GRE - Info NOT Content - Quantitative Reasoning

Question 1

Quant Reasoning assesses

Answer 1

basic mathematical skills

understanding of elementary mathematical concepts

ability to reason quantitatively and to model and solve problems with quantitative methods

Question 2

Quant Reasoning - Some of the questions in the measure are posed in real-life settings, while others are posed in purely mathematical settings. The skills, concepts and abilities are tested in the four content areas below:

Answer 2

Arithmetic topics include properties and types of integers, such as divisibility, factorization, prime numbers, remainders and odd and even integers; arithmetic operations, exponents and roots; and concepts such as estimation, percent, ratio, rate, absolute value, the number line, decimal representation and sequences of numbers.

Algebra topics include operations with exponents; factoring and simplifying algebraic expressions; relations, functions, equations and inequalities; solving linear and quadratic equations and roots; solving simultaneous equations and inequalities; setting up equations to solve word problems; and coordinate geometry, including graphs of functions, equations and inequalities, intercepts and slopes of lines.

Geometry topics include parallel and perpendicular lines, circles, triangles — including isosceles, equilateral and 30°-60°-90° triangles — quadrilaterals, other polygons, congruent and similar figures, three-dimensional figures, area, perimeter, volume, the Pythagorean theorem and angle measurement in degrees. The ability to construct proofs is not tested.

Data analysis topics include basic descriptive statistics, such as mean, median, mode, range, standard deviation, interquartile range, quartiles and percentiles; interpretation of data in tables and graphs, such as line graphs, bar graphs, circle graphs, boxplots, scatterplots and frequency distributions; elementary probability, such as probabilities of compound events and independent events; random variables and probability distributions, including normal distributions; and counting methods, such as combinations, permutations and Venn diagrams. These topics are typically taught in high school algebra courses or introductory statistics courses. Inferential statistics is not tested.

Question 3

there are some important assumptions about numbers and figures that are listed in the Quantitative Reasoning section directions:

Answer 3

All numbers used are real numbers.
All figures are assumed to lie in a plane unless otherwise indicated.
Geometric figures, such as lines, circles, triangles and quadrilaterals, are not necessarily drawn to scale. That is, you should not assume that quantities such as lengths and angle measures are as they appear in a figure. You should assume, however, that lines shown as straight are actually straight, points on a line are in the order shown, and more generally, all geometric objects are in the relative positions shown. For questions with geometric figures, you should base your answers on geometric reasoning, not on estimating or comparing quantities by sight or by measurement.
Coordinate systems, such as xy-planes and number lines, are drawn to scale; therefore, you can read, estimate or compare quantities in such figures by sight or by measurement.
Graphical data presentations, such as bar graphs, circle graphs and line graphs are drawn to scale; therefore, you can read, estimate or compare data values by sight or by measurement.

Question 4

Quantitative Reasoning Question Types

The Quantitative Reasoning measure has four types of questions:

Answer 4

Quantitative Comparison Questions
Multiple-choice Questions — Select One Answer Choice
Multiple-choice Questions — Select One or More Answer Choices
Numeric Entry Questions
Each question appears either independently as a discrete question or as part of a set of questions called a Data Interpretation set. All of the questions in a Data Interpretation set are based on the same data presented in tables, graphs or other displays of data.

In the computer-based test, you are allowed to use a basic calculator — provided on-screen — on the Quantitative Reasoning measure. Read more about using the calculator.

Question 5

Quantitative Comparison Questions

Description

Answer 5

Questions of this type ask you to compare two quantities – Quantity A and Quantity B – and then determine which of the following statements describes the comparison:

Quantity A is greater.
Quantity B is greater.
The two quantities are equal.
The relationship cannot be determined from the information given.

Question 6

Quantitative Comparison Questions Tips for Answering

Answer 6

1. Become familiar with the answer choices. Quantitative Comparison questions always have the same answer choices, so get to know them, especially the last choice, “The relationship cannot be determined from the information given.” Never select this last choice if it is clear that the values of the two quantities can be determined by computation. Also, if you determine that one quantity is greater than the other, make sure you carefully select the corresponding choice so as not to reverse the first two choices.
2. Avoid unnecessary computations. Don’t waste time performing needless computations in order to compare the two quantities. Simplify, transform or estimate one or both of the given quantities only as much as is necessary to compare them.
3. Remember that geometric figures are not necessarily drawn to scale. If any aspect of a given geometric figure is not fully determined, try to redraw the figure, keeping those aspects that are completely determined by the given information fixed but changing the aspects of the figure that are not determined. Examine the results. What variations are possible in the relative lengths of line segments or measures of angles?
4. Plug in numbers. If one or both of the quantities are algebraic expressions, you can substitute easy numbers for the variables and compare the resulting quantities in your analysis. Consider all kinds of appropriate numbers before you give an answer: e.g., zero, positive and negative numbers, small and large numbers, fractions and decimals. If you see that Quantity A is greater than Quantity B in one case and Quantity B is greater than Quantity A in another case, choose “The relationship cannot be determined from the information given.”
5. Simplify the comparison. If both quantities are algebraic or arithmetic expressions and you cannot easily see a relationship between them, you can try to simplify the comparison. Try a step-by-step simplification that is similar to the steps involved when you solve the equation 5=4x + 3 for x, or that is similar to the steps involved when you determine that the inequality
   3y + 2
   ——— ), less than (

Question 7

Quantitative Comparison Questions
Sample Questions
Directions: Compare Quantity A and Quantity B, using additional information centered above the two quantities if such information is given, and select one of the following four answer choices:

(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.

A symbol that appears more than once in a question has the same meaning throughout the question.

1. Quantity A
   The least prime number greater than 24

Quantity B
The greatest prime number less than 28

(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.

Answer 7

Explanation

For the integers greater than 24, note that 25, 26, 27, and 28 are not prime numbers, but 29 is a prime number, as are 31 and many other greater integers. Thus, 29 is the least prime number greater than 24, and Quantity A is 29. For the integers less than 28, note that 27, 26, 25, and 24 are not prime numbers, but 23 is a prime number, as are 19 and several other lesser integers. Thus, 23 is the greatest prime number less than 28, and Quantity B is 23. Thus, the correct answer is Choice A, Quantity A is greater.

Question 8

Quantitative Comparison Questions
Sample Questions
Directions: Compare Quantity A and Quantity B, using additional information centered above the two quantities if such information is given, and select one of the following four answer choices:

(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.

A symbol that appears more than once in a question has the same meaning throughout the question.

2. Lionel is younger than Maria.
   Quantity A
   Twice Lionel’s age

Quantity B
Maria’s age

(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.

Answer 8

Explanation

If Lionel’s age is 6 years and Maria’s age is 10 years, then Quantity A is greater, but if Lionel’s age is 4 years and Maria’s age is 10 years, then Quantity B is greater. Thus, the relationship cannot be determined. The correct answer is Choice D, the relationship cannot be determined from the information given.

Question 9

Quantitative Comparison Questions
Sample Questions
Directions: Compare Quantity A and Quantity B, using additional information centered above the two quantities if such information is given, and select one of the following four answer choices:

(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.

3. Quantity A
   54% of 360

Quantity B
150

(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.

Answer 9

Explanation

Without doing the exact computation, you can see that 54 percent of 360 is greater than of 360, which is 180, and 180 is greater than Quantity B, 150. Thus, the correct answer is Choice A, Quantity A is greater.

Question 10

Quantitative Comparison Questions
Sample Questions
Directions: Compare Quantity A and Quantity B, using additional information centered above the two quantities if such information is given, and select one of the following four answer choices:

(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.

INSERT FIGURE 1 FROM DESKTOP

4.Quantity A
PS

Quantity B 
SR
(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.

Answer 10

Explanation

From Figure 1, you know that PQR is a triangle and that point S is between points P and R, so PS

Question 11

Quantitative Comparison Questions
Sample Questions
Directions: Compare Quantity A and Quantity B, using additional information centered above the two quantities if such information is given, and select one of the following four answer choices:

(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.

5. y = 2x⌃2 + 7x - 3
   Quantity A is x and Quantity B is y

(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.

Answer 11

Explanation

If x=0 then y=2(0⌃2) + 7(0) - 3= -3, so in this case, x > y; but if x = 1, then y = 2(1⌃2) + 7(1) - 3 = 6 so in that case,
y > x. Thus, the correct answer is Choice D, the relationship cannot be determined from the information given.

Note that plugging numbers into expressions may not be conclusive. However, it is conclusive if you get different results after plugging in different numbers: the conclusion is that the relationship cannot be determined from the information given. It is also conclusive if there are only a small number of possible numbers to plug in and all of them yield the same result, say, that Quantity B is greater.

Now suppose there are an infinite number of possible numbers to plug in. If you plug many of them in and each time the result is, for example, that Quantity A is greater, you still cannot conclude that Quantity A is greater for every possible number that could be plugged in. Further analysis would be necessary and should focus on whether Quantity A is greater for all possible numbers or whether there are numbers for which Quantity A is not greater.

Question 12

Quantitative Comparison Questions
Sample Questions
Directions: Compare Quantity A and Quantity B, using additional information centered above the two quantities if such information is given, and select one of the following four answer choices:

(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.

6. The following sample questions focus on simplifying the comparison.

y > 4

Quantity A:
3y + 2
————
5

Quantity B:
y
(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.

Answer 12

Explanation

Set up the initial comparison:
3y + 2
———— is ? to y (, or =)
5

Then simplify:

Step 1: Multiply both sides by 5 to get 3y + 2 ? 5y

Step 2: Subtract 3y from both sides to get 2 ? 2y

Step 3: Divide both sides by 2 to get 1 ? y

The comparison is now simplified as much as possible. In order to compare 1 and y, note that you are given the information y > 4 (above Quantities A and B). It follows from y > 4 that y > 1 or so that in the comparison 1 ? y, the placeholder ? represents less than ( -y. So the relationship in the final, simplified inequality may be the opposite of the relationship between Quantities A and B. This is another reason to consider the impact of each step carefully.

Question 13

Quantitative Comparison Questions
Sample Questions
Directions: Compare Quantity A and Quantity B, using additional information centered above the two quantities if such information is given, and select one of the following four answer choices:

(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.

7. Quantity A
   2⌃30 - 2⌃29
   —————————
   2

Quantity B
2⌃ 28

(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

Answer 13

Explanation

Set up the initial comparison:

2⌃30 - 2⌃29
———————— ? 2⌃28
2

Then simplify:

Step 1: Multiply both sides by 2 to get 2⌃30 - 2⌃29 ? 2⌃29

Step 2: Add 2⌃29 to both sides to get 2⌃30 ? 2⌃29 + 2⌃29

Step 3: Simplify the right-hand side using the fact that (2)(2⌃29) = 2⌃30 to get 2⌃30 ? 2⌃30

The resulting relationship is equal to (=). In reverse order, each simplification step implies equal to in the preceding comparison. So Quantities A and B are also equal. Thus, the correct answer is Choice C, the two quantities are equal.

Question 14

Quantitative Comparison Questions
Sample Questions
Directions: Compare Quantity A and Quantity B, using additional information centered above the two quantities if such information is given, and select one of the following four answer choices:

(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.

8. Quantity A
   x⌃2 + 1

Quantity B
2x - 1

(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.

Answer 14

Explanation

Set up the initial comparison:
x⌃2 + 1 ? 2x - 1

Then simplify by noting that the quadratic polynomial
x⌃2 - 2x + 1 can be factored:
Step 1: Subtract 2x from both sides to get x⌃2 - 2x + 1 ? -1

Step 2: Factor the left-hand side to get (x - 1)⌃2 ? -1

The left-hand side of the comparison is the square of a number. Since the square of a number is always greater than or equal to 0, and 0 is greater than the simplified comparison is the inequality (x-1)⌃2 > -1 and the resulting relationship is greater than (>). In reverse order, each simplification step implies the inequality greater than (>) in the preceding comparison. Therefore, Quantity A is greater than Quantity B. The correct answer is Choice A, Quantity A is greater.

Question 15

Quantitative Comparison Questions
Sample Questions
Directions: Compare Quantity A and Quantity B, using additional information centered above the two quantities if such information is given, and select one of the following four answer choices:

(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.

8. w > 1

Quantity A
7w - 4

Quantity B
2w + 5

(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.

Answer 15

Explanation

Set up the initial comparison:

7w - 4 ? 2w + 5

Then simplify:

Step 1: Subtract 2w from both sides and add 4 to both sides to get 5w ? 9

Step 2: Divide both sides by 5 to get w ? 9 ( or 9/5)
—
5

The comparison cannot be simplified any further. Although you are given that w > 1 you still don’t know how w compares to 9/5 or 1.8. For example, if w=1.5 then w1.8 In other words, the relationship between w and 9/5 cannot be determined. Note that each of these simplification steps is reversible, so in reverse order, each simplification step implies that the relationship cannot be determined in the preceding comparison. Thus, the relationship between Quantities A and B cannot be determined. The correct answer is Choice D, the relationship cannot be determined from the information given.

The strategy of simplifying the comparison works most efficiently when you note that a simplification step is reversible while actually taking the step. Here are some common steps that are always reversible:

Adding any number or expression to both sides of a comparison
Subtracting any number or expression from both sides
Multiplying both sides by any nonzero number or expression
Dividing both sides by any nonzero number or expression
Remember that if the relationship is an inequality, multiplying or dividing both sides by any negative number or expression will yield the opposite inequality. Be aware that some common operations like squaring both sides are generally not reversible and may require further analysis using other information given in the question in order to justify reversing such steps.

Question 16

Strategies of simplifying the comparison that are reversible

Answer 16

The strategy of simplifying the comparison works most efficiently when you note that a simplification step is reversible while actually taking the step. Here are some common steps that are always reversible:

Adding any number or expression to both sides of a comparison
Subtracting any number or expression from both sides
Multiplying both sides by any nonzero number or expression
Dividing both sides by any nonzero number or expression
Remember that if the relationship is an inequality, multiplying or dividing both sides by any negative number or expression will yield the opposite inequality. Be aware that some common operations like squaring both sides are generally not reversible and may require further analysis using other information given in the question in order to justify reversing such steps.

Question 17

Multiple-choice Questions — Select One Answer Choice
Description
These questions are multiple-choice questions that ask you to select only one answer choice from a list of five choices.

Tips for Answering

Answer 17

Tips for Answering
Use the fact that the answer is there. If your answer is not one of the five answer choices given, you should assume that your answer is incorrect and do the following:
Reread the question carefully – you may have missed an important detail or misinterpreted some information.
Check your computations – you may have made a mistake, such as mis-keying a number on the calculator.
Reevaluate your solution method – you may have a flaw in your reasoning.
Examine the answer choices. In some questions you are asked explicitly which of the choices has a certain property. You may have to consider each choice separately or you may be able to see a relationship between the choices that will help you find the answer more quickly. In other questions, it may be helpful to work backward from the choices, say, by substituting the choices in an equation or inequality to see which one works. However, be careful, as that method may take more time than using reasoning.
For questions that require approximations, scan the answer choices to see how close an approximation is needed. In other questions, too, it may be helpful to scan the choices briefly before solving the problem to get a better sense of what the question is asking. If computations are involved in the solution, it may be necessary to carry out all computations exactly and round only your final answer in order to get the required degree of accuracy. In other questions, you may find that estimation is sufficient and will help you avoid spending time on long computations.

Question 18

Sample quest 1
If 5x + 32 = 4 - 2x what is the value of x ?

(A) -4
(B) -3
(C) 4
(D) 7
(E) 12

Answer 18

Explanation

Solving the equation for x, you get 7x = 28 and so x = -4. The correct answer is Choice A, -4.

Question 19

Sample Ques 2
Which of the following numbers is farthest from the number 1 on the number line?

(A) -10
(B) -5
(C) 0
(D) 5
(E) 10

Answer 19

Explanation

Circling each of the answer choices in a sketch of the number line (Figure 4) shows that of the given numbers, -10 is the greatest distance from 1.

INSERT FIG 4 NUM LINE PIC

Another way to answer the question is to remember that the distance between two numbers on the number line is equal to the absolute value of the difference of the two numbers. For example, the distance between -10 and 1 is |-10 - 1| = 11 and the distance between 10 and 1 is |10 - 1| = |9| = 9 The correct answer is Choice A, -10.

Question 20

SAMPLE QUES 3

INSERT FIG 5

The figure above shows the graph of a function f defined by f(x) = |2x| + 4 for all numbers x. For which of the following functions g, defined for all numbers x, does the graph of g intersect the graph of f ?

(A) g(x) = x-2
(B) g(x) = x+3
(C) g(x) = 2x-2
(D) g(x) = 2x + 3
(E) g(x) = 3x - 2

Answer 20

Explanation

You can see that all five choices are linear functions whose graphs are lines with various slopes and y-intercepts. The graph of Choice A is a line with slope 1 and y-intercept shown in Figure 6.

INSERT FIG 6 HERE

It is clear that this line will not intersect the graph of f to the left of the y-axis. To the right of the y-axis, the graph of f is a line with slope 2, which is greater than slope 1. Consequently, as the value of x increases, the value of y increases faster for f than for g, and therefore the graphs do not intersect to the right of the y-axis. Choice B is similarly ruled out. Note that if the y-intercept of either of the lines in Choices A and B were greater than or equal to 4 instead of less than 4, they would intersect the graph of f.

Choices C and D are lines with slope 2 and y-intercepts less than 4. Hence, they are parallel to the graph of f (to the right of the y-axis) and therefore will not intersect it. Any line with a slope greater than 2 and a y-intercept less than 4, like the line in Choice E, will intersect the graph of f (to the right of the y-axis). The correct answer is Choice E, g(x) = 3x - 2

Question 21

Sample Ques 4
A car got 33 miles per gallon using gasoline that cost $2.95 per gallon. Approximately what was the cost, in dollars, of the gasoline used in driving the car 350 miles?

(A) $10
(B) $20
(C) $30
(D) $40
(E) $50

Answer 21

Explanation

Scanning the answer choices indicates that you can do at least some estimation and still answer confidently. The car used 350/33 gallons of gasoline, so the cost was
(350/33)(2.95) dollars. You can estimate the product (350/33)(2.95) by estimating 350/33 a little low, 10, and estimating 2.95 a little high, 3, to get approximately (10)(3) = 30 dollars. You can also use the calculator to compute a more exact answer and then round the answer to the nearest 10 dollars, as suggested by the answer choices. The calculator yields the decimal 31.287…, which rounds to 30 dollars. Thus, the correct answer is Choice C, $30.

Question 22

Sample Ques 5
A certain jar contains 60 jelly beans — 22 white, 18 green, 11 yellow, 5 red and 4 purple. If a jelly bean is to be chosen at random, what is the probability that the jelly bean will be neither red nor purple?

(A) 0.09
(B) 0.15
(C) 0.54
(D) 0.85
(E) 0.91

Answer 22

Explanation

Since there are 5 red and 4 purple jelly beans in the jar, there are 51 that are neither red nor purple, and the probability of selecting one of these is 51/60. Since all of the answer choices are decimals, you must convert the fraction to its decimal equivalent, 0.85. Thus, the correct answer is Choice D, 0.85.

Question 23

Description
These questions are multiple-choice questions that ask you to select one or more answer choices from a list of choices. A question may or may not specify the number of choices to select.

Tips for Answering

Answer 23

Tips for Answering
Note whether you are asked to indicate a specific number of answer choices or all choices that apply. In the latter case, be sure to consider all of the choices, determine which ones are correct, and select all of those and only those choices. Note that there may be only one correct choice.
In some questions that involve conditions that limit the possible values of numerical answer choices, it may be efficient to determine the least and/or the greatest possible value. Knowing the least and/or greatest possible value may enable you to quickly determine all of the choices that are correct.
Avoid lengthy calculations by recognizing and continuing numerical patterns.

Question 24

Sample Questions
Directions: Select one or more answer choices according to the specific question directions.

If the question does not specify how many answer choices to select, select all that apply.

The correct answer may be just one of the choices or as many as all of the choices, depending on the question.
No credit is given unless you select all of the correct choices and no others.
If the question specifies how many answer choices to select, select exactly that number of choices.

SQ 1
Which two of the following numbers have a product that is between -1 and 0 ?

Indicate both of the numbers.

(A) -20
(B) -10
(C) 2⌃-4
(D) 3⌃-2

Answer 24

Explanation

For this question, you must select a pair of answer choices. The product of the pair must be negative, so the possible products are
(-20)(2⌃-4), (-20)(3⌃-2), (-10)(2⌃-4), and (-10)(3⌃-2). The product must also be greater than -1. The first product is -20/2⌃4 = -20/16 -1, so you can stop there. The correct answer consists of Choices B (-10 ) and C ( 2-4).

Question 25

Sample Questions
Directions: Select one or more answer choices according to the specific question directions.

If the question does not specify how many answer choices to select, select all that apply.

The correct answer may be just one of the choices or as many as all of the choices, depending on the question.
No credit is given unless you select all of the correct choices and no others.
If the question specifies how many answer choices to select, select exactly that number of choices.

SQ 2:
Which of the following integers are multiples of both 2 and 3 ?

Indicate all such integers.

(A) 8
(B) 9
(C) 12
(D) 18
(E) 21
(F) 36

Answer 25

Explanation

You can first identify the multiples of 2, which are 8, 12, 18 and 36, and then among the multiples of 2 identify the multiples of 3, which are 12, 18 and 36. Alternatively, if you realize that every number that is a multiple of 2 and 3 is also a multiple of 6, you can identify the choices that are multiples of 6. The correct answer consists of Choices C (12), D (18) and F (36).

Question 26

Sample Questions
Directions: Select one or more answer choices according to the specific question directions.

If the question does not specify how many answer choices to select, select all that apply.

The correct answer may be just one of the choices or as many as all of the choices, depending on the question.
No credit is given unless you select all of the correct choices and no others.
If the question specifies how many answer choices to select, select exactly that number of choices.

SQ 3:
Each employee of a certain company is in either Department X or Department Y, and there are more than twice as many employees in Department X as in Department Y. The average (arithmetic mean) salary is $25,000 for the employees in Department X and $35,000 for the employees in Department Y. Which of the following amounts could be the average salary for all of the employees of the company?

Indicate all such amounts.

(A) $26,000
(B) $28,000
(C) $29,000
(D) $30,000
(E) $31,000
(F) $32,000
(G) $34,000

Answer 26

Explanation

One strategy for answering this kind of question is to find the least and/or greatest possible value. Clearly the average salary is between $25,000 and $35,000, and all of the answer choices are in this interval. Since you are told that there are more employees with the lower average salary, the average salary of all employees must be less than the average of $25,000 and $35,000, which is $30,000. If there were exactly twice as many employees in Department X as in Department Y, then the average salary for all employees would be, to the nearest dollar, the following weighted mean,

(2)(25,000) + (1)(35,000)
————————————— = 28,333 dollars
2 + 1
where the weight for $25,000 is 2 and the weight for $35,000 is 1. Since there are more than twice as many employees in Department X as in Department Y, the actual average salary must be even closer to $25,000 because the weight for $25,000 is greater than 2. This means that $28,333 is the greatest possible average. Among the choices given, the possible values of the average are therefore $26,000 and $28,000. Thus, the correct answer consists of Choices A ($26,000) and B ($28,000).

Intuitively, you might expect that any amount between $25,000 and $28,333 is a possible value of the average salary. To see that $26,000 is possible, in the weighted mean above, use the respective weights 9 and 1 instead of 2 and 1. To see that $28,000 is possible, use the respective weights 7 and 3.

Question 27

Sample Questions
Directions: Select one or more answer choices according to the specific question directions.

If the question does not specify how many answer choices to select, select all that apply.

The correct answer may be just one of the choices or as many as all of the choices, depending on the question.
No credit is given unless you select all of the correct choices and no others.
If the question specifies how many answer choices to select, select exactly that number of choices.

SQ 4:
Which of the following could be the units digit of 57⌃n, where n is a positive integer?

Indicate all such digits.

(A) 0
(B) 1
(C) 2
(D) 3
(E) 4
(F) 5
(G) 6
(H) 7
(I) 8
(J) 9

Answer 27

Explanation

The units digit of 57⌃n is the same as the units digit of 7⌃n for all positive integers n. To see why this is true for n = 2, compute 57⌃2 by hand and observe how its units digit results from the units digit of 7⌃2. Because this is true for every positive integer n, you need to consider only powers of 7. Beginning with n = 1 and proceeding consecutively, the units digits of 7, 7⌃2, 7⌃3, 7⌃4 and 7⌃5 are 7, 9, 3, 1 and 7, respectively. In this sequence, the first digit, 7, appears again, and the pattern of four digits, 7, 9, 3, 1, repeats without end. Hence, these four digits are the only possible units digits of 7⌃n and therefore of 57⌃n. The correct answer consists of Choices B (1), D (3), H (7) and J (9).

Question 28

Numeric Entry Questions
Description
Questions of this type ask you either to enter your answer as an integer or a decimal in a single answer box or to enter it as a fraction in two separate boxes — one for the numerator and one for the denominator. In the computer-based test, use the computer mouse and keyboard to enter your answer.

Tips for Answering

Answer 28

Tips for Answering
Make sure you answer the question that is asked. Since there are no answer choices to guide you, read the question carefully and make sure you provide the type of answer required. Sometimes there will be labels before or after the answer box to indicate the appropriate type of answer. Pay special attention to units such as feet or miles, to orders of magnitude such as millions or billions, and to percents as compared with decimals.
If you are asked to round your answer, make sure you round to the required degree of accuracy. For example, if an answer of 46.7 is to be rounded to the nearest integer, you need to enter the number 47. If your solution strategy involves intermediate computations, you should carry out all computations exactly and round only your final answer in order to get the required degree of accuracy. If no rounding instructions are given, enter the exact answer.
Examine your answer to see if it is reasonable with respect to the information given. You may want to use estimation or another solution path to double-check your answer.

Question 29

Sample Questions
Directions: Enter your answer as an integer or a decimal if there is a single answer box OR as a fraction if there are two separate boxes — one for the numerator and one for the denominator.

To enter an integer or a decimal, either type the number in the answer box using the keyboard or use the Transfer Display button on the calculator.

First, click on the answer box — a cursor will appear in the box — and then type the number.
To erase a number, use the Backspace key.
For a negative sign, type a hyphen. For a decimal point, type a period.
To remove a negative sign, type the hyphen again and it will disappear; the number will remain.
The Transfer Display button on the calculator will transfer the calculator display to the answer box.
Equivalent forms of the correct answer, such as 2.5 and 2.50, are all correct.
Enter the exact answer unless the question asks you to round your answer.
To enter a fraction, type the numerator and the denominator in the respective boxes using the keyboard.

For a negative sign, type a hyphen; to remove it, type the hyphen again. A decimal point cannot be used in a fraction.
The Transfer Display button on the calculator cannot be used for a fraction.
Fractions do not need to be reduced to lowest terms, though you may need to reduce your fraction to fit in the boxes.

SQ 1
One pen costs $0.25 and one marker costs $0.35. At those prices, what is the total cost of 18 pens and 100 markers?

Answer 29

$(an empty box appears next to the dollar sign. I will put the answer in the box on the exam)

Explanation

Multiplying $0.25 by 18 yields $4.50, which is the cost of the 18 pens; and multiplying $0.35 by 100 yields $35.00, which is the cost of the 100 markers. The total cost is therefore Equivalent decimals, such as $39.5 or $39.500, are considered correct. Thus, the correct answer is $39.50 (or equivalent).

Note that the dollar symbol is in front of the answer box, so the symbol $ does not need to be entered in the box. In fact, only numbers, a decimal point and a negative sign can be entered in the answer box.

Question 30

Sample Questions
Directions: Enter your answer as an integer or a decimal if there is a single answer box OR as a fraction if there are two separate boxes — one for the numerator and one for the denominator.

To enter an integer or a decimal, either type the number in the answer box using the keyboard or use the Transfer Display button on the calculator.

First, click on the answer box — a cursor will appear in the box — and then type the number.
To erase a number, use the Backspace key.
For a negative sign, type a hyphen. For a decimal point, type a period.
To remove a negative sign, type the hyphen again and it will disappear; the number will remain.
The Transfer Display button on the calculator will transfer the calculator display to the answer box.
Equivalent forms of the correct answer, such as 2.5 and 2.50, are all correct.
Enter the exact answer unless the question asks you to round your answer.
To enter a fraction, type the numerator and the denominator in the respective boxes using the keyboard.

For a negative sign, type a hyphen; to remove it, type the hyphen again. A decimal point cannot be used in a fraction.
The Transfer Display button on the calculator cannot be used for a fraction.
Fractions do not need to be reduced to lowest terms, though you may need to reduce your fraction to fit in the boxes.

SQ 2
Rectangle R has length 30 and width 10, and square S has length 5. The perimeter of S is what fraction of the perimeter of R ?

Answer 30

/

Explanation

The perimeter of R is 30 + 10 + 30 + 10 = 80 and the perimeter of S is (4)(5) = 20. Therefore, the perimeter of S is 20/80 of the perimeter of R. To enter the answer 20/80 you should enter the numerator 20 in the top box and the denominator 80 in the bottom box. Because the fraction does not need to be reduced to lowest terms, any fraction that is equivalent to 20/80 is also considered correct, as long as it fits in the boxes. For example, both of the fractions 2/8 and 1/4 are considered correct. Thus, the correct answer is 20/80 (or any equivalent fraction).

Question 31

Sample Questions
Directions: Enter your answer as an integer or a decimal if there is a single answer box OR as a fraction if there are two separate boxes — one for the numerator and one for the denominator.

To enter an integer or a decimal, either type the number in the answer box using the keyboard or use the Transfer Display button on the calculator.

First, click on the answer box — a cursor will appear in the box — and then type the number.
To erase a number, use the Backspace key.
For a negative sign, type a hyphen. For a decimal point, type a period.
To remove a negative sign, type the hyphen again and it will disappear; the number will remain.
The Transfer Display button on the calculator will transfer the calculator display to the answer box.
Equivalent forms of the correct answer, such as 2.5 and 2.50, are all correct.
Enter the exact answer unless the question asks you to round your answer.
To enter a fraction, type the numerator and the denominator in the respective boxes using the keyboard.

For a negative sign, type a hyphen; to remove it, type the hyphen again. A decimal point cannot be used in a fraction.
The Transfer Display button on the calculator cannot be used for a fraction.
Fractions do not need to be reduced to lowest terms, though you may need to reduce your fraction to fit in the boxes.

SQ 3
Figure 7
RESULTS OF A USED-CAR AUCTION

Small Cars Large Cars

Number of cars offered 32 23
Number of cars sold 16 20
Projected sales total for
cars offered (in thousands) $70 $150
Actual sales total (in thousands) $41 $120

For the large cars sold at an auction that is summarized in the table above, what was the average sale price per car?

Answer 31

Explanation

From Figure 7, you see that the number of large cars sold was 20 and the sales total for large cars was $120,000 (not $120). Thus the average sale price per car was $120,000/20 = $6000 The correct answer is $6,000 (or equivalent).

(Note that the comma in 6,000 will appear automatically in the answer box in the computer-based test.)

Question 32

Sample Questions
Directions: Enter your answer as an integer or a decimal if there is a single answer box OR as a fraction if there are two separate boxes — one for the numerator and one for the denominator.

To enter an integer or a decimal, either type the number in the answer box using the keyboard or use the Transfer Display button on the calculator.

First, click on the answer box — a cursor will appear in the box — and then type the number.
To erase a number, use the Backspace key.
For a negative sign, type a hyphen. For a decimal point, type a period.
To remove a negative sign, type the hyphen again and it will disappear; the number will remain.
The Transfer Display button on the calculator will transfer the calculator display to the answer box.
Equivalent forms of the correct answer, such as 2.5 and 2.50, are all correct.
Enter the exact answer unless the question asks you to round your answer.
To enter a fraction, type the numerator and the denominator in the respective boxes using the keyboard.

For a negative sign, type a hyphen; to remove it, type the hyphen again. A decimal point cannot be used in a fraction.
The Transfer Display button on the calculator cannot be used for a fraction.
Fractions do not need to be reduced to lowest terms, though you may need to reduce your fraction to fit in the boxes.
SQ 4
A merchant made a profit of $5 on the sale of a sweater that cost the merchant $15. What is the profit expressed as a percent of the merchant’s cost?

Give your answer to the nearest whole percent.

Answer 32

Explanation

The percent profit is (5/15)(100) = 33.333… = 33.3… percent, which is 33%, to the nearest whole percent. Thus, the correct answer is 33% (or equivalent).

If you use the calculator and the Transfer Display button, the number that will be transferred to the answer box is 33.333333, which is incorrect since it is not given to the nearest whole percent. You will need to adjust the number in the answer box by deleting all of the digits to the right of the decimal point (using the Backspace key).

Also, since you are asked to give the answer as a percent, the decimal equivalent of 33 percent, which is 0.33, is incorrect. The percent symbol next to the answer box indicates that the form of the answer must be a percent. Entering 0.33 in the box would give the erroneous answer 0.33%.

Question 33

Sample Questions
Directions: Enter your answer as an integer or a decimal if there is a single answer box OR as a fraction if there are two separate boxes — one for the numerator and one for the denominator.

To enter an integer or a decimal, either type the number in the answer box using the keyboard or use the Transfer Display button on the calculator.

First, click on the answer box — a cursor will appear in the box — and then type the number.
To erase a number, use the Backspace key.
For a negative sign, type a hyphen. For a decimal point, type a period.
To remove a negative sign, type the hyphen again and it will disappear; the number will remain.
The Transfer Display button on the calculator will transfer the calculator display to the answer box.
Equivalent forms of the correct answer, such as 2.5 and 2.50, are all correct.
Enter the exact answer unless the question asks you to round your answer.
To enter a fraction, type the numerator and the denominator in the respective boxes using the keyboard.

For a negative sign, type a hyphen; to remove it, type the hyphen again. A decimal point cannot be used in a fraction.
The Transfer Display button on the calculator cannot be used for a fraction.
Fractions do not need to be reduced to lowest terms, though you may need to reduce your fraction to fit in the boxes.

SQ 5
Working alone at its constant rate, machine A produces k liters of a chemical in 10 minutes. Working alone at its constant rate, machine B produces k liters of the chemical in 15 minutes. How many minutes does it take machines A and B, working simultaneously at their respective constant rates, to produce k liters of the chemical?

Answer 33

__ minutes

Machine A produces k/10 liters per minute, and machine B produces k/15 liters per minute. So when the machines work simultaneously, the rate at which the chemical is produced is the sum of these two rates, which is k/10 + k/15 = k(25/150) = k/6 liters per minute. To compute the time required to produce k liters at this rate, divide the amount k by the rate k/6 to get k / k/6 = 6 Therefore, the correct answer is 6 minutes (or equivalent).

One way to check that the answer of 6 minutes is reasonable is to observe that if the slower rate of machine B were the same as machine A’s faster rate of k liters in 10 minutes, then the two machines, working simultaneously, would take half the time, or 5 minutes, to produce the k liters. So the answer has to be greater than 5 minutes. Similarly, if the faster rate of machine A were the same as machine B’s slower rate of k liters in 15 minutes, then the two machines, would take half the time, or 7.5 minutes, to produce the k liters. So the answer has to be less than 7.5 minutes. Thus, the answer of 6 minutes is reasonable compared to the lower estimate of 5 minutes and the upper estimate of 7.5 minutes.

Question 34

Data Interpretation Sets
Description
Data Interpretation questions are grouped together and refer to the same table, graph or other data presentation. These questions ask you to interpret or analyze the given data. The types of questions may be Multiple-choice (both types) or Numeric Entry.

Tips for Answering

Answer 34

Tips for Answering
Scan the data presentation briefly to see what it is about, but do not spend time studying all of the information in detail. Focus on those aspects of the data that are necessary to answer the questions. Pay attention to the axes and scales of graphs; to the units of measurement or orders of magnitude (such as billions) that are given in the titles, labels and legends; and to any notes that clarify the data.
Bar graphs and circle graphs, as well as other graphical displays of data, are drawn to scale, so you can read or estimate data visually from such graphs. For example, you can use the relative sizes of bars or sectors to compare the quantities that they represent, but be aware of broken scales and of bars that do not start at 0.
The questions are to be answered only on the basis of the data presented, everyday facts (such as the number of days in a year) and your knowledge of mathematics. Do not make use of specialized information you may recall from other sources about the particular context on which the questions are based unless the information can be derived from the data presented.

Question 35

Data Interpretation Sets
Sample Questions
Directions: Questions 1 to 3 are based on the following data.

Figure 8
ANNUAL PERCENT CHANGE IN DOLLAR AMOUNT OF SALES AT FIVE RETAIL STORES FROM 2006 TO 2008
Table with information on the annual percent change in dollar amount of sales by store for the years 2006 through 2008
Store Percent Change Percent Change
from 2006 to 2007 from 2007 to 2008
P 10 –10
Q –20 9
R 5 12
S –7 –15
T 17 –8

1. If the dollar amount of sales at Store P was $800,000 for 2006, what was the dollar amount of sales at that store for 2008 ?

(A) $727,200
(B) $792,000
(C) $800,000
(D) $880,000
(E) $968,000

Answer 35

Explanation

According to Figure 8, if the dollar amount of sales at Store P was $800,000 for 2006, then it was 10 percent greater for 2007, which is 110 percent of that amount, or $880,000. For 2008 the amount was 90 percent of $880,000, which is $792,000. The correct answer is Choice B, $792,000.

Note that an increase of 10 percent for one year and a decrease of 10 percent for the following year does not result in the same dollar amount as the original dollar amount because the base that is used in computing the percents is $800,000 for the first change but $880,000 for the second change.

Question 36

See Q 1 for data set (card 35) At Store T, the dollar amount of sales for 2007 was what percent of the dollar amount of sales for 2008 ?

Give your answer to the nearest 0.1 percent.

Answer 36

Explanation

If A is the dollar amount of sales at Store T for 2007, then 8 percent of A, or 0.08A is the amount of decrease from 2007 to 2008. Thus A - 0.08A = 0.92A is the dollar amount for 2008. Therefore, the desired percent can be obtained by dividing A by 0.92A which equals A/0.92A = 1/0.92 = 1.0869565… Expressed as a percent and rounded to the nearest 0.1 percent, this number is 108.7%. Thus, the correct answer is 108.7% (or equivalent).

Question 37

See Q 1 for data set (card 35)
Based on the information given, which of the following statements must be true?

Indicate all such statements.

(A) For 2008 the dollar amount of sales at Store R was greater than that at each of the other four stores.
(B) The dollar amount of sales at Store S for 2008 was 22 percent less than that for 2006.
(C) The dollar amount of sales at Store R for 2008 was more than 17 percent greater than that for 2006.

Answer 37

Explanation

For Choice A, since the only data given in Figure 8 are percent changes from year to year, there is no way to compare the actual dollar amount of sales at the stores for 2008 or for any other year. Even though Store R had the greatest percent increase from 2006 to 2008, its actual dollar amount of sales for 2008 may have been much smaller than that for any of the other four stores, and therefore Choice A is not necessarily true.

For Choice B, even though the sum of the two percent decreases would suggest a 22 percent decrease, the bases of the percents are different. If B is the dollar amount of sales at Store S for 2006, then the dollar amount for 2007 is 93 percent of B, or 0.93B and the dollar amount for 2008 is given by (0.85)(0.93)B which is 0.7905B. Note that this represents a percent decrease of 100 - 79.05 = 20.95 percent, which is not equal to 22 percent, and so Choice B is not true.

For Choice C, if C is the dollar amount of sales at Store R for 2006, then the dollar amount for 2007 is given by 1.05C and the dollar amount for 2008 is given by (1.12)(1.05)C which is 1.176C Note that this represents a 17.6 percent increase, which is greater than 17 percent, so Choice C must be true.

Therefore, the correct answer consists of only Choice C (The dollar amount of sales at Store R for 2008 was more than 17 percent greater than that for 2006).

Question 38

Add”Using the calculator” once pictures can be added to cards

Answer 38

The following guidelines are specific to the on-screen calculator in the computer-based test:

When you use the computer mouse or the keyboard to operate the calculator, take care not to mis-key a number or operation.
Note all of the calculator’s buttons, including Transfer Display.
The Transfer Display button can be used on Numeric Entry questions with a single answer box. This button will transfer the calculator display to the answer box. You should check that the transferred number has the correct form to answer the question. For example, if a question requires you to round your answer or convert your answer to a percent, make sure that you adjust the transferred number accordingly.
Take note that the calculator respects order of operations, as explained below.