Quantitative Reasoning

Answer: −6

1. − 6
Follow PEMDAS, and be careful when dealing with negative numbers.

6(−3 + 1)− 6(3 − 4) = 6(−2)− 6(−1) =
−12 −(−6) = −12 + 6 = −6

Answer: 2

2. 2

Using PEMDAS, first multiply 4 by 2. The expression is now |6 − 8|. Then subtract 8 from 6 to get −2. Finally, take the absolute value of −2 to get 2.

Answer: 4

3. 4

Dividing by a fraction is equal to multiplying by the reciprocal of that fraction. So, dividing by is equal to multiplying by :

Answer: 3

4. 3

The least common multiple of 6 and 15 is 30. Every multiple of 30 will also be a multiple of both 6 and 15. That means there are only three two-digit multiples of both 6 and 15: 30, 60, and 90.

Answer: 2

5. 2

When multiplying integers, at least one integer must be even to get an even result. If a and c are odd, then b must be even for a × b × c to be even. That means the smallest value of b, which must be positive, is 2.

Answer: 4

6. 4

The positive factors of 54 are 1, 2, 3, 6, 9, 18, 27, and 54. Of those, four (1, 3, 9, and 27) are odd.

Answer: 23

7. 23

Any multiple of 10 will have prime factors of 5 × 2. In this case, 46,000 = 46 × 10 × 10 × 10 = 23 × 2 × 5 × 2 × 5 × 2 × 5 × 2. The number 23 is prime and cannot be broken down further.

Answer: 6

8. 6

Whenever there are exponents in a fraction, see if it is possible to get a common base in both the numerator and the denominator. Here, the denominator has a base of 6. There are two ways to simplify the numerator to get a base of 6:

Either way, the final result is .

Answer: 1

9. 1

Numbers that leave a remainder of 2 when divided by 5 are numbers that are 2 greater than a multiple of 5, e.g., 7 (5 + 2), 12 (10 + 2), 17 (15 + 2) . . . Similarly, numbers that leave a remainder of 3 are 3 greater than a multiple of 5, e.g., 8, 13, 18 . . .

Pick any two valid numbers to test a − b. If a = 7 and b = 8, then a − b = 56. When 56 is divided by 5, the result is 11 with a remainder of 1 (as 56 is 1 greater than 55, a multiple of 5).

This will work out for any values of a and b. On Test Day, it may be tempting to try a second set of values. However, when the correct answer is a single number, there is no need to do so. The GRE will not play tricks, and every valid set of numbers will lead to the same result. Have confidence with the numbers you selected and move on.

Answer: 4

10. 4

|x| is the distance between 0 and x on a number line. If |x| < 6, then x must be less than 6 units away from 0. If x is an integer, that means it is any integer from −5 to 5. If |x| > 3, then x must be more than 3 units away from 0. That means x can be 4 or greater, or it can be −4 or less. To satisfy both conditions, x can be −5, −4, 4, or 5.

Answer: 9,940

11. 9,940
The largest 4-digit number is 9,999. When 9,999 is divided by 71, the result is 140 with a remainder of 59. That means 9,999 is 59 greater than the nearest multiple of 71. Subtracting 59 from 9,999 will produce the greatest 4-digit multiple of 71. Another way to approach this, after doing the division, is to recognize that 71 − 141 would produce a number larger than 9,999, so the largest 4-digit multiple of 71 would be 71 − 140.

Answer: −30

12. − 30

Use FOIL to multiply:

Answer: 42

13. 42

The mixture of radicals and exponents can be confusing. Exponents are usually easier to deal with, so start by converting each radical into an exponential term:

63, 56, and 7 are all multiples of 7, so factor out 7 from each term:

. Use the rules of exponents to finish:

Answer: 1,076

14. 1,076

When the digits are reversed, the new number will have the same number of digits. No two 3-digit numbers add up to 7,777 (as the largest 3-digit number is 999, and 999 plus 999 is only 1,998), so x must be a 4-digit number. Let A, B, C, and D represent the digits of x. Adding x to its reverse would look like this.

ABCD + DCBA = 7777

The smallest 4-digit numbers would begin with 1, so A should be 1.

1BCD + DCB1 = 7777

For the sum to end in 7, D would have to be 6.

1BC6 + 6CB1 = 7777

After that, the smallest possible value of B would be 0. In that case, C would have to be 7.

1076 + 6701 = 7777

Thus, 1,076 is the smallest possible value of x.

Answer: 16

15. 16

Start by converting .125 to a fraction. Once that’s done, use the rules of radicals and exponents to simplify:

Answer:

1.

In order to add or subtract fractions, first convert them so that they have a common denominator. The denominators in this question (3, 5, 2, and 15) are all factors of 30, so multiply the numerator and denominator of each fraction by the number that will result in a denominator of 30: .

Now add all the numerators: .

Answer: |x⁴|

2. |x⁴|

If x is between −1 and 1, it will be a fraction. As positive proper fractions are raised to greater powers, they become smaller and smaller. Negative fractions raised to an even power are positive and raised to an odd power are negative. However, in either case, their absolute value continues to decrease as the exponent gets greater. Since this question asks about absolute values, it does not matter whether the fraction is negative or positive. Because 4 is less than 5, |x⁴| > |x⁵|.

Answer: 15

3. 15

Convert the part-to-part ratio to the part-to-whole ratio for red pens: . Set up the proportion , and cross multiply to get 5R = 75 and R = 15.

Answer: 20%

4. 20%

Set up the proportion . Cross multiply to get 1,700 = 85x. Divide both sides by 85 to find that x = 20.

Answer:

5.

First, multiply the two fractions in the numerator by multiplying the two numerators and the two denominators to get which reduces to . Invert the fraction in the denominator and multiply: .

Answer: 7

6. 7

The formula for computing averages is . Rearrange this to Sum of values = Number of values × Average. Including x, there are 6 values, so 6 + 3 − 2 + 5 + 11 + x = 6 × 5. Thus, 23 + x = 30 and x = 7.

Another way to handle this question is with the balance approach. The average is 5, so the known numbers above this are 6 and 11. These are 1 and 6 above the average, respectively. The known values below are 3 and −2, which are 2 and 7 below the average, respectively. Without x, then, the values are 1 + 6 − 2 − 7 = −2, which means 2 below the average overall. Thus, x must be 2 above the average in order to balance, making x equal to 5 + 2 = 7.

Answer: 3 hours and 54 minutes

7. 3 hours and 54 minutes

Use the Time-Speed-Distance formula in the format to determine the time for each leg of Abdul’s trip. For the first part, . For the second leg, . The distance for the third part of the trip is not given, but it can be calculated by subtracting the two known distances from the total: D₃ = 200 − 30 − 50 = 120. Apply the equation for time: . Rather than trying to work with a common denominator to add the times, convert to minutes: . Divide 234 minutes by 60 minutes in an hour to get 3 hours with 54 minutes remaining.

Answer: $99.14

8. $99.14

The first discount is 15%. Since 15% of $120 is $18, this markdown reduces the price to $102. The second discount is applied to the reduced price: 10% of $102 is $10.20, so the price, before tax, would be $102 − $10.20 = $91.80. The 8% tax on that amount is $7.344, so the final price would be $91.80 + $7.34 = $99.14, since a customer would not pay a fraction of a cent. (Note: If you did not want to open the calculator for the final calculation, you could use the distributive property to calculate the tax: 8% of $91.80 is .08(90 + 1 + 0.8) = 7.2 + .08 +.064, which rounds to $7.34.)

Answer: 3 hours

9. 3 hours

The simplified formula for combined work is . This question provides values for T and the time to complete the task for one worker. Convert the total time to 72 minutes and Jennifer’s solo time to 120 minutes and plug these values into the equation: . Cross multiply: 72(120 + B) = 120B. Distribute the multiplication: 72(120) + 72B = 120B. Group the terms with the variable: 72(120) = 120B − 72B = 48B. Divide both sides by 24 to get 3(120) = 2B, so B = 3(60) minutes, which is 3 hours.

Answer: 20,000

10. 20,000

Multiply the values in the numerator and in the denominator, paying close attention to the decimal placement. To multiply 0.003 × 10⁴, move the decimal point 4 places to the right, which requires adding a zero. The numerator becomes 30 × 2.4 = 72. After multiplying the first terms, the denominator is 0.36 × 10−². To simplify the division, convert this to 36 × 10⁻⁴. A negative exponent in a denominator is equivalent to the same positive exponent in the numerator, so the expression simplifies to .

Answer: 27

11. 27

Restate the ratios as fractions: initially , so y = 4x. If x were increased by 1, the new ratio would be . Cross multiply to get y = 3x + 3. Set the two values of y equal to each other: 4x = 3x + 3, which simplifies to x = 3. Since y = 4x, y = 4(3) = 12. Because the question states the increase of x as a hypothetical statement, use the initial value for x to calculate z: 3 + 2(12) = 27.

Answer: 12 hours

12. 12 hours

Since there are 3 workers in this question, use the formula for adding rates of multiple workers: . The question states that the three women could compete the task together in 4 hours, and it provides the information that K = 8. It then states that Latasha can complete the task in half the time that Melanie can. Set M = 2L and plug the values into the equation to get . Multiply each term by the least common multiple of the denominators, 8L, to clear the fractions: . This simplifies to 2L = L + 8 + 4, so L = 12.

Answer: 300

13. 300

Since there will not be any partial coins, the number of coins in the bag must be divisible by 5, 3, and 4. The least common multiple (LCM) of those numbers is 3 × 4 × 5 = 60. If there were 60 coins in the bag, there would be pennies, nickels, and dimes. The total number of coins other than quarters is 24 + 20 + 15 = 59 coins, so there could only be 1 quarter. The question specifies that there are 5 quarters, so there must be 5 × 60 = 300 coins.

The question could also be approached using algebra, by setting up the equation , where C represents the total number of coins. Multiply through by the LCM of 60 to get 60C = 24C + 20C + 15C + 300. Combine like terms: 60C = 59C + 300, so C = 300.

Answer: 18:11

14. 18:11

To facilitate computations, ratios should be stated in fraction format, so write . Now solve by manipulating the ratios so that they share a numerator or denominator. Multiply both parts of A:B by 2 to get . Flip the ratio B:C so that . Since both fractions have the same denominator, A and C can be compared directly, and the ratio of A to C . Next, convert this to . Flip the ratio C:D to get . Again, since the denominators are equal, .

Alternatively, “chain” the ratios to find the answer. Looking at just the variables rather than the numbers, note that because the Bs and Cs cancel out. Plug the values of the variables into this equation:

Answer: 25 mpg

15. 25 mpg

Make the necessary conversions step-by-step. One way to start is by converting km to miles: . (Don’t worry about carrying through all the calculations along the way; wait to see if something cancels out later.) Now convert this rate to gallons/mile: . If you notice that 3.8 is 40 × 0.095, you could simplify this fraction to . The question asks for the rate in miles per gallon, so invert this rate to get . If you use the calculator, the result is 0.04 gal/mi. Invert that by calculating 1 ÷ 0.04 = 100 ÷ 4 = 25 mpg.

Answer: 1

1. 1

None of the operations in parentheses can be completed, so the first step is to distribute the factors across the terms in parentheses: 3x − 18 + 8 − 2 + 4x = 7 − 4x − 8. Combine like terms: 7x − 12 = −1 − 4x. Add 4x and 12 to each side of the equation: 7x + 4x − 12 + 12 = −1 + 12 − 4x + 4x. So, 11x = 11 and x = 1.

Answer: (a − 2b)(c + d)

2. (a − 2b)(c + d)

Notice that the first 2 terms each contain the variable c, so convert these to (c)(a − 2b) by factoring out c. The last 2 terms each contain the variable d, so convert these to (d)(a − 2b). The expression can be restated as (c)(a − 2b) + (d)(a − 2b). Now (a − 2b) is a common term, so factor it out to get (a − 2b)(c + d).

Answer: 4

3. 4

Substitute 3 for x in the expression: 3³ − 3² − 7(3 − 1) = 27 − 9 − 7(2) = 4.

Answer: 2

4. 2

Add 7 to both sides of the inequality to get 3x ≤ 6. Divide both sides by 3 to determine that x ≤ 2. Since “≤” means “greater than or equal to,” the minimum value of x is 2.

Answer: −16

5. −16

Substitute 2x for y in the equation: 3(2x + 6x) − 7(2(2x) + 3) = 11. Simplify: 3(8x) − 7(4x + 3) = 11. Complete the multiplications: 24x − 28x − 21 = 11. Finally, add 21 to both sides of the equation and combine like terms to get −4x = 32. Divide both sides by −4 to get x = −8. The question asks for the value of y, which is 2x, so multiply −8 by 2 to get y = −16.

Answer:

6.

First, get rid of the fraction by multiplying everything by 4: . The equation becomes 4xy + 7 + 3x − 8y = 4. Add 8y and subtract 7 from both sides to get 4xy + 3x = 4 + 8y − 7. Factor out x and combine like terms: x(4y + 3) = 8y − 3. Finally, divide both sides by (4y + 3), so .

Answer: 2, −2

7. 2, −2

This is the difference of two squares, a pattern equation that is likely to appear on the GRE. Remember that a² − b² = (a + b)(a − b). Since 9x² = (3x)² and 36 = (6)², the equation becomes (3x + 6)(3x − 6) = 0. If 3x + 6 = 0, then 3x = −6 and x = −2; if 3x − 6 = 0, then 3x = 6 and x = 2. A shortcut would be to add 36 to both sides to get 9x² = 36 then dividing both sides by 9 to get x² = 4. Remember that x can be either +2 or −2!

Answer: 3

8. 3

This question presents 2 linear equations with 2 variables. Systems of equations such as this can be solved by combination or substitution. To combine the 2 equations, adapt one of the equations to make the absolute value of one of its variable’s coefficients the same as the absolute value of that variable’s coefficient in the other equation. Here, the terms of the first equation can be multiplied by 2 to get 2x + 8y = 38. Subtract the second equation from this so that the 2x cancels out:

To solve using substitution, rearrange the first equation to get x = 19 − 4y. Substitute that for x in the second equation: 2(19 − 4y) − y = 11, so 38 − 8y − y = 11, 27 = 9y, and y = 3.

Answer: 6

9. 6

Symbolism questions such as this are algebra questions that can be solved by substitution. In order to find the value of (r), first determine the value of r. Since 3r − 4 = 5, 3r = 9 and r = 3. Substitute 3 for x to get . Simplify to .

Answer: 14

10. 14

A series of numbers beginning with the third number in the series is composed of the previous number in the series plus the second prior number plus 3. So, in order to calculate the value of any number in the series, the value of the 2 preceding numbers is needed. To find the value of s₄, the values of s₃ and s₂ must first be determined. The question provides values for s₁ and s₂, so s₃ = s₂ + s₁ + 3, which is 4 + 0 + 3 = 7. So s₄ = 7 + 4 + 3 = 14.

Answer: 5

11. 5

The question has three variables and three linear equations, so you can solve for any of the variables as long as the equations are distinct. Start by simplifying the first equation: x − 2z = 2y − 2z, so x = 2y. Now substitute 2y wherever x appears in the other equations to end up with two equations with two variables: 2(2y) − 6y + z = 1 simplifies to z − 2y = 1. Similarly, 3(2y) + y − 2z = 4 becomes 7y − 2z = 4. Multiply the first of these two equations by 2 to get 2z − 4y = 2. Now add the equations:

Plug that value into 2z − 4y = 2: 2z − 8 = 2, so 2z = 10 and z = 5.

Answer: 3

12. 3

When confronted with a function of a function, as in this question, start from the inside and work outward. Substitute 2 for y to get ##(2) = 2² − 2(2) + 3 = 4 − 4 + 3 = 3. Now evaluate .

Answer:

13.

First rearrange the equation into standard quadratic form so that it can be factored: 6x² − 11x − 10 = 0. Identify the factors of the last term and those of the coefficient of the first term. The factors of −10 are (−10 and 1), (10 and −1), (−5 and 2), and (5 and −2). The factors of 6 are (1 and 6), (−1 and −6), (2 and 3), and (−2 and −3). Find the pairs that will result in an algebraic sum of −11, which is the coefficient of the middle term. Note that 3 × (−5) = −15 and 2 × 2 = 4; −15 + 4 = −11. So (2x − 5)(3x + 2) are the correct factors. Set each factor equal to zero to find the possible values of x. If 2x − 5 = 0, then ; if 3x + 2 = 0, then .

Answer: 85

14. 85

To predict a forward value for the sequence, determine the equation that quantifies the relationship between any value and “the two prior values.” Looking at the overall trend, the values always increase, and the rate of increase accelerates. Start with s_3, which could be the simple sum of s₁ and s₂. However, moving along to s₄, that value is greater than the sum of s₂ and s₃, so there must be a multiplier or exponent in the formula. Continuing forward, try to identify a pattern relating to how much greater each value is than the sum of the two prior values. One way to do this is to set up a table showing the “missing” quantities.

Look at the values on the far right: 1, 3, 5, and 11. Those are the values of s_{x − 2}. So the equation for this series is sx = sx_{− 1} + sx_{− 2} + sx_{− 2} = sx_{− 1} + 2(sx_{− 2}). Apply this formula to calculate the value of s₉: s₉ = 43 + (2)21 = 43 + 42 = 85.

Answer: 37

15. 37

Represent the three people’s number of tokens by the first letters of their names and translate the information given into algebraic equations: R = T + 3, C = 2R, and C = 3T − 1. Since there are three distinct linear equations and three variables, you can solve this system of equations. Substitute 2R for C in the third equation: 2R = 3T − 1. Double the first equation to get 2R = 2T + 6. This gives two different equations for 2R, so set them equal to each other: 2T + 6 = 3T − 1. Subtract 2T from both sides and add 1 to both sides to get T = 7. This means that R has 10 tokens, 3 more than T, and C has twice as many as R, which is 20. The total number of tokens among the three people is 7 + 10 + 20 = 37.

Answer: 14

1. 14

The range of a group of numbers is the positive difference between the largest and smallest values of the group. For these numbers, that is 8 − (−6) = 8 + 6 = 14.

Answer: −4

2. −4

Arrange the numbers in ascending order: −5, −4, −2, 0, 2, 3, 7, 7. Because there are 8 numbers, the median will be the average of the 4th and 5th numbers, which is . The mode is 7 since that is the only value that appears more than once. The range is 7 − (−5) = 12. Plugging these values into the equation given in the question yields 1 + 7 − 12 = −4.

Answer: 85

3. 85

The total number of students in the table is 40. There were 100% − 80% = 20% who scored at the 80th percentile or above, so that was 0.2 × 40 = 8 students. Count down starting with the highest-scoring category: the total number of students who scored over 94, 90–94, or 85–89 was 3 + 2 + 3 = 8. So the minimum score to obtain a grade of A was 85.

Answer: 120

4. 120

The two types of bulbs that are less than 60 watts are 25 and 40 watts. The percentage of the bulbs in those two power levels is 10% + 20% = 30%. Since the total number of bulbs is 400, the number of bulbs less than 60 watts is 0.30 × 400 = 120.

Answer: 2009−2010

5. 2009–2010

Examine the chart to find the greatest vertical difference between two adjacent years. This is also the line segment that has the greatest slope. Sales in 2009 were $450,000, and sales in 2010 were $525,000. This increase of $75,000 was greater than for any other equivalent period shown.

Answer: 19

6. 19

An average (mean) is the sum of all the values in a group divided by the number of values. In this question, the average is given but the sum is not. Rearrange the formula for averages: sum = (number of values) × (average), which is 6 × 15 = 90. Since there are 6 numbers in the group, one-third of that is 2. If those 2 numbers are increased by 12 each, then the sum is increased by 24. Divide the new sum of 90 + 24 = 114 by 6 to obtain the new average (mean) of 19. (Perhaps you recognized that if one-third of the numbers were increased by 12, the overall average would increase by 12 ¸ 3 = 4 to the new value of 19.)

Answer: 1:4

7. 1:4

Box plots are drawn to represent quartiles. Therefore, the range from Q₃ to G represents the 75th percentile and above, and the range from L to Q₁ represents the 25th percentile and below. G − Q₃ is 8 − 7 = 1 and Q₁ − L is 1 −(−3) = 4, so the ratio is 1:4.

Answer: 1.75

8. 1.75

Calculate the average by multiplying each value by the number of times it occurred, adding those products, and dividing by the total number of minutes, which is 60. First, get the products: 0(15) + 1(19) + 2(12) + 3(6) + 4(3) + 5(2) + 6(0) + 7(2) + 8(1) = 0 + 19 + 24 + 18 + 12 + 10 + 0 + 14 + 8 = 105. Divide by 60 to obtain the average: 105 ÷ 60 = 1.75.

Answer: 26%

9. 26%

Relative frequency is the number of values with the characteristic of interest expressed as a percent of the total number of values. For this question, that is the number of customers between noon and 2:00 pm expressed as a percent of the total customers. First, find the total number of customers: 22 + 38 + 60 + 44 + 33 + 27 + 31 + 38 + 49 + 58 = 400. The total number of customers between noon and 2:00 pm is 60 + 44 = 104, so the percent of the total during that time is (104 ÷ 400) × 100% = (104 ¸ 4)% = 26%.

Answer: 2016

10. 2016

The numbers of left-handed students are shown in the dark-shaded parts of the columns and the right-handed students are in the light-shaded portions. The correct answer will be the year in which the dark-shaded bar is the tallest relative to the light-shaded bar. This would also be the year in which the dark-shaded portion is tallest relative to the total height of the bar, since the former is a part-to-part ratio and the latter is a part-to-whole ratio. This comparison may be faster to visualize than to calculate. A quick glance at the chart shows that the ratios for 2015 and 2019 are much smaller than the others. Compare 2016 and 2017: both had the same number of “lefties,” but there were more “righties” in 2017. Similarly, compare 2016 and 2018: both had the same total number of aspiring calligraphers, but there were fewer left-handed students in 2018. Therefore, 2016 had the highest ratio of left-handed students.

Answer: 0.5

11. 0.5

Notice that each list consists of 8 numbers, 6 of which are 3, and 2 that have other values. However, in each list those other numbers average 3 (1 and 5 in the first list, 2 and 4 in the second), so the mean of each list is 3. This fact greatly simplifies calculating the differences from the mean for each list. For list A, the differences are 2, 0, 0, 0, 0, 0, 0, and 2; those of list B are 1, 0, 0, 0, 0, 0, 0, and 1. The sum of the squares of these differences is 2² + 2² = 4 + 4 = 8 for list A, and 1² + 1² = 1 + 1 = 2 for list B. The averages of the squared differences are 8 ¸ 8 = 1 and 2 ÷ 8 = 0.25. The standard deviations of the two lists are the non-negative square roots of these averages. For list A, the square root of 1 is 1; for list B, the square root of 0.25 is 0.50. Therefore, the difference between the two standard deviations is 1 − 0.5 = 0.5.

Answer: 38.5

12. 38.5

Because of the numbers involved, this question can be efficiently approached using the balance method for averages. Since the overall average was 37.4, the 7th graders’ average of 36.4 was 1.0 below the school average. Those students were cumulatively 1.0 × 110 = 110 points below the overall average, so the 8th graders had to be cumulatively 110 points above average. Since there were 100 of them, they had to average 110 ÷ 100 = 1.1 above 37.4, which is 38.5.

Answer: 8

13. 8

The average of the group is 3 and there are 8 values (including x and y), so the sum of the values is 3 × 8 = 24. The sum of the known values is (−1) + 3 + 0 + (− 2) + 4 + 2 = 6, so x + y = 24 − 6 = 18. There are now two equations: x − y = 2 and x + y = 18. To solve using substitution, rearrange the first equation to x = y + 2. Substitute that for x in the second equation: (y + 2) + y = 18. So 2y = 16 and y = 8. Alternatively, solve for y by combining the two equations:

Answer: 18.5 parts/billion

14. 18.5 parts/billion

Although there is some variation, the overall trend of the graph is linear and upward. The best estimate of the slope is the overall trend from 1950 to 2010. In 2010, the concentration was 18.0 and in 1950 it was 12.0. Thus, over a period of 60 years, the increase was 18.0 − 12.0 = 6.0, which equates to 0.1 per year. The increase projected from 2010 to 2015 is 5 × 0.1 = 0.5. Add that amount to the 2010 value to get 18.0 + 0.5 = 18.5 as the estimated value for 2015.

Answer: 10

15. 10

In order to determine the range of the group of numbers, the values of x and y are needed. Rearrange the equation for x to: x² + 8x + 16 = 0. This factors to (x + 4)² = 0, so the only value for x is −4. At this point, the smallest known value in the group is −4, and the greatest is 6. Set up the equation for y in standard quadratic format: y² − y − 6 = 0. This factors out to (y − 3)(y + 2), so y can be either 3 or −2. Since both of these values are inside the range of values already known, they have no effect on the overall range of the group, which is 6 −(−4) = 10.

Answer: 120

1. 120

Since the books are just being put in a suitcase, order doesn’t matter, and the combinations formula can be used.

Answer: 360

2. 360

Here, order does matter since the candles are distinctly colored and being placed into slots on the candelabras. There are 6 possible candles for the first slot, 5 for the second, 4 for the third, and 3 for the 4th. To find the total number of possibilities, multiply each of the possibilities for the four slots together (6 × 5 × 4 × 3) to get 360.

Answer:

3.

There are 6 equally likely outcomes for one roll of a fair die. One of these outcomes is 6, so the probability of rolling a 6 is . The question asks for the probability of rolling a 6 on the first roll and the probability of rolling a 6 on the second roll. These events are independent, so multiply the two probabilities: .

Answer:

4.

A roll of 2 or 4 or 6 would meet the criterion in the question. These are mutually exclusive outcomes, so add their probabilities: .

Answer: 18

5. 18

The number of options Pablo has for the beverage is simply 3, because he can only select one item of the 3 that are available to him. To calculate the number of options for the grain bars, use the combinations formula, because the order in which Pablo selects the 2 grain bars does not matter: . Since Pablo gets to choose a beverage and two grain bars, and for each of the 3 beverages he can choose from 6 different options of grain bars, multiply the two numbers of choices: 3 × 6 = 18.

Answer: 9

6. 9

The formula based on the inclusion-exclusion principle for sets states that |A ∪ B| = |A| + |B|−|A ∩ B|. Substitute the numbers given in the question: A ∪ B| = 7 + 5 − 3 = 9.

Answer:

7.

There are two possible outcomes for each flip of the coin: heads or tails. You might be tempted to think that the total number of possible outcomes for 4 consecutive flips would be 4 × 2 = 8, but remember that the coin is flipped once and then a second time and then a third time and then a fourth time, so the total number of possible outcomes is actually 2⁴ = 16. If only one head is the result, that could occur on any one of the 4 flips, so there are 4 desired outcomes. The probability is thus .

Answer:

8.

The most efficient way to approach this question is to determine the probability that Latisha will not draw a blue marble in 3 attempts and subtract that from 1 to get the probability that she will. There are 6 total marbles, of which 4 are not blue, so the probability that Latisha will not draw a blue marble on the first attempt is . For the second attempt there will only be 5 marbles remaining, 3 of which are not blue, so the probability of not drawing a blue marble on the second attempt is . By the time Latisha tries a third time, there will be 2 blue marbles among the 4 that are left, so the probability of drawing a marble that is not blue is . In order not to draw any blue marbles, Latisha will have to be unsuccessful on the first and second and third attempts, so the probability of that happening is . Therefore, the probability that Latisha will be successful is .

Answer: 12

9. 12

The formula for overlapping sets is Total = Group A + Group B − Both + Neither. Since the question defines crossover as country and pop, the rock songs can be considered “neither.” Plug the given values into the equation: 60 = 30 + 18 − Crossover + 24. Add Crossover to both sides of the equation and subtract 60 from both sides to get: Crossover = 30 + 18 + 24 − 60 = 12.

Answer: 6

10. 6

After 1 blue chip was removed, there were 9 chips left. If the probability of drawing another blue chip from those remaining 9 was then , there must have been blue chips, and 9 − 3 = 6 red chips remaining. Since no red chips were drawn, the original number of red chips must also have been 6.

Answer: 30%

11. 30%

Set up a table to organize the information and derive the values needed to answer this overlapping sets question. The question states that 60% was the total count for A. Put that at the bottom of the “For A” column. The question further states that two-thirds of those voters, which is 40%, also voted for B, so enter that value in the chart as well, at the top of the “For A” column. The total of all the categories must be 100%, as shown in the bottom right cell of the table. Now use these figures to calculate other cells. If 60% of the total voted for A, then 40% of the total did not vote for A. That goes at the bottom of the “Not for A” column. The question does not state what percentage voted for B, but it does mention that B was “chosen by a majority,” so enter >50% in the Total column, in the “For B” row. Since 40% of that total is already represented in the “For B/For A” cell, the “For B/Not for A” cell must be greater than 10%, so enter that in the table. Finally, look at the middle cell. If 40% of the total did not vote for A, and at least 10% were “For B,” then less than 30% did not vote for either A or B. Put that in the middle cell.

Answer: 60

12. 60

This question represents a pattern of a counting problem with certain conditions or restrictions added. There are 6! ways to arrange 6 different items. However, in this case, many of those arrangements will appear identical. Consider the configuration in the question, AAABBC. If the A’s were not identical there would be 3! = 6 different-appearing ways to arrange them. However, since the A’s are identical, all 6 of those arrangements have the same appearance. So the number of arrangements must be reduced by a factor of 6. Similarly, there are 2 identical ways to set up any configuration of the B’s. The number of different-appearing arrangements is thus . This simplifies to .

Answer: 288

13. 288

This question involves the “groups of groups” pattern. First consider how many ways the groups (squads) can be arranged. Since there are 3 distinct squads, that is 3! = 3 × 2 × 1 = 6 different ways. For the squad that is permitted to choose any order they wish, there are 4! = 4 × 3 × 2 × 1 = 24 different ways they can line up. The squad that lines up by height can only have 2 variations and the remaining squad only has one way to line up within the squad. Therefore, the total number of ways that the platoon can line up is 6 × 24 × 2 × 1 = 288.

Answer: 0.6

14. 0.6

Use the formula for two independent events to calculate the probability that event B occurs. Designate the probability that A occurs as P_A, that B occurs as P_B, and the probability that at least one occurs as P_{A or B}. From the question, P_A = 0.5 and P_{A or B} = 0.8. The formula for P_{A or B} is: P_{A or B} = P_A + P_B − P_{A and B}. Since the events are independent, P_{A and B} = P_A × P_B. Thus, the formula can be written as P_A or B = P_A+ P_B − (P_A × P_B). Plug in the known values to get:

An alternative approach to this question would be to use the fact that the probability of neither event occurring is 1 − 0.8 = 0.2. Since this is equivalent to P_{Not A} × P_{Not B,} set up the equation: 0.2 = (1.0 − 0.5) × P_{Not B}. So 0.4 is P_{Not B} and 0.6 is P_B.

Answer:

15.

The desired outcome is R-S-T in any order. The probability of rolling an R (P_R) on any roll is . Similarly, and Thus the probability of rolling one of each is . There are 3! = 6 different orders in which R-S-T can be rolled, so the total probability of rolling R-S-T in any order is

Answer: (−5,−8)

1. (−5,−8)

The midpoint of a line segment is found by taking the average of the x-coordinates of the endpoints and the average of the y-coordinates of the endpoints. For m and n, the average of the x-coordinates is [−2 + (−8)] ¸ 2 = (−10) ÷ 2 = −5. The average of the y-coordinates is (−6 + −10) ÷ 2 = −16 ÷ 2 = −8.

Answer: 60

2. 60

The area of a triangle equals (base)(height) ÷ 2. For a right triangle, the base and height are the legs. This is an 8:15:17 right triangle, so the legs are 8 and 15. Thus, the area is (8)(15) ¸ 2 = 4(15) = 60.

Recognizing the 8:15:17 right triangle pattern was helpful, but the missing leg could also have been found via the Pythagorean theorem:

Answer:

3.

The area of a circle equals πr². Set this equal to 36 and solve for r:

This is the radius. The diameter is twice that, or .

Answer: 52

4. 52

Angles on one side of a straight line add up to 180°, so the interior angle of the triangle that is next to the 122° angle must equal 180° − 122°, or 58°. The angles of a triangle add up to 180° as well, so now the final angle of the triangle must be 180° − 70° − 58° = 52°. Angle a is vertical to this last angle, and vertical angles are always congruent. Thus, angle a is also 52°.

Answer: 200

5. 200

If the area of half a circle is 50π, then the area of the entire circle is twice that, or 100π. For a circle, area = πr², so the radius of the semicircle is 10. (100π = πr², so r² = 100 and r = 10.) This is also the height of the rectangle. The diameter of the semicircle is twice that, or 20. This is also the width of the rectangle. The area of a rectangle is width × height, which is 20 × 10 = 200.

Answer:

6.

In an equilateral triangle, all sides are equal and all angles are equal to 60°. The area is equal to . The base is equal to 12, but the height needs to be calculated. Drawing a height will create two 30°-60°-90° triangles.

The ratio of sides in a 30°-60°-90° triangle is For each smaller triangle, the side opposite the 30° angle (x) is 6. That means the height (opposite the 60° angle) is equal to . With the height determined, the area of the original equilateral triangle is equal to .

Answer: 55°

7. 55°

Because both pairs of lines are parallel, all of the acute angles are congruent and all of the obtuse angles are congruent. As a result, every obtuse angle is supplementary to every acute angle. Since p is obtuse and q is acute, it must be that p + q = 180°. Thus, q = 180° − 125° = 55°.

Answer: 1

8. 1

To find the point(s) of intersection between two functions in the coordinate plane, set them equal to each other, and solve:

This equation has only one solution: −5. Thus, there is only one point of intersection between the two parabolas.

Answer:

9.

Call the radius r and the length of a side of the square s. The question asks for the ratio of r to s, or . The area of the circle is πr² and the area of the square is s². The question states that these are equal:

πr² = s²

Take the square root of both sides to simplify:

Finally, divide both sides by s and by to find the ratio of r to s:

Answer: 1,600

10. 1,600

Draw a diagram to visualize the region:

This is a trapezoid. The area of a trapezoid equals the average of the bases times the height:

Answer: 121

11. 121

All four sides of a rhombus have equal length, so the length of each side here is 44 ÷ 4 = 11. The area of a rhombus equals base × height, and there’s no way that the height (the distance between two opposite sides) could possibly be longer than the base (the length of one side). The height would be maximized when it is equal to the length of one side, which occurs when adjacent sides are perpendicular, i.e., when the rhombus is a square. Thus, a rhombus’s area is maximized when the rhombus is a square, and the maximum area of the rhombus in the question is simply 11 × 11, or 121.

Answer:

12.

Let x be the radius of circle O. In that case, the following is true of circle O:

Circumference of circle O: 2πx

Area of circle O: πx²

The question says that circle P has an area four times as big, or 4πx². That means the following is true of circle P:

Radius of circle P:

Circumference of circle P: 2(2x)π = 4πx

The circumference of circle O is 2πx, which is half the circumference of circle P, 4πx.

Answer: 432 − 36π

13. 432 − 36π

The radius of circle A is also the height of rectangle ABCD, so AB = 12. The length of rectangle ABCD is three times this height, so BC = 3 × 12 = 36. That means the area of rectangle ABCD is 12 × 36 = 432.

The sector of the circle that’s inside the rectangle is one-fourth of the entire circle. That’s because angle A is a 90° angle, and 90° is one-quarter of 360°, the total number of degrees in a circle. The total area of the circle is πr² = π(12)² = 144π. One-fourth of that is 144π ÷ 4 = 36π.

The shaded area is this sector area subtracted from the total area of the rectangle, or 432 − 36π.

Answer:

14.

The area of a parallelogram is base × height, so . By the properties of 30°-60°-90° triangles, AE = 6. (We know triangle ADE is a 30°-60°-90° triangle because angle A is 60°, and angle E, formed by altitude DE, is 90°.) It follows that EB = AB − AE = 14 − 6 = 8. The area of a rectangle is also base × height, so the area of rectangle EDFB is .

Answer: 196

15. 196

The volume of a right cylinder is V = πr²h. In this case, the volume is 637π and the height is 13, so plug those values in for V and h, respectively, and solve for r:

The diameter of the circle is 2r, or 2(7) = 14. This is also the length of a side of square WXYZ. The area of a square is the length of one side squared, so the area of square WXYZ is 14², or 196.

Answer: B

1. B

Comparing the two quantities piece by piece, you find that the only difference is the third piece: −2 in Quantity A and −1 in Quantity B. You don’t know the value of x, but whatever it is, x² in Quantity A must have the same value as x² in Quantity B, and 2x in Quantity A must have the same value as 2x in Quantity B. Because any quantity minus 2 must be less than that quantity minus 1, Quantity B is greater than Quantity A. (B) is the correct answer.

Answer: C

2. C

Make the quantities look alike. If x = 2y, then −x = −2y. Replace the exponent −x in Quantity B with −2y. Thus, Quantity B becomes . A value with a negative exponent in the denominator is equivalent to the same value with a positive exponent in the numerator, so Quantity B can be restated as 2²y. Since 4 = 2², Quantity A can be written as (2²)y = 2²y. The quantities are identical, and (C) is correct.

Answer: D

3. D

Do the same thing to both quantities to make them look like the centered information. When you multiply both quantities by 5s, you get qrs in Quantity A and 15 in Quantity B. Because qrs could be any integer greater than 12, qrs could be greater than, equal to, or less than 15. (D) is correct.

Answer: D

4. D

Because angle X = angle Y, at least two sides of the triangle are equal. You can draw two diagrams with X and Y as the base angles of a triangle. In one diagram, make the triangle tall and narrow so that angle X and angle Y are very large and angle Z is very small. In this case, Quantity B is greater. In the second diagram, make the triangle short and wide so that angle Z is much larger than angle X and angle Y. In this case, Quantity A is greater. Because more than one relationship between the quantities is possible, the correct answer is (D).

Answer: C

5. C

Since ∠ ABC is 45°, ∠ BCA is 180° − 90° − 45° = 45°, so ABC is an isosceles right triangle. Because ED = DF, triangle DEF is also an isosceles right triangle. The formula for the area of a triangle is Area = base × height. Since the triangles in this question are isosceles right triangles, the formula for the area of a triangle can be restated as base² by substituting the base in for the height, since they are the same value. Given that the area of triangle ABC is four times the area of triangle DEF, set up the equation Even though there are two variables but only one equation, you can solve by Picking Numbers because only the ratio matters, not the actual values. Try 2 for the value of AC to get . Given the side ratios of an isosceles right triangle, , side . So Quantity A becomes Multiply both the numerator and the denominator by to get This is the same as Quantity B, so (C) is correct.

Answer: D

6. D

In a set of 35 consecutive integers, the median will be the number in the middle. Of the remaining 34 values, 17 will be less than the median and 17 will be greater.

Imagine that the range of values is 1 through 35. In this case, the first value is odd, and the values 1 through 17 are below the median, 18 is the median, and 19 through 35 are above the median. So there are 9 odd numbers and 8 even numbers below the median, the median itself is even, and there are 9 odd numbers and 8 even numbers above the median.

Now imagine that the range of values is 2 through 36. In this case, the first value is even, the median is 19, there are 9 even numbers and 8 odd numbers below the median, and there are 9 even numbers and 8 odd numbers above the median.

So, if the first integer is odd, there’s an 8 in 35 chance of picking an even number less than the median and a 9 in 35 chance of picking an odd number greater than the median. Quantity B would be greater than Quantity A. However, if the first integer is even, the odds are reversed, with a 9 in 35 chance of picking an even number less than the median and an 8 in 35 chance of picking an odd number greater than the median. That would make Quantity A greater than Quantity B. Because more than one relationship is possible, (D) is the correct answer.

Answer: A

7. A

The stimulus provides no actual prices for the different tickets, so use A to represent the regular price of an adult’s ticket and C to represent the regular price of a child’s ticket. The tickets are full price on Saturday, so Quantity A could be calculated as: 120A + 100C.

On Sunday afternoon, there’s a 20% discount on all tickets. That means the tickets will be 80% of their original price. Quantity B would thus be calculated as:

150(.8A) + 120(.8C) = 120A + 96C

Comparing the two results, both quantities have an equal total in sales from adults, but Quantity A has 100C while Quantity B has 96C. That means Quantity A is larger, making (A) the correct answer.

Answer: A

8. A

Using FOIL, Quantity A can be rewritten as x² − 4. Quantity B needs to be simplified. In the numerator, x² − 25 can be factored into (x + 5)(x − 5). In the denominator, x² − 6x + 5 can be factored into (x − 5)(x − 1). The fraction can then be simplified by canceling out common factors in the numerator and denominator:

So, Quantity B is equal to −5. Without knowing what x is, it might seem that Quantity A and Quantity B cannot be compared. However, when x is squared, the result cannot be negative. The smallest value it could have is 0. Subtracting 4, the smallest possible value of Quantity A is −4. That means Quantity A must be greater than or equal to −4. Any such number will always be greater than −5, so Quantity A will always be greater than Quantity B, no matter what x is. That makes (A) the correct answer.

Answer: D

9. D

It may be tempting to jump straight into the probability formula here:

each apartment with a 4 in it (those starting with 4 as well as ending in 4) and the number of total apartments would give , which, when simplified, equals Quantity B. However, notice that the centered information gives information about apartments, while Quantity A is based on selecting a random tenant. There’s no information on how many tenants are in each apartment. The above math works if each apartment has an equal number of tenants. However, if each apartment with a 4 in the number has three tenants while other apartments only have one, it would be more likely that a randomly selected tenant lives in an apartment with a 4 in the number. Since more than one relationship is possible, the correct answer is (D).

Answer: C

10. C

It’s important to realize that the total distance need not be calculated to compare the two quantities. For this question, let d represent the total distance, in miles, traveled. Quantity A will thus be equal to d.

Quantity B is the average speed for the entire trip. The average speed, in miles per hour, is calculated by taking the total mileage (d) and dividing that by the total number of hours. The car starts at 30 miles per hour and slows down 5 miles per hour every 10 minutes. That means it will travel first at 30 miles per hour, then 25 miles per hour, then 20, then 15, then 10, then 5. After that, it will stop. It will travel for 10 minutes at each of those 6 speeds, for a total of 60 minutes, which is 1 hour. That means the average speed will be equal to . Regardless of the value of d (which could be calculated, but would only take up valuable time for this question), the two quantities are equal. That makes (C) the correct answer.

Answer: E

1. E

The other variables all build upon u, so use the Picking Numbers strategy: pick a small number for u and find the values for r, s, and t. For instance, if u = 1, then t = 2u, so t = 2; s = 5t, so s = 10; and r = 3s, so r = 30.

So, The correct answer is (E).

Answer: 100

2. 100

Since ℓ₁ and ℓ₂ are parallel, the rules for angles created when parallel lines are cut by a transversal apply. If ∠ a has a measure of 40°, then ∠ z is also 40°. Because the sum of angles that make up a straight line is 180°, angles d and w are each 180° − 40° = 140°. Furthermore, since vertical (opposite) angles are equal, ∠ y is 140° and ∠ c is 40°. Thus, d + w − (y + c) = 140 + 140 − (140 + 40) = 280 − 180 = 100.

Answer: C

3. C

The question asks for the number of sea lions, and there are fewer sea lions than penguins, so starting small is a good idea. Backsolving works well here. Start with (B). If there are 36 sea lions, then there are 36 + 84 = 120 penguins, and the ratio of sea lions to penguins is . This ratio is less than , so the answer must be larger. Try (D). If there are 72 sea lions, then there are 72 + 84 = 156 penguins, and the ratio of sea lions to penguins is . Since this ratio is too large, the correct answer must be (C).

Answer: B, E

4. B, E

The question asks which numbers have more than two distinct prime factors. Start with 20 and break it down into prime factors: 2 × 2 × 5. Since 20 has only 2 distinct prime factors (2 and 5), (A) can be eliminated. Next, notice that 100 is just 20 × 5, so it has the same two distinct prime factors as 20. (C) can therefore be eliminated. Similarly, 200 is just 100 × 2, so 200 also has the same two distinct prime factors as 20 and 100. Eliminate (D). Now break down 30 into prime factors: 2 × 3 × 5. There are three distinct prime factors here, so (B) is one of the correct answers. Since 210 is 30 × 7, it has four distinct prime factors (2 × 3 × 5 × 7), so (E) is also one of the correct answers.

Answer: C

5. C

Each square shares at least one side with another square. That means all of the squares have the same side length. Segment AB would pass through two of the squares, creating two isosceles right triangles in each square. Since each square is the same size, the hypotenuse of each of those triangles would be units. Recall that with isosceles right triangles, the ratio of the side lengths is , with representing the hypotenuse. That means the actual hypotenuse of corresponds to , and x, representing one side of a square, is 4. There are 10 sides around the perimeter of the figure, so the entire figure has a perimeter of 10 × 4 = 40 units. (C) is the correct answer.

Answer: D

6. D

It helps to mark up the figure as you work, so start by redrawing it on your scratch sheet. The area of a rectangle is length × width. The length is the sum of the three top segments: BE + EG + CG = 1.5 + 2 + 1.5 = 5. The width of the rectangle would be the height of triangle AFD less the height of triangle EFG. With an area of x and a base of 2, you can solve for the height of EFG: . The height of EFG would be x.

The height of AFD can be solved by recognizing that triangles EFG and AFD are similar. They share their top angle, and sides AF and DF cut through the parallel lines of the rectangle, creating two sets of corresponding angles. Similar triangles have proportional sides. The base of EFG is 2 and the base of AFD is 5, so the proportion is 2:5. The heights would be in the same proportion: , where h is the height of AFD. Cross multiply to get 2h = 5x. Divide by 2 to solve for h: . That means the width of the rectangle would be . That makes the area of the rectangle , making (D) the correct answer.

Answer: B

7. B

This question can be deftly handled using the balance approach. The last score was 25, which is 4 points above the final average. That means the previous games must have been a total of 4 points below the final average. The average for each previous game was 20, which is 1 point below the final average. It would take 4 games at 1 point below average to balance out the 4 points above average achieved on the last game. That makes (B) the correct answer.

This can also be solved algebraically. Let g represent the number of games she played before last night’s game. If she averaged 20 points per game by then, she scored a total of 20g points. After last night’s game, she played one more game for a total of g + 1 games. She also scored 25 additional points for a total of 20g + 25 points. Her new average is 21, which is found by dividing her current total points by the total number of games she has played:

Cross multiply to get 21(g + 1) = 20g + 25. Distribute the 21 to get 21g + 21 = 20g + 25. From there, isolate g to get g = 4. That means she played 4 games before last night’s game.

Answer: C

8. C

The total number of possible outcomes is the number of possible outcomes from Set A (3) multiplied by the number of possible outcomes from Set B (3). That’s a total of 9 possible outcomes; 5 of those are even: ((1 + 5 = 6; 1 + 7 = 8; 2 + 6 = 8; 3 + 5 = 8; 3 + 7 = 10)) and only 2 are prime (1 + 6 = 7; 2 + 5 = 7). So, the probability of getting an even result is , while the probability of getting a prime result is . Subtract those figures to figure out how much greater the probability is of getting an even result: . (C) is the correct answer.

Note that (D) is a trap for those who only look at the possible sums. There are five distinct sums that are possible: 6, 7, 8, 9, 10. However, there is only one way that a sum of 6 can result (1 + 5), while there are two ways that a sum of 7 can result (1 + 6 and 2 + 5). Thus, the probability of getting each sum is not the same. Each individual outcome of selecting a number from each set must be considered.

Answer: C

9. C

With smaller numbers, it would be straightforward to simply list out all of the positive factors and count how many were odd. With 768, however, that would be overly time-consuming. Remember, though, that any non-prime number greater than 1 can be expressed as the product of prime factors, and finding prime factors is more manageable, even with large numbers.

A prime factor tree would show that the prime factorization of 768 consists of eight 2s and a 3. Since multiplying the 3 by any combination of 2s would result in an even factor, 3 itself is the only positive odd factor that can be generated through prime factorization. (B), however, is a trap for those who forget that 1 is a factor of every number. 768 thus has two positive odd factors, 1 and 3, and (C) is the correct answer.

Answer: B

10. B

The distance covered in a full lap around the circle would be equal to the circumference of that circle. The circumference of a circle is equal to 2πr. With a radius of , the circumference of the circular path is .

If Evan ran t times around the path, the total distance run would be . Evan ran a total of 5 km, so . Divide both sides by to get t = 3, which means Evan ran 3 times around the path. (B) is the correct answer.

Answer: B

1. B

Referring to the first bar chart, the average weekly earnings of a high school graduate are $675 and those of a worker with some college are $750. So the difference in weekly earnings is $750 − $675 = $75. Since the comparison is being made to the worker with some college, the percentage change is , which is (B), the correct answer. If you used $675 as the denominator, you would have chosen (C), 11%. Note that (E), 90%, is what percent of the earnings of the person with some college is earned by the high school graduate, not what percent less.

Answer: C

2. C

The number of full-time workers in City C is 200,000 × 0.40 = 80,000. The two dark gray bars on the left are the percentages of those workers without college degrees in City C; that is, 10% + 25% = 35%. Thus, the number of full-time workers in City C with no college at all is 0.35 × 80,000 = 28,000, which is (C).

Answer: D

3. D

If the total weekly earnings of the workers with some college is $22,500,000 and the average weekly earnings of that cohort are $750, then there are such workers. Refer to the second graph to see that these 30,000 people make up one-quarter of the workforce in City D, so the total workforce there is 4 × 30,000 = 120,000. Full-time workers with less than a high school education are 5% of that total, which is 120,000 × 0.05 = 6,000 people. Since this category of workers earns $500 per week, their total weekly earnings are 6,000 × $500 = $3,000,000. (D) is correct.

Answer: B

4. B

This is a weighted average question. Since the numbers of workers in each category are given in percentages, pick 100 as the number of workers in each city. Because you only need to find the difference between the two weighted averages, there are shortcuts that greatly simplify the calculations. In the “Less than high school” category, there would be 10% of 100 or 10 workers in C and there would be just 5 in D. Those 5 “extra” workers in C earn 5 × $500 = $2,500 weekly. There are the same number of high school graduates in both cities, so there will be no difference. Move along to the “Some college” group: there are 30 in C and 25 in D. The weekly earnings for 5 such people is 5 × $750 = $3,750. In both of these categories of workers, the total earnings in City D are less than those in City C. Finally, there are 10 more of the high-earners in City D. Those people make 10 × $1,200 = $12,000 per week. So, the total weekly earnings in City D are $12,000 − $3,750 − $2,500 = $5,750 greater than in City C. Remember that this comparison was based upon 100 workers, so the difference in the averages is which is (B). If you didn’t use the shortcut and actually computed the weighted averages, they are $921.25 and $863.75, which gives the same $57.50 difference.

Answer: 65

5. 65

Hannah’s current earnings are 20% greater than the category average, which makes them 1.2 × $750 = $900 per week. If she takes a 50% pay cut for 30 weeks, her lost wages will be Adding that to the college expenses of $6,000, Hannah needs to make up a total of $19,500. When she gets her degree, Hannah will be making $1,200 per week. If she hadn’t cut back her hours to attend school, she would still be making $900 per week, so she is recouping her $19,500 at the rate of $1,200 − $900 = $300 per week. So, it would take Hannah to recoup her costs.

Answer: A, B, D

6. A, B, D

Calculate the differences for the years 2000 through 2005:

(A): 2000: 9% − 3% = 6%. Correct.

(B): 2001: 9.5% − 3.5% = 6%. Correct.

(C): 2002: 8.5% − 4.5% = 4%. Incorrect.

(D): 2003: 8.5% − 3% = 5.5%. Correct.

(E): 2004: 8.5% − 5.5% = 3%. Incorrect.

(F): 2005: 7% − 7% = 0%. Incorrect.

Note that precise calculation is not necessary for 2004 and 2005; eyeballing the difference in the graph should be sufficient for these years.

Answer: C

7. C

First, find the changes from year to year. Note that downward changes must be represented by negative values; making all the values positive yields distractor choice (E).

2000 to 2001: 0.5%

2001 to 2002: 1%

2002 to 2003: −1.5%

2003 to 2004: 2.5%

2004 to 2005: 1.5%

2005 to 2006: −1%

2006 to 2007: −0.5%

2007 to 2008: 0.5%

2008 to 2009: 2.5%

Then, calculate the average of these changes:

The correct answer is (C).

Answer: E

8. E

Look up the unemployment values for 2006:

Country X: 6%
Country Y: 5%

Six is 1.2 times 5, and 1.2 = 120%. Thus, the unemployment rate of Country X (6) is 120% of that of Country Y (5). Critical thinking also leads to (E) without having to do any calculations, as the correct answer has to be greater than 100% and the other four choices are less than 100%. Many of the trap choices hinge on a misunderstanding of the question. Five is about 16.7% smaller than 6 (A), and 6 is 20% greater than 5 (B). (D) is backward, giving Y as a percent of X.

Answer: A

9. A

The 2003 unemployment rate for Country Y shown in the graph is 8.5%, so 6% is a decrease of 2.5 percentage points. At this point, you could calculate both averages and compare them directly, but it’s much faster to think critically. There are 10 years shown, so each year contributes one tenth of the average. Thus, a decrease of 2.5 percentage points would decrease the overall average by 2.5 ÷ 10, or 0.25 percentage points, (A).

Answer: B

10. B

Recall the percent change formula:

Strategic elimination is much faster here than precise calculation. For Country X, comparing (A) and (B), 2000 and 2003 had the same rate, but the rate in 2004 was higher than the rate in 2002, so (A) is wrong. Also for Country X, comparing (B) and (C), the increase from 2003 to 2004 was the same as the increase from 2008 to 2009 (2.5 percentage points in both cases), but the value in 2003 was much lower, so a 2.5 percentage point change represented a more significant change in 2003 to 2004. (C) is wrong.

For Country Y, (D) and (E) feature decreases that are less than 50%: and , respectively. (B) is , which is much greater than 50% and thus the right answer.

Answer: C

1. C

Quantity A is a permutation because order matters. The number of ways 3 people chosen from a group of 6 can be arranged in a line, where order matters, is 6 × 5 × 4 = 120. Quantity B is a combination because order does not matter. The number of ways 3 people can be selected from a group of 10, where order does not matter, is:

The two quantities are equal.

Answer: C

2. C

There are three pairs of vertical angles in the figure: a and d, b and e, and c and f, so the values in each pair are equal. The sum of the angles that make up a straight line is 180°, so a + b + c = 180. The centered information states that a + b = 2c, so substitute 2c for a + b to get 2c + c = 180. Thus, 3c = 180 and c = 60. Given that e = 2d, substitute b for e and a for d to get b = 2a. Substituting again, a + 2a + 60 = 180, so 3a = 120, and a = 40. So b = 2a = 80. Now that the values for all the angles are known, compare the two quantities. Quantity A, a°, is 40°. Quantity B is 2(b° − f °). Using the fact that c = f, this is 2(80° − 60°) = 2(20°) = 40°, so the quantities are equal. (C) is correct.

Answer: D

3. D

Pick a value for p and see what effect it has on r and s. If p = 1, r = (7 × 1) + 3 = 10, and s = (3 × 1) + 7 = 10, and the two quantities are equal. But if p = 0, r = (7 × 0) + 3 = 3, and s = (3 × 0) + 7 = 7, and Quantity A is less than Quantity B. Because there are at least two different possible relationships, the answer is (D).

Answer: A

4. A

Use the Picking Numbers strategy to answer this question. Suppose the original selling price of the shirt, x, is $100. After a 10% increase in price, the shirt would sell for 110% of $100, which is $110. If there is a 10% decrease next, the shirt would sell for 90% of the current price. That would be 90% of $110: 0.9 × $110 = $99. This price is less than the original amount, x, so Quantity A is greater.

Answer: B

5. B

Since this is a Quantitative Comparison question, you do not need to know the exact value of z, just whether it is less than, equal to, or greater than 33. So, instead of solving for z, just plug in 33 as the initial number of calls on hold. Since were answered, that would have left of the original callers on hold. Adding 15 more would bring the total to 37, which is greater than the 35 callers who were actually holding. Thus, the initial number was less than 33 and (B) is correct. (For the record, z = 30.)

Answer: A

6. A

There are n people in a room. One-third of them leave the room. So, there are people in the room. Four people enter the room, so you have people. There are now of the original number of people in the room, therefore Now solve for n.

So, n = 24 and Quantity A is larger.

Answer: B

7. B

Since point R is the intersection of the line with the y-axis, plug x = 0 into the equation for the line: 0 = 12 − 2y, so y = 6. The coordinates of point R are (0,6). Similarly, for point S, x = 12 − 2(0) = 12. The coordinates of point S are (12,0). For many line questions, you will need to restate the equation in standard y = mx + b form, but that is not necessary here. This is a right triangle because the two legs are on the x- and y-axes.

Quantity B is 30°, so think about the characteristics of a 30°-60°-90° triangle, one of which is that the short leg is half the length of the hypotenuse. With legs of 6 and 12, the hypotenuse of triangle RST is greater than 12. Therefore, since the short leg is 6, which is less than half the length of the hypotenuse, triangle RST must be “flatter” than a 30°-60°-90° triangle, and angle RST must be less than 30°. (B) is correct.

Answer: A

8. A

Start by simplifying the quantity in Quantity A: is the same as , which is 4x. Subtract x from both quantities, and you’re left with 3x in Quantity A and in Quantity B. Now divide both sides by x, and you’re left with 3 in Quantity A and in Quantity B. Square both quantities, and you get 9 in Quantity A and x in Quantity B. Since x is an integer between 1 and 9, exclusive, Quantity A is larger. If the algebra seems too abstract, go ahead and use the Picking Numbers strategy. If x equals 4, then Quantity A equals (2 + 2)2 = 16, and Quantity B equals 4 + 8 = 12.

Answer: B, C, D, E

9. B, C, D, E

If then 3x = 2y and Substitute into the equation , 2x + 3x = 30, 5x = 30, x = 6. Then, and y² = 81. So any answer with greater than 81 under the radical will be greater than y. Therefore, the correct choices are (B), (C), (D), and (E).

Answer: A, B, E

10. A, B, E

The best place to start here is with pairs of positive integers that have a product of 10. The numbers 5 and 2 have a product of 10, as do 10 and 1. But remember that integers may be negative, so −1 and −10 are possible, as well as −2 and −5. The mean of −1 and −10 is −5.5; the mean of −2 and −5 is −3.5. The mean of 2 and 5 is 3.5. The correct answers are (A), (B), and (E).

Answer: B, D

11. B, D

The prime factorization of 420 is 2 × 2 × 3 × 5 × 7. The sum of these prime factors is 2 + 2 + 3 + 5 + 7 = 19. Now test each of the choices. The prime factorization of 52 is 2 × 2 × 13, giving a sum of 2 + 2 + 13 = 17. For 104, note that this is just 52 × 2. Just add one more 2 to the sum of the prime factors of 52 to get 17 + 2 = 19. For 150, the prime factorization is 2 × 3 × 5 × 5, giving a sum of 2 + 3 + 5 + 5 = 15. For 176, the prime factorization is 2 × 2 × 2 × 2 × 11, giving a sum of 2 + 2 + 2 + 2 + 11 = 19. Finally, note that 450 is 150 × 3. Since you already did the prime factorization of 150, just add one more 3 to get a sum of 15 + 3 = 18. Thus, (B) and (D) are the correct answers.

Answer: 9

12. 9

Use the definition of average to write the sum of the first n bowlers’ scores: , n × average = sum of scores. Substitute the values given in the question and you have 160n = sum of scores for the initial set of bowlers. Now write the formula for the average again, using the additional score of 170. Now there are n + 1 bowlers.

Cross multiply and use algebra to solve for n.

There were 9 bowlers in the original group.

Answer: E

13. E

First, identify the numbers in set T: 3, 5, 7, 11, 13. The average of the numbers in set T is Its standard deviation is given in the question stem as 3.71. In order to increase the standard deviation of a set of numbers, you must add a value that is more than one standard deviation away from the mean. One standard deviation below the mean for set T is 7.8 − 3.71 = 4.09, and one standard deviation above the mean is 7.8 + 3.71 = 11.51. Any value outside this range 4.09 ≤ x ≤ 11.51 would increase set T’s standard deviation, since it would make the set more “spread out” from the mean than it currently is. The only choice that does that is choice (E).

Answer: C

14. C

Solving this problem involves several steps, but none is too complicated. The circle has its center at point T. Start with the triangle on the right whose vertices are at T and two points on the circumference of the circle. This makes two of its sides radii of the circle, which we’re told each have a length of 3. Because all radii must have equal length, this makes the triangle an isosceles triangle. In addition, you’re told one of the base angles of this triangle has a measure of 60°. Thus, the other base angle must also have a measure of 60° (since the base angles in an isosceles triangle have equal measure). The sum of the two base angles is 120°, leaving 180° − 120° or 60° for the other angle, the one at point T (making ΔTUV an equilateral triangle with sides of 3).

Now, angle RTS is opposite this 60° angle, so its measure must also be 60°. Therefore, ΔRST is another equilateral triangle, and its sides are 3. Therefore, the length of RS is 3, choice (C).

Answer: B

15. B

The probability formula is:

When one die is rolled, there are six possible outcomes. When two dice are rolled, the number of possible outcomes is 6 × 6, or 36. Getting a total value of 7 can be achieved in the following ways: (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1). There are six possible ways.

So the probability of rolling a total of 7 is which can be reduced to choice (B).

Answer: 20

16. 20

First, 2 fields are plowed in 5 hours. But that’s with 4 farmers plowing. One farmer would take four times as long, or 5 × 4 = 20 hours, to plow 2 fields. At that rate, one farmer would take 20 × 3 = 60 hours to plow the remaining 6 fields. Three farmers, then, could do the job in of the time, or 20 hours.

Answer: C

17. C

The residential usage (in billions) in 2000 was about 22; the usage was about 52 in 2010. Since 52 was the highest usage over this time period and 22 was the lowest, the range is the difference between these numbers. Therefore, 52 − 22 = 30, and the range is about 30 billion gallons. The correct answer is (C).

Answer: C

18. C

The two amounts were closest to each other in 2002. The residential amount appears to be about 28; the total appears to be about 32: 28 ÷ 32 = 0.875. Choice (C) is the closest.

Answer: D

19. D

The three usages with the greatest amounts per person are faucets, washers, and showers, totaling 30 gallons per day. Multiply by 10,000 to get 300,000, choice (D).

Answer: B

20. B

The residential consumption (in billions) in 2010 was approximately 52. Take half of that amount, 26, to represent the amount of water used by households with efficient appliances and plumbing. Let W represent the amount of water these households would have used otherwise.

Set up a percent equation to solve for W. Remember, the savings were 35%, so subtract 35 from 100 to find the percent that would have been used.

The savings in billions of gallons was 40 − 26 = 14. The correct answer is (B).

Answer: A

1. A

This question requires no computation but only a general understanding of how averages work and what the word “median” means. The median of a group of numbers is the “middle number”; it is the value above which half of the numbers in the group fall and below which the other half fall. If you have an even number of values, the median is the average of the two “middle” numbers; if you have an odd number of values, the median is one of the values. Here, in Quantity B, the median is 101. In Quantity A, if the numbers were 100, 101, and 102, then the average would also be 101, but because the third number, 103, is greater than 102, then the average must be greater than 101. Quantity A is greater than 101, and Quantity B equals 101; Quantity A is larger.

Answer: A

2. A

Start with the information you are given. You know that the length of the chord is 15. What does that mean? Well, because you don’t know exactly where A and B are, it doesn’t mean too much, but it does tell you that the distance between two points on the circle is 15. That tells you that the diameter must be at least 15. If the diameter were less than 15, then you couldn’t have a chord that was equal to 15, because the diameter is always the longest chord in a circle. The diameter of the circle is 15 or greater, so the circumference must be at least 15π. That means that Quantity A must be larger than Quantity B.

Answer: C

3. C

The equation in the centered information looks complicated, but we’ll take it one step at a time. Because Quantity A has only h in it, solve the equation for h, leaving h on one side of the equal sign and r on the other side. First, substitute the value for x into the first equation; then solve for h in terms of r.

	Substitute 1 for x.
	Divide both sides by
	Take the positive square root of both sides, using the information that r and h are positive.
	Divide both sides by r to get h alone.
	The two quantities are equal.

Answer: C

4. C

Draw an xy-plane and label the points given to help solve this problem. You know where points B and C are; they’re on the x-axis. You don’t know where A is, however, which may make you think that the answer is choice (D). But you’re given more information: you know that the triangle has an area of 18. The area of any triangle is one-half the product of the base and the height. Make side BC the base of the triangle; you know the coordinates of both points, so you can find their distance apart, which is the length of that side. C is at the origin, the point (0,0); B is at the point (6,0). The distance between them is the distance from 0 to 6 along the x-axis, or just 6. So that’s the base. What about the height? Because you know that the area is 18, you can plug what you know into the area formula.

That’s the other dimension of the triangle. The height is the distance between the x-axis and point A. Now you know that A must be somewhere in the first quadrant, since both the x- and y-coordinates are positive. Don’t worry about the x-coordinate of the point, because that’s not what’s being compared; you care only about the value of y. You know that the distance from the x-axis to the point is 6, because that’s the height of the triangle and that y must be positive. Therefore, the y-coordinate of the point must be 6. That’s what the y-coordinate is: a measure of the point’s vertical distance from the x-axis. (Note that if you hadn’t been told that y was positive, there would be two possible values for y: 6 and −6. A point that’s 6 units below the x-axis would also give a triangle with height 6.) You still don’t know the x-coordinate of the point, and in fact you can’t figure that out, but you don’t care. You know that y is 6; therefore, the two quantities are equal.

Answer: D

5. D

With symbolism problems like this, it sometimes helps to put the definition of the symbol into words. For this symbol, you can say something like “x Φ y means take the sum of the two numbers and divide that by the difference of the two numbers.” One good way to do this problem is to Pick Numbers. You know that p is positive and q is negative. So suppose p is 1 and q is −1. Figure out what p Φ q is first. You start by taking the sum of the numbers, or 1 + (−1) = 0. That’s the numerator of the fraction, and you don’t really need to go any further than that. Whatever their difference is, because the numerator is 0, the whole fraction must equal 0. (The difference can’t be 0 also, since p ≠ q.) So that’s p Φ q. Now what about q Φ p? Well, that’s going to have the same numerator as p Φ q: 0. The only thing that changes when you reverse the order of the numbers is the denominator of the fraction. So q Φ p has a numerator of 0, and that fraction must equal 0 as well.

So you’ve found a case where the quantities are equal. Try another set of values and see whether the quantities are always equal. If p = 1 and q = −2, then the sum of the numbers is 1 + (−2) or −1. So that’s the numerator of the fraction in each quantity. Now for the denominator of p Φ q, you need p − q = 1 − (−2) = 1 + 2 = 3. Then the value of p Φ q is The denominator of q Φ p is q − p = −2 − 1 = −3. In that case, the value of q Φ p is The relationship between quantities is different; therefore, the answer is (D).

Answer: D

6. D

Picking Numbers will help you solve this problem. For and , so Quantity A is larger. For and so Quantity B is larger. The relationship between quantities is different; therefore, the answer is (D).

Answer: B

7. B

The sum of the interior angles of a triangle is 180°. One of the angles of the large triangle formed by the outside perimeter is a right angle, or 90° angle. The remaining 90° must come from the sum of x, y, and the angle adjacent to y. Therefore, the sum of x and y alone must be less than 90°. Quantity B is greater, making the answer (B).

Answer: A

8. A

You could solve this system of equations by substitution or combination. To use combination, multiply all the terms of t − s = −4 by 4 to get −4s + 4t = −16. Now add this result to the other equation:

So t = 6. Plug this value into the second equation: 6 − s = −4, to get s = 10. Thus, Quantity A is greater, and (A) is correct.

Answer: D

9. D

Divide both sides of the inequality by 6. You’re left with (10)n > 10,001. The number 10,001 can also be written as 10⁴ + 1, so you know that (10)n > 10⁴ + 1. Therefore, Quantity A, n, must be 5 or greater. Quantity B is 6. Because n could be less than, equal to, or greater than 6, you need more information.

Answer: B

10. B

Try to set the quantities equal. Could the tens digit of y be 4? If it is, and the thousands digit is 2.5 times the units digit, then the thousands digit must be . . . 10? That can’t be right. A digit must be one of the integers 0–9; 10 isn’t a digit. Therefore, 4 is too big to be the tens digit of y. In fact, the only possible value for the tens digit of y is 2. Quantity B is greater than Quantity A.

Answer: D

11. D

To find the average, add the quantities together and divide by 4: (2x + 3) + (5x − 4) + (6x − 6) + (3x − 1) = 16x − 8 and . The correct choice is (D).

Answer: E

12. E

When a transversal cuts a pair of parallel lines, in this case ℓ₁ and ℓ₂, the angles are always supplementary and their sum is 180. So, the sum (x + y) is equal to the sum (a + b). The exact values of the individual angle measures cannot be determined from the figure. The answer is (E).

Answer: D

13. D

First, find the value of x, using the fact that there are 180° in a straight line. Set the sum of the angle measures equal to 180: 5x + 4x + 3x = 180, 12x = 180, and x = 15. Angle SRU equals 4x + 5x = 9x, which is 135°. Choice (D) is correct.

Answer: B, C, D, E

14. B, C, D, E

You know that the ratio of oranges to apples is 9 to 10 and that there are at least 200 apples. The ratio tells you that there are more apples than oranges. At the minimum, there must be 180 oranges to satisfy the proportion . There could be more than 200 apples, so any number of oranges greater than 180 for which the ratio 9:10 applies is also correct. All of the choices are multiples of 9, so the correct choices are (B), (C), (D), and (E).

Answer: C, D, E

15. C, D, E

The area of the shaded region is the area of the square minus the area of the portion of the circle that is inside the square. The area of a square is its side squared. The area of square ABCD is 4² = 4 × 4, which is 16. Now find the area of the portion of the circle that is inside the square. Because the diameter of the circle is a side of the square, you know that exactly one-half of the circle’s area is inside the square. Also, because the diameter of the circle is twice the radius, the radius of the circle is or 2. The area of a circle with a radius r is πr². The area of the complete circle in this question is π(2)², which is 4π. So half the area of this circle is 2π. Thus, the area of the shaded region is 16 − 2π.

That means that 16 − 4π and 16 − 16π are less than 16 − 2π, so they cannot be correct choices. However, the sum of 16 and any positive number is greater than 16 and also greater than 16 − 2π. So, the correct choices are (C), (D), and (E).

Answer: 8

16. 8

Simplify each inequality to restate them in terms of x and y. Divide all the terms of −30 ≤ 6x ≤ 60 by 6 to get −5 ≤ x ≤ 10. To simplify 40 ≤ 2y + 4 ≤ 8, subtract 4 from each term to get 36 ≤ 2y ≤ 4. Then, divide through by 2 to see that 18 ≤ y ≤ 2. The maximum value of x − y will be the greatest possible value of x minus the least possible value of y. This is 10 − 2, so the correct answer is 8.

Answer: D

17. D

Although there are 6 numbers on the die, when it comes to evens and odds, there are only two outcomes of a roll. Thus, for three rolls, there are 2 × 2 × 2 = 8 equally possible outcomes: EEE, EEO, EOE, EOO, OOO, OOE, OEO, and OEE. Three of these, EOO, OOE, and OEO, result in exactly one even number. Use the probability formula to determine the probability, which is .

The decimal equivalent is 0.375, so (D) is correct.

Answer: B

18. B

The first graph shows the beginning and ending prices. Start by “eyeballing” the two bars for each of the five funds. Fund E was flat, so eliminate that one. Fund D had a slight increase, but since the question asks for the greatest percent increase and Fund D had a relatively high beginning price, it does not have the greatest percent increase.

You might be able to infer that Fund B had the greatest percent increase since its starting value was less than those of Fund A and Fund C, but here are the calculations to confirm that. Use the percent change formula to compare the increases of these three funds:

. Fund A had an original price of $24 and a new price of $28, so the percent change was

For Fund B:

For Fund C:

So, Fund B had the greatest percentage increase, and (B) is correct.

Answer: 50,000

19. 50,000

Extract the per share prices of the two funds from the first chart and multiply each by the number of shares shown on the second chart. For Fund C, that is $20/share × 1,000 shares = $20,000. For Fund E, the value is $50/share × 600 shares = $30,000. Thus, the total is $20,000 + $30,000 = $50,000.

Answer: A

20. A

The dollar value increase of each fund is the price increase per share times the number of shares held. For Fund A, that was ($28 − $24)(600) = ($4)(600) = $2,400. Fund B increased by ($16 − $12)(500) = ($4)(500) = $2,000. The increase in Fund C was ($22 − $20)(1,000) = ($2)(1,000) = $2,000. The value of the investment in Fund D increased by ($42 − $40)(300) = ($2)(300) = $600. Fund E’s price did not increase, so the correct choice is (A).

Answer: B

1. B

The diagonal of a square of side 4 is The circumference of a circle is π times the diameter. So, the circumference of this circle is Now write Quantity B, as and you can compare the quantities piece by piece. The factors of 4 and are the same in both quantities, but π is less than 5. So, Quantity B is larger.

Answer: D

2. D

You could pick numbers here, but using logic would be more efficient. Since b + c is negative, then b, as the smaller of the two, must be negative. Since a is less than b, it must also be negative. But c could be negative or positive with a smaller absolute value than the absolute value of b, or c could even be zero. The product of a and c will be greater than 0 if both are negative. However, if c is positive, the product of a and c would be negative. And if c is zero, then the two quantities are equal. All three relationships are possible; therefore, (D) is correct.

Answer: C

3. C

There are 12 positive integer factors of 96: 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, and 96. There are 12 positive integer factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72. The two quantities are equal.

Answer: A

4. A

If x > 0, then which also equals must be greater than 1. On the other hand, must be less than 1. This is because when x > 0, the numerator x is smaller than the denominator, so the ratio Therefore, when x > 0, and Quantity A is greater.

Answer: C

5. C

For this question, notice the relationship between the bases, 2 and 4. When comparing exponents, it’s easiest to work with equal bases.

You know that 4 = 2². Therefore, 4q = (2²)q = 2²q. Now you have 2p = 2²q, so p = 2q. The quantities are equal, choice (C).

Answer: A

6. A

Before you go to the trouble of multiplying the terms, let’s see if there’s a shortcut. For the GRE, make sure you know the common unit conversions for time. There are 60 seconds in a minute and 60 minutes in an hour, so there are 7 × 60 × 60 seconds in 7 hours. There are 24 hours in a day and 7 days in a week, so there are 7 × 24 × 52 hours in 52 weeks. Let’s rewrite the quantities:

Taking away the common values gives you:

You still shouldn’t do the math, however. The best strategy is to compare piece by piece, which shows that Quantity A is larger than Quantity B.

Answer: C

7. C

The sum of the measures of the angles on one side of a straight line is 180°. The equation for these angles can be written as (2t + 12) + (t² − 2t + 24) = 180. Removing the parentheses to combine like terms results in the equation t² + 36 = 180. This equation simplifies to t² = 144, which means that t is ±12. However, since angle measurements on the GRE are always positive, t is 12. (C) is correct.

Answer: A

8. A

Since x > 2, you know x > 0 and you can divide both quantities by x without changing their relationship. Quantity A is then x² and Quantity B is 4. Since x > 2, the least value for x² is greater than 2² = 4. Therefore, (A) is correct.

Answer: 5

9. 5

Let’s first find the value of 9♣. Then we’ll find the value of (9♣) ♦ 3.

Answer: 1.5

10. 1.5

The area of a rectangle is its length times its width.

The area of rectangle A is 12 × 5 = 60.

The area of rectangle B is 9 × 10 = 90.

So the area 60 of rectangle A must be multiplied by a number, which you can call x, to obtain the area 90 of rectangle B.

Then 60x = 90. So .

Answer: 30

11. 30

Here’s a problem where it really pays to have learned the Pythagorean triplets. The Pythagorean theorem will work because this is a right triangle, but it is much more strategic to see if any of the Pythagorean triplets work first. Start by testing the 3:4:5 and 5:12:13 triplets. The 3:4:5 is not the correct triplet because, while 12 is a multiple of 3 and 4, the other side lengths (either 16 and 20 if 12 is the shorter leg, or 9 and 15 if 12 is the longer leg) do not fit into the expressions given for AB and BC. (Remember that geometric figures on the GRE are not necessarily drawn to scale.) Next check 5:12:13. If AC is 12, then AB must be 5 and BC must be 13. This fits the given information as 5 + 8 = 13. Now it is time to find the area (be careful to not just answer with the value of x).

The area of a triangle is one-half the base times the height. The area of a right triangle is , because one leg can be considered to be the base and the other leg can be considered to be the height. So the area of triangle ABC is

The answer is 30.

Answer: A, B, C, D

12. A, B, C, D

The average formula is as follows:

Therefore,

Sum of the terms = Average × Number of terms

The sum of the scores of the four students whose average was 85 is 85(4) = 340. Let’s call the fifth student’s score x. If the new average is to be greater than 84 and less than 86 and the sum of the scores of all five students is 340 + x, then . If you multiply all parts of the inequality by 5, you get 420 < 340 + x < 430. Subtracting 340 from all parts of the inequality, you get 80 < x < 90, making (A), (B), (C), and (D) the correct choices.

Answer: 5

13. 5

This question can be broken into two equations with two unknowns, Meg’s age now (M) and Rolf’s age now (R). Equation (i) shows the relationship now; equation (ii) shows the relationship three years ago.

Substitute 2R for M in equation (ii) and solve for R:

Rolf is 5 years old now.

Answer: C

14. C

The cost of manufacturing x crayons is (570 + 0.5x) cents. Because each crayon sells for 10 cents, x crayons will sell for 10x cents. You want the smallest value of x such that 10x cents is at least 570 + 0.5x cents. So you must solve the equation 10x = 570 + 0.5x for the value of x that will recoup the investment.

The minimum number of crayons is 60, choice (C).

Alternatively, you could have avoided setting up an algebraic equation by Backsolving, starting with either (B) or (D).

Answer: A

15. A

You can write that . By canceling a factor of x from the numerator and denominator of you have .

So, . The answer is (A).

Answer: A, B, C

16. A, B, C

The first thing to find for this question is the set of integers that will work for set J. You need x³ to be a multiple of the LCM (least common multiple) of 72 and 216. Since 216 is 72 × 3, the LCM is 216. If this does not stand out immediately, use prime factorization to find the LCM. The prime factorization of 72 is (2³) × (3²) and the prime factorization of 216 is (2³) × (3³), which confirms that 216 is the LCM for 72 and 216. Setting x³ equal to 216 means that x = 6. This confirms that J is the set of all positive integers that are multiples of 6. The factors of 6 are 1, 2, 3, and 6, so (A), (B), and (C) are the correct choices. Be careful with 12: it is a multiple of 6, not a factor.

Answer: B

17. B

The goal is to find the average of (x + 2), (2x − 1), and (x + 4), which requires finding a value for x. In order to find x, set the equation equal to 0 by subtracting 8 from both sides. The resulting equation, 2x² + 6x − 8 = 0, factors into (2x − 2)(x + 4) = 0. This means that x can be either 1 or −4. However, x was stipulated to be positive, so x = 1.

When you substitute 1 for x, the equation becomes , so (B) is the correct answer.

Answer: C

18. C

Before you answer any graph question, begin by examining the graphs. Here you have two graphs, a segmented bar graph representing team revenue breakdowns for five teams and a pie chart showing the distribution of venue revenues for Team X.

You’re now ready to attack the question, which asks you to find the team with the median venue revenue for 2018 and to determine what percent of that team’s total revenue is media revenue. This question must refer to the first graph, and the first part of the question—finding the team with the median venue revenue—is straightforward. Median refers to the number in the middle. By looking at the white portions of the bars in the top graph, you see that Team Z has the median venue revenue. The fastest approach to the answer here (and throughout graph questions generally) is to approximate. The downside to bar graphs is that it’s often very hard to get a read on the values. The upside is that if you approximate, often you don’t have to read the values. Here you need to determine what percent of Team Z’s bar is represented by media revenue (the segment in the middle—always be especially careful to isolate the correct piece of data). By approximating, you can see that the middle segment is about half of the entire bar. Thus the correct answer has to be close to 50%. The only answer choice that works is (C), 45%.

Answer: B, C, D, E

19. B, C, D, E

Percent change problems are extremely popular graph questions, and as long as you set them up correctly, they are a great opportunity. This question asks for the approximate percent increase in Team Y’s total revenue from 2018 to 2019, so you need to figure out (roughly) the amount of increase, place that over the original amount, and then convert the fraction into a percent. You are given the total revenue for 2019 as at least $150 million, so you need to locate the total revenue for 2018 from the bar graph. It looks to be approximately $120 million, so the amount of increase is $30 million (or more), and the original amount is $120 million. Now let’s apply the formula:

So, any percent greater than or equal to 25% is the answer. The answers are (B), (C), (D), and (E).

Answer: E

20. E

When looking at the bar graph, you see from the lowest portion of the bar for Team X that venue revenues of Team X were approximately $40 million (call it 40m, for short). From the pie chart, the venue revenues of Team X from merchandise sales were approximately 20% of 40 million dollars, the venue revenues from ticket sales were approximately 10%, and the venue revenues from food sales were approximately 35%. The venue revenues of Team X from merchandise, in dollars, were approximately 0.2(40m) = 8m. The venue revenues of Team X from ticket sales, in dollars, were approximately 0.1(40m) = 4m. So the venue revenues of Team X from merchandise sales and ticket sales, in dollars, were approximately 8m + 4m = 12m. The venue revenues of Team X from food sales, in dollars, were approximately 0.35(40m) = 14m. The percent of the venue revenues of Team X that were from merchandise sales and ticket sales, out of the venue revenues of Team X that were from food sales, is approximately .

To the nearest percent, 85.7% is 86%. Choice (E) is correct.