Question 1
Difficulty: Easy · Percent Correct: 79.5% · Type: Numeric Entry
Luke drives the first 300 miles of a trip at 60 miles an hour. How fast does he have to drive, in miles per hour, on the final 200 miles of the trip if the total time of the trip is to equal 7 hours?
On the first leg of the trip, he travels 300 miles at 60 mph. You can find the time by using the following formula:
T2=2 D_
V 2=2 _30060 2=25 hr
If the first leg takes 5 hours, and if Luke wants to complete the trip in 7 hours, then he will have to complete the second leg in just 2 hours. This second leg has a distance of 200 miles. This would require a speed of
V)=2 _D
T 2=2 _2002 2=2100 mph
Question 2
Difficulty: Easy · Percent Correct: 78.2% · Type: Quantitative Comparison
Column A Column B
Sum of integers from 1 to 40 inclusive 800
To evaluate column A, look at pairs of integers, starting with the greatest and least in the list. The sum of 1 and 40 is 41. Similarly, the sum of 2 and 39 is 41. If you continue in that pattern, you can create 20 pairs of numbers that add to 41:
1 + 40 2 + 39 3 + 38 ⋮ 19 + 22 20 + 21
Since each pair sums to 41, and there are 20 pairs, the total sum described in column A is 41 × 20 = 820. Since 820 is greater than 800, the answer is choice (A).
The quantity in Column A is greater The quantity in Column B is greater The two quantities are equal
The relationship cannot be determined from the information given
Word problems Answer: 100 Answer: A A B C D
Question 3
Difficulty: Easy · Percent Correct: 65.5% · Type: Numeric Entry
If Michael can shovel all the snow off a standard driveway in 12 minutes, and Eamon can shovel all the snow off a standard driveway in 36 minutes, then working together, how many minutes would it take for them both to shovel all the snow off a standard driveway?
You can’t add or subtract times to complete the driveway, but the big idea is that you can add rates.
(Michael’s rate) + (Eamon’s rate) = (their combined rate) Michael's rate2=2 _121 2(1 driveway / 12 minutes)8
Eamon's rate2=2 _361 (1 driveway / 12 minutes)
These rates are in the unlikely units of “driveways per minute.” You add these to get the combined rate.
combined rate2=2 _121 2+ 2 _1
36 2=2 _363 2+2 _361 2= 2 _364 2=2 _91
Combined, they would complete one driveway every 9 minutes.
Question 4
Difficulty: Medium · Percent Correct: 63.4% · Type: Multiple Choice
Sue planted 4 times as many apple seeds as she planted orange seeds. 15% of the apple seeds grew into trees, and 10% of the orange seeds grew into trees. If a total of 420 apple trees and orange trees grew from the seeds, how many orange seeds did Sue plant? 540 600 660 720 760
Shouldn’t the ratio of apple:orange be 4:1 or 4A=O? You know there are 4 times as many apple seeds as there are orange seeds. So if there are 4 apple seeds, then there is 1 orange seed. When setting up the equation, you need to multiply orange seeds by 4 to make the number equal to apple seeds.
If you tried 4A = 1O, then with our example you would get 16 = 1. So you need to flip it to be 1A = 4O, and then you get 4 = 4.
Alternatively, you might rephrase the statement in your own words or draw a small picture before writing an equation. A group of apples is 4 times the size of a group of oranges, i.e., A = 4 × O.
Answer: 9
Answer: B
A Let A2=2number of apple seeds
Let O2=2number of orange seeds A2=24O # of apple trees2=20.15A # of orange trees2=20.1O 0.15(×100)A2+20.1(×100)O2=2420(×100)8 815A2+210O2=242,000
}
B C D EGRE Quantitative Reasoning
Or, visually, AAAA:O
Notice that those aren’t set equal to each other—it’s a ratio. If you want to set these equal to each other, you have to make the oranges side 4 times the size.
(AAAA) = (O) × 4
Question 5
Difficulty: Hard · Percent Correct: 55.8% · Type: Multiple Choice
A container holds 4 quarts of alcohol and 4 quarts of water. How many quarts of water must be added to the container to create a mixture that is 3 parts alcohol to 5 parts water by volume? 4 __ 3 5 __ 3 7 __ 3 8 __ 3 10 __ 3 Question 6
Difficulty: Hard · Percent Correct: 38.2% · Type: Numeric Entry
The sum of the pre-tax costs of Item A and Item B is $300. In Alumba, each item would be charged a flat 7%. In Aplandia, Item A is subject to 5% tax and Item B is subject to 10% tax. If the tax in Aplandia on the purchase of both items is exactly $3 more than it is in Alumba, then what is the pre-tax price of Item A?
In Alumba, the flat tax would be 7% of $300, or $21.
In Aplandia, A is subject to 5% tax, B subject to 10%, and the total tax comes to $24. Let A be the price of Item A. Then the price of Item B is (300 − A).
0.05A + 0.10(300 – A) = 24 0.05A + 30 – 0.1A = 24 –0.05A = –6 0.05A = 6 0.1A = 12 A = 120 Answer: D A + = 4 qt 4 qt x qt 4 + x qt 4 qt Water Alcohol _alcoholwater 2:2 _4 42+2x 2=2 _35 3(42+2x)2=24(5) 122+23x2=220 3x2=28 x2=2 _83 B C D E Answer: $120
Shouldn’t you calculate the tax rate in Alumba as (7% A) + (7% B) instead of 7% of the total? Would using (7% A) + (7% B) mean that you have a 14% total tax rate? The individual price of the items doesn’t matter for determining the total tax rate in Alumba. Assessing a tax of 7% individually (on each item) or on the entire purchase will give you the same tax amount, which is $21 for Alumba.
Think of the flat 7% tax as something you could distribute. Let’s say the two items cost X and Y. You can express this 7% tax as a tax on the entire purchase:
0.07(X + Y)
Or you can express this tax, by distributing the multiplication, as a tax on each part of the purchase:
0.07X + 0.07Y
The tax rate remains at 7% for the overall purchase, but it turns out that when you tax something as a group, each part of the group is also taxed at 7%.
You can also prove this using real numbers. Let’s say Item A costs $10 and item B costs $20. First let’s evaluate the tax if it were applied to the whole purchase:
$10 + $20 = $30 0.07 × $30 = $2.10 Total Tax: $2.10
Now let’s evaluate the tax on each item individually and compare this to our first answer. Item A: $10 × 0.07 = $0.70
Item B: $20 × 0.07 = $1.40 Total Tax: $1.40 + $0.70 = $2.10
As you can see, both methods arrive at the same answer, so you can definitely
conclude that the tax doesn’t become 14% just because you tax two items in a purchase at 7%.
Question 7
Difficulty: Hard · Percent Correct: 54.2% · Type: Multiple Choice
The nth term ( t n ) of a certain sequence is defined as tn2=2tn−12+24. If t12=2−7 then t712= 273 277 281 283 287 Answer: A A B C D E
GRE Quantitative Reasoning
The first thing to realize is that this formula: tn2=2tn−12+24
would imply that: t22=2t12+24
t32=2t22+24, etc.
In other words, to get each new term, we’re simply adding 4 to the previous term. We start with the first term of –7, and then we add a bunch of 4s. How many 4s do we add? Well, for the second term, we add 4 once; for the third term, we add 4 twice from the start. Hence, for the nth term, we add 4 a total of (n – 1) times from the start.
Thus, for the 71st term, we would start with the first term of –7 and add 4 a total of seventy times.
t712=2−72+2(42×270)2=22802−272=2273
The answer is (A).
Question 8
Difficulty: Medium · Percent Correct: 49.7% · Type: Multiple Answer
a, b, and c are positive integers. If b equals the square root of a, and if c equals the sum of a and b, which of the following could be the value of c?
Indicate all such values. 21 30 45 72 100 331
Because b is an integer, we know that b and (b2+21) are consecutive integers. In other words, c is the product of two consecutive integers. Knowing that, we can reword the original question prompt—“Which of the following can be written as the product of two consecutive integers?”—and find the possible values of c.
Answer: B D A b)=2 √ _a 8 8 b 2 2=2a c2=2a2+2b =2b22+2b =2b(b2+21)2=2even number B C D E F
Question 9
Difficulty: Easy · Percent Correct: 62.3% · Type: Multiple Answer
In set s, there are four numbers. Three of the numbers are 13, 29, and 41, and the fourth number is x. If the mean of the set is less than 25, what could be the value of x?
Indicate all possible values of x. 13 15 17 19 21 23 25 27
The easiest way to approach this problem is to think about the sum. average2=2 sum of list_N 8 8 sum of list2=2N2×2(average)
The second, rewritten form is useful to you. average < 25
sum = 4 × (average) < 100 13 + 29 + 41 + x < 100 83 + x < 100
x < 17
The missing number must be less than 17, so it could be 13 or 15.
Question 10
Difficulty: Easy · Percent Correct: 69.2% · Type: Numeric Entry
In Alioth Industries, 20% of the employees have advanced degrees and the others have bachelor’s degrees. The average salary for an employee with an advanced degree is $350,000, and the average salary for an employee with a bachelor’s degree is $100,000. What is the average salary, in dollars, for all the employees at Alioth Industries? This is a weighted average. Let’s make all the salaries 1,000 times smaller, just to simplify calculations. This would give us the problem:
0.2 × (350) + 0.8 × (100) = 70 + 80 = 150 The average for all employees is $150,000.
Answer: A B A B C D E F G H Statistics Answer: $150,000
GRE Quantitative Reasoning
Question 11
Difficulty: Medium · Percent Correct: 71.1% · Type: Quantitative Comparison
The average (arithmetic mean) of 7 different numbers is 5
Column A Column B
Median of the 7 numbers 5
The quantity in Column A is greater The quantity in Column B is greater The two quantities are equal
The relationship cannot be determined from the information given
We know that the average of seven different numbers is 5. The word “different” implies that no number appears twice on the list. For example, having a set in which all the entries equal 5 would be excluded by this restriction.
Certainly, we could have a symmetrical set of consecutive integers centered on 5. This would be the set {2, 3, 4, 5, 6, 7, 8}. Both the mean and the median of this set are 5. This choice for the set makes the columns equal, suggesting the answer (C) for at least this choice.
If we can select another set that fits the criteria, mean = 5, and makes one column bigger than the other, then we will have chosen two different possibilities that give two different relationships, and bam! right away we would know that (D) would be the answer. Can we find another set that fits the requirements?
Well, we could subtract one from each of the first six numbers to get the
consecutive integers 1 through 6. To keep the same average and same sum, we would have to add six to the highest number giving us 8 + 6 = 14. This gives us another set with an average of 5, the set {1, 2, 3, 4, 5, 6, 14}. This set has a median of 4, which is less than the 5.
This is an idea we discuss in depth in the “picking numbers” section of the quantitative comparison strategies chapter starting on page 196, but here we’ll simply point out that if we can make different choices that indicate different
relationships, this automatically means that a single relationship between the columns is not uniquely determined by the situation, and that necessitates an answer of (D).
Answer: D
A B C D
Question 12
Difficulty: Hard · Percent Correct: 39.2% · Type: Multiple Answer
Marcia has 2 liters (L) of a 60% concentrated solution of phosphoric acid. She wants to add 3L of less concentrated phosphoric acid solutions, so that she has 5L of a solution with a concentration less than 50%. Which could she add?
Indicate all possible combinations of solutions. 3L of a 40% concentrated solution
3L of a 42% concentrated solution 3L of a 44% concentrated solution 3L of a 46% concentrated solution 3L of a 48% concentrated solution
2L of a 40% solution and 1L of a 45% solution 1L of a 40% solution and 2L of a 45% solution
Subtract 50 from all the concentration numbers, just to make the calculations easier to handle. In this view, she has 2L of a concentration 10, and wants to add 3L of
Concentration X to get 5L of concentration less than 0. First, let’s focus on getting 5L of concentration exactly 0.
10 + 10 + X + X + X = 0 20 + 3X = 0
3X = –20 X = ___–203 = –6.67
Now, change back to real concentrations. This X would have concentration of 50 – 6.67 = 43.33%
Three liters of a solution with concentration 43.33% would produce 5L of exactly 50% concentration. You want a concentration less than 50%, so you need to add 3L of a solution that has a concentration lower than 43.33%.
(A) and (B) work.
(C) and (D) and (E) don’t work.
Now, the last two choices are tricky. For these two, subtract 40 from all the concentration values, to make the calculations easier.
For (F), you want to combine 2L of concentration 0 with 1L of concentration 5. That’s an average of (02+202+25)/32= 1.67. In other words, in real concentrations, mixing these would be equivalent to 3L of solution with concentration 41.67%. This is less than 43.33%, so (F) works.
For (G), you want to combine 1L of concentration 0 with 2L of concentration 5. That’s an average of (02+252+25)/32= 3.33. In other words, in real concentrations, mixing these would be equivalent to 3L of solution with concentration 43.33%. This is exactly the concentration of the 3L that would produce a resultant solution of exactly 50% concentration, but you want a concentration less than 50% in the final solution, so (G) doesn’t work.
Answers: A B F A B C D E F G
GRE Quantitative Reasoning
Why did we subtract 50? What we’re doing here is comparing the percentages to 50%. So, in other words, we’re setting 50% to 0. The reason for doing this is that the question asks for a final solution of less than 50%.
So you write each 60% as “+10” because it’s 10 more than 50%. And you have 2L of “+ 10” and you want to add 3L of X to get something less than 0 (which you mean to represent less than 50%).
10 + 10 + x + x + x < 0 x < – __:203 x < –6
The concentration you want should be 6.67 less than 50%, or 43.33% phosphoric acid. This means that the average concentration of the 3L solution should be less than 43.33% phosphoric acid.
If that’s confusing, here is another way to do the problem:
2L × 60% phosphoric acid = 2 × 0.6 = 1.2L of pure phosphoric acid
Now you’re adding 3L to get less than 50% of a 5L solution. 50% of 5 is 2.5L of pure phosphoric acid.
So if you’re starting with 1.2L of the acid and you must have less than 2.5L in our final solution, then you must add less than
2.5 – 1.2 = 1.3L of pure phosphoric acid
In other words, the 3L added must contain less than a total of 1.3L pure phosphoric acid. So you look at the answer choices and multiply the liters by the percents to see which are less than 1.3L of pure acid. For example, 3L of 42% is okay because you get the following:
3 × 0.42 = 1.26L of acid, which is less than 1.3
Question 13
Difficulty: Hard · Percent Correct: 36.5% · Type: Numeric Entry
In a certain set of numbers, 12.5 is 1.5 units of standard deviation above the mean, and 8.9 is 0.5 units of standard deviation below the mean. What is the mean of the set? Give your answer to the nearest 0.1.
Both the mean and the standard deviation are unknown, so we have two unknowns. If we assign a variable to each, then each statement in the prompt allows us to set up an equation, and we would have two equations with two unknowns. Let mean = M and standard deviation = S. Then,
“12.5 is 1.5 units of SD above the mean”8 812.5 = M + 1.5S “8.9 is 0.5 units of SD below the mean”8 88.9 = M – 0.5S Answer: 9.8
We discussed two equations with two unknowns in the algebra section on page 451. Since M appears with a coefficient of 1 in both equations, we could eliminate M and find S by subtracting the equations.
Now that we have a value for S, we can plug this into either equation to find the M. M + (1.5)(1.8) = 12.5
M + 2.7 = 12.5
M = 12.5 – 2.7 = 9.8
This value is already rounded to the nearest 0.1, so we can enter this as is. The answer is 9.8. 12.52=2M2+21.5S −2[8.92=2M2−20.5S] 12.52−28.92=21.5S −2(−0.5S)2=21.5S +20.5S 3.62=22S 1.82=2S
GRE Quantitative Reasoning