = 4, it follows that

12 = x, y squared = x y times y = 4y, and so y = 3. Substituting y = 3 in the

equation xy = 4 gives 3x = 4, or x = 4 thirds. Thus the correct answer is Choice E,

4 thirds.


document.doc


The total cost of 8 bagels at a bakery is x dollars. At this cost per bagel, which of the following represents the total cost, in dollars, of y bagels?

A. the fraction 8 over x y

B. the fraction 8x over y

C. the fraction 8y over x

D. the fraction x y over 8

E. the fraction x over 8y



The total cost of 8 bagels is x dollars, so one bagel costs x over 8 dollars. Therefore the

total cost of y bagels is y times x over 8, or the fraction x y over 8 dollars.

Thus the correct answer is Choice D, the fraction x y over 8.

This explanation uses the following strategy.Strategy 1: Translate from Words to an Arithmetic or Algebraic Representation

document.doc


Which of the following is equal to the fraction with numerator 2 raised to the x minus y

power, and denominator 2 raised to the x + y power for all integers x and y ?

A. 4 raised to the negative x power

B. 4 raised to the negative y power

C. 2 raised to the x y power

D. 4 raised to the x power

E. 4 raised to the y power



Simplifying the fraction the fraction with numerator 2 raised to the x minus y power, and

denominator 2 raised to the x + y power yields the

fraction with numerator 2 raised to the x minus y power, and denominator 2 raised to the x + y power = 2 raised to the x minus y +, open parenthesis, x + y, close parenthesis, power = 2 raised to the x minus y minus x minus y, power = 2 raised to the negative 2y power.

document.doc

The only answer choice with 2 as the base is Choice C, 2 raised to the x y power which is

clearly not the correct answer. Since all the other answer choices have 4 as the base, it is a good

idea to rewrite 2 raised to the negative 2y power, as an expression with 4 as the base, as

follows.

2 raised to the negative 2y power = the fraction 1 over, 2 raised to the 2y power, = the fraction 1 over, two squared, raised to the y power = the fraction 1 over, 4 raised to the y power = 4 raised to the negative y power.

Thus the correct answer is Choice B, 4 raised to the negative y power.

This explanation uses the following strategies.Strategy 5: Simplify an Arithmetic or Algebraic RepresentationStrategy 8: Search for a Mathematical Relationship.


How many integers are in the solution set of the inequality x squared minus 10 is

less than 0?

A. Two

B. Five

C. Six

D. Seven

E. Ten


document.doc


The inequality x squared minus 10 is less than 0 is equivalent to x squared is

less than 10. By inspection, the positive integers that satisfy this inequality are 1, 2, and 3. Note that 0 and the negative integers negative 1, negative 2, and negative 3 also satisfy the inequality, and there are no other integer solutions. So there are seven integers in the solution set: negative 1, negative 2, negative 3, 0, 1, 2, and 3. Thus the correct answer is Choice D, seven.



A group of 5,000 investors responded to a survey asking whether they owned stocks and whether they owned bonds. Of the group, 20 percent responded that they owned only one of the two types of investments. If r is the number of investors in the group who owned stocks but not bonds, which of the following represents the number of investors in the group who owned bonds but not stocks, in terms of r ?

A. 5,000 minus r

B. 1,000 minus r

C. r minus 1,000

D. 1,000r

E. 0.2 times open parenthesis 5,000 minus r close parenthesis


document.doc

Explanation for Question 12.Twenty percent of the 5,000 investors that responded to the survey said they owned either stocks

or bonds, but not both. So the number of investors in that group is 0.20 times

5,000, or 1,000. Given that r members of that group owned stocks but not bonds, the number of investors in that group who owned bonds but not stocks is 1,000 minus r. Thus the correct answer is Choice B, 1,000 minus r.

This explanation uses the following strategies.Strategy 1: Translate from Words to an Arithmetic or Algebraic RepresentationStrategy 8: Search for a Mathematical Relationship


If the fraction with numerator r + s, and denominator 4 + 5, is equal to the fraction

r over 4, plus the fraction s over 5, which of the following statements must be true?

A. r = s

B. 5r = 4s

C. 5r equals negative 4s

D. 25r = 16s

E. 25r equals negative 16s


document.doc


You can simplify each side of the equation the fraction with numerator r + s, and

denominator 4 + 5, is equal to the fraction r over 4, plus the fraction s over 5, as follows.

Left-hand Side of Equation: the fraction with numerator r + s, and denominator 4 +

5, = the fraction r + s over 9,

Right-hand Side of Equation:

the fraction r over 4 + the fraction s over 5 = the fraction with numerator 5r + 4s, and denominator 20.

So the equation the fraction with numerator r + s, and denominator 4 + 5, = the

fraction r over 4 + the fraction s over 5 can be rewritten as the fraction r + s over

9, = the fraction with numerator 5r + 4s, and denominator 20.

Cross multiplying in the rewritten equation gives 20 times open

parenthesis r + s, close parenthesis, = 9 times, open parenthesis, 5r + 4s, close parenthesis, which simplifies to 20r + 20s = 45r + 36s, or negative 16s = 25r. Thus the correct answer is Choice E, 25r = negative 16s.

This explanation uses the following strategy.Strategy 5: Simplify an Arithmetic or Algebraic Representation

document.doc

Question 14.This question has five answer choices, labeled A through E. Select all the answer choices that apply.

In the xy-plane, triangular region R is bounded by the lines x = 0, y = 0, and 4x + 3y = 60. Which of the following points lie inside region R ?

Indicate all such points.

A. the point with coordinates 2 comma 18

B. the point with coordinates 5 comma 12

C. the point with coordinates 10 comma 7

D. the point with coordinates 12 comma 3

E. the point with coordinates 15 comma 2

Select and indicate all the answer choices that apply. The correct answer to a question of this type could consist of as few as one or as many as all five of the answer choices.

Explanation for Question 14.Consider the three lines that bound triangular region R. The line x = 0 is the y-axis, and the line

y = 0 is the x-axis. The line 4x + 3y = 60 intersects the x-axis at the point with coordinates

15 comma 0 and intersects the y-axis at the point with coordinates 0 comma 20. The

following figure shows region R.

Figure for Explanation of Algebra Question 14

document.doc

Begin skippable part of figure descriptionIn the figure, triangular region R is above the x-axis, to the right of the y-axis and below the line 4x + 3y = 60.

End skippable part of figure description.From the figure, you can see that all points inside region R have positive coordinates and lie below the line 4x + 3y = 60. Note that the equation 4x + 3y = 60 can be rewritten in the form

y = 20 minus 4 thirds x. In this form, you can see that points inside region R satisfy

the inequality y is less than 20 minus 4 thirds x. Since all of the answer choices have

positive coordinates, you need only to check whether the coordinates in each answer choice satisfy

the inequality y is less than 20 minus 4 thirds x. or equivalently 4x +3y

is less than 60.

Choice A, the point with coordinates 2 comma 18:

4x +3y = 4 times 2, +, 3 times 19 = 62, which is greater than 60.

So Choice A is not in region R.

Choice B, the point with coordinates 5 comma 12:

4x +3y = 4 times 5, +, 3 times 12 = 56, which is less than 60.

So Choice B is in region R.

Choice C, the point with coordinates 10 comma 7:


So Choice C is not in region R.

Choice D, the point with coordinates 12 comma 3:

4x +3y = 4 times 12, +, 3 times 3 = 57, which is less than 60.

So Choice D is in region R.

Choice E, the point with coordinates 15 comma 2:


So Choice E is not in region R.

Thus the correct answer consists of choices B and D; that is, the point with

coordinates 5 comma 12 and the point with coordinates 12 comma 3.

document.doc

This explanation uses the following strategies.Strategy 3: Translate from an Algebraic to a Graphical RepresentationStrategy 8: Search for a Mathematical Relationship

Question 15.This question has five answer choices, labeled A through E. Select all the answer choices that apply.

At the beginning of a trip, the tank of Diana’s car was filled with gasoline to half of its capacity. During the trip, Diana used 30 percent of the gasoline in the tank. At the end of the trip, Diana added 8 gallons of gasoline to the tank. The capacity of the tank of Diana’s car was x gallons. Which of the following expressions represent the number of gallons of gasoline in the tank after Diana added gasoline to the tank at the end of the trip?

Indicate all such expressions.

A. the fraction x over 2, end fraction, minus, the fraction 3x over 20, end fraction, +

8

B. the fraction 7x over 20, end fraction, + 8

C. the fraction 3x over 20, end fraction, + 8

D. the fraction x over 2, end fraction, plus, the fraction 3x over 20, end fraction,+ 8

E. the fraction 7x over 20, end fraction, minus 8

Select and indicate all the answer choices that apply. The correct answer to a question of this type could consist of as few as one or as many as all five of the answer choices.

document.doc

Explanation for Question 15.The capacity of the car’s tank was x gallons of gasoline. Before the trip, the tank was half full and

therefore contained x over 2 gallons of gasoline. During the trip, 30 percent of the

gasoline in the tank was used, so the number of gallons left was 70 percent of x over 2 or

7 tenths times x over 2 = 7x over 20. After Diana added 8 gallons of gasoline to

the tank, the total number of gallons in the tank was the fraction 7x over 20, end fraction,

+ 8. Thus one correct choice is Choice B; that is, the fraction 7x over 20, end fraction,

+ 8.

However, the question asks you to find all of the answer choices that represent the number of gallons of gasoline in the tank at the end of the trip. So you need to determine whether any of the

other choices are equivalent to the fraction, 7x over 20, end fraction, + 8. Of the answer

choices, only Choices A and C have the same constant term as Choice B: 8. So these are the only

choices that need to be checked. Choice A, the fraction x over 2, end fraction, minus,

the fraction 3x over 20, end fraction, + 8 can be simplified as follows.

the fraction x over 2, end fraction, minus, the fraction 3x over 20, end fraction, + 8 = the fraction 10x over 20, end fraction, minus, the fraction 3x over 20, end fraction, + 8 = the fraction 7x over 20, end fraction, + 8

So Choice A is equivalent to the fraction 7x over 20, end fraction, + 8, Choice C,

the fraction 3x over 20, end fraction, + 8 is clearly not equivalent to the fraction 7x over

20, end fraction, + 8. Thus the correct answer consists of choices A and B; that is,

the fraction x over 2, end fraction, minus, the fraction 3x over 20, end

fraction, + 8 and the fraction 7x over 20, end fraction, + 8.

document.doc

This explanation uses the following strategies.Strategy 1: Translate from Words to an Arithmetic or Algebraic RepresentationStrategy 5: Simplify an Arithmetic or Algebraic RepresentationStrategy 8: Search for a Mathematical Relationship

Question 16.This question does not have any answer choices; it is a numeric entry question. To answer this question, enter a number in the answer space provided.

Machine A, working alone at its constant rate, produces x pounds of peanut butter in 12 minutes. Machine B, working alone at its constant rate, produces x pounds of peanut butter in 18 minutes. How many minutes will it take machines A and B, working simultaneously at their respective constant rates, to produce x pounds of peanut butter?

The answer space is followed by the word “minutes.”

To answer this question, enter a number in the answer space provided. The number can include a decimal point, and can be positive, negative, or zero. The number entered cannot be a fraction.

Explanation for Question 16.Translating the given information into an algebraic expression, you see that machine A produces

x over 12 pounds of peanut butter in 1 minute, and machine B produces x

over 18 pounds of peanut butter in 1 minute. Therefore, working simultaneously, machine A and

machine B produce x over 12 + x over 18 pounds of peanut butter in 1 minute.

Letting t be the number of minutes it takes machines A and B, working simultaneously, to produce x pounds of peanut butter, you can set up the equation

open parenthesis, x over 12 + x over 18, close parenthesis, times t = x.

To solve the equation for t, divide this equation by x, to get

open parenthesis, 1 over 12 + 1 over 18, close parenthesis, times t = 1.

document.doc

This simplifies to the equation

5 over 36,end fraction, times, t = 1.

which further simplifies to

t = 36 over 5 = 7.2.

The correct answer is 7.2 minutes.

This explanation uses the following strategies.Strategy 1: Translate from Words to an Arithmetic or Algebraic RepresentationStrategy 8:Search for a Mathematical Relationship

Question 17.This question does not have any answer choices; it is a numeric entry question. To answer this question, enter a number in the answer space provided.

The function f has the property that f of x = f of, open parenthesis x + 1 close

parenthesis, for all numbers x. If f of 4 = 17, what is the value of f of 8?

To answer this question, enter a number in the answer space provided. The number can include a decimal point, and can be positive, negative, or zero. The number entered cannot be a fraction.

Explanation for Question 17.The property that f of x = f of, open parenthesis, x + 1, close parenthesis, for all

numbers x implies that f of 4 = f of 5, f of 5 = f of 6, f

of 6 = f of 7, and f of 7 = f of 8. Therefore, since f of 4 = 17, it follows

that f of 8 = 17. Thus the correct answer is 17.

This explanation uses the following strategy.Strategy 7: Find a Pattern

This is the end of Chapter 4 – Algebra Answer Key with Answers and Explanations.

document.doc

section 4 - educational testing service · web viewthe answer space is followed by the word...

Documents