Arithmetic
For questions in the Quantitative Comparison format (“Quantity A” and “Quantity B” given), the answer choices are always as follows:
(A)Quantity A is greater.
(B)Quantity B is greater.
(C)The two quantities are equal.
(D)The relationship cannot be determined from the information given.
For questions followed by a numeric entry box 
, you are to enter your own answer in the box. For questions followed by a fraction-style numeric entry box 
, you are to enter your answer in the form of a fraction. You are not required to reduce fractions. For example, if the answer is 
, you may enter 
or any equivalent fraction.
All numbers used are real numbers. All figures are assumed to lie in a plane unless otherwise indicated. Geometric figures are not necessarily drawn to scale. You should assume, however, that lines that appear to be straight are actually straight, points on a line are in the order shown, and all geometric objects are in the relative positions shown. Coordinate systems, such as xy-planes and number lines, as well as graphical data presentations, such as bar charts, circle graphs, and line graphs, are drawn to scale. A symbol that appears more than once in a question has the same meaning throughout the question.
| 1. | Quantity A 39 – (25 – 17) |
Quantity B 39 – 25 – 17 |
| 2. | Quantity A 14 – 3(4 – 6) |
Quantity B (4)(–3)(2)(–1) |
| 3. | Quantity A –5 × 1 ÷ 5 |
Quantity B –6 × 1 ÷ 6 |
4.What is the value of 5 – (4 – (3 – (2 – 1))) ?
| 5. | Quantity A |
Quantity B (–2)2 |
| 6. | Quantity A 53 – 52 |
Quantity B 5 |
| 7. | Quantity A –10 – (–3)2 |
Quantity B –[10 + (–3)2] |
| 8. | Quantity A (30,000,000)(2,000,000) |
Quantity B (15,000,000)(4,000,000) |
9.What is the sum of the numbers in the grid below?
10. Molly worked at an amusement park over the summer. Every two weeks, she was paid according to the following schedule: at the end of the first 2 weeks, she received $160. At the end of each subsequent 2-week period, she received $1, plus an additional amount equal to the sum of all payments she had received in previous weeks. How much money was Molly paid during the full 10 weeks of summer?
A book with 80,000 words costs $24 and a short story with 1,000 words costs $1.
| 11. | Quantity A Cost per word of the book |
Quantity B Cost per word of the short story |
| Ticket Prices at the Natural History Museum | ||
| Weekdays | Weekends & Holidays | |
| Child (ages 5–18) | $7 | $9 |
| Adult (ages 19–64) | $14 | $16 |
| Senior (ages 65+) | $8 | $10 |
| *Children under age 5 attend free | ||
| 12. | Quantity A The price for tickets at the Natural History Museum on a weekday for one 12-year-old and one 39-year-old |
Quantity B The price for tickets at the Natural History Museum on a weekend for one 4-yearold, two 8-year-olds, and one senior over 65 years old, after applying a coupon for $10 off the total cost |
On a certain train, tickets cost $6 each for children and $9 each for adults. The total train ticket cost for a certain group of six passengers was between $44 and $50.
| 13. | Quantity A The number of children in the group |
Quantity B The number of adults in the group |
14. If 617 is divided by 49, the sum of the tens digit and the tenths digit of the resulting number is what value?
(A)1
(B)5
(C)6
(D)7
(E)9
| 15. | Quantity A The number of days between May 30, 1917, and May 15, 1996, inclusive |
Quantity B The number of days between May 15, 1912, and May 30, 1991, inclusive |
Alfred’s Coffee Shop offers a “buy six cups of coffee, get one free” discount, and Boris’s Coffee Shop offers 15% off all orders of six or more cups of coffee. At both shops, the regular price of a single cup of coffee is $2.60.
| 16. | Quantity A The total cost for one order of seven single cups of coffee from Alfred’s |
Quantity B The total cost for one order of seven single cups of coffee from Boris’s |
17. In a certain ancient kingdom, the standard unit of measure was the “crown,” equal to 10 standard modern inches. An alternative unit of measure was the “scepter,” equal to 14 standard modern inches. If a tower measured 70 crowns tall, how many scepters tall was it?
(A)35
(B)49
(C)50
(D)75
(E)98
18. A total of $450 was donated to charity by 25 employees. If 15 employees donated at least $12 but less than $19 and 9 employees donated at least $19, what is the maximum amount, in dollars, that the last employee could have donated?
19. A tank has a capacity of 200 pints. How many gallons of water would it take to fill the tank to 
of its capacity? (1 gallon = 8 pints)
gallons
1 kilogram = 2.2 pounds
| 20. | Quantity A The number of kilograms in 44 pounds |
Quantity B The number of pounds in 44 kilograms |
21. If the formula for converting degrees Fahrenheit to degrees Celsius is 
, what is the value of F when C is 30?
(A)
(B)
(C)86
(D)
(E)112
22. On a trip, Joe’s car traveled an average of 36 miles per gallon of fuel. Approximately how many kilometers did the car travel on 10 liters of fuel? (5 miles = approximately 8 kilometers; 1 gallon = approximately 4 liters)
kilometers
23. How many 1-inch square tiles would it take to cover the floor of a closet that has dimensions 5 feet by 4 feet? (1 foot = 12 inches)
(A)20
(B)240
(C)1,440
(D)2,160
(E)2,880
Child A ate 
of a kilogram of chocolate and Child B ate 300 grams of chocolate. (1 kilogram = 1000 grams)
| 24. | Quantity A The weight, in grams, of the chocolate that Child A ate |
Quantity B Twice the weight, in grams, of the chocolate that Child B ate |
25. Out of 5.5 billion bacteria grown for an experiment, 1 in 75 million has a particular mutation. Approximately how many of the bacteria have the mutation?
(A)7
(B)73
(C)733
(D)7,333
(E)73,333
26. A particular nation’s GDP (Gross Domestic Product) is $4.5 billion. If the population of the nation is 1.75 million, what is the per capita (per person) GDP, rounded to the nearest dollar?
(A)$3
(B)$25
(C)$257
(D)$2,571
(E)$25,714
27. Global GDP (Gross Domestic Product) was $69.97 trillion in 2011. If the world population for 2011 was best estimated at 6,973,738,433, approximately what was the global GDP per person?
(A)$10
(B)$100
(C)$1,000
(D)$10,000
(E)$100,000
28. The runners on a cross country team ran a 5-mile race at average (arithmetic mean) speeds ranging from 4 miles per hour to 7 miles per hour, inclusive. Which of the following are possible race completion times for individual members of the team?
Indicate all such times.
- 36 minutes
- 48 minutes
- 60 minutes
- 75 minutes
- 90 minutes
- 120 minutes
Arithmetic Answers
1. (A). First simplify inside the parentheses:
39 – (25 – 17) =
39 – 8 =
31
You could also distribute the minus sign to get 39 – 25 + 17 if you prefer. Quantity B is equal to –3, so Quantity A is greater. If you noticed right away that the minus sign would distribute in Quantity A but not Quantity B, you could have picked (A) without doing any arithmetic.
2. (B). This question is testing PEMDAS (Parentheses/Exponents, then Multiplication/Division, then Addition/Subtraction), at least in Quantity A. Make sure that you simplify inside the parentheses, and then multiply, before subtracting:
14 – 3(4 – 6) =
14 – 3(–2) =
14 + 6 =
20
Quantity B is (4)(–3)(2)(–1) = 24.
3. (C). The two quantities are equal. Note that in Quantity A:
–5 × 1 ÷ 5 =
–5 ÷ 5 =
–1
In Quantity B:
–6 × 1 ÷ 6 =
–6 ÷ 6 =
–1
4. 3. Make sure to begin with the innermost parentheses:
5 – (4 – (3 – (2 – 1))) =
5 – (4 – (3 – 1)) =
5 – (4 – 2) =
5 – (2) =
3
5. (B). In Quantity A, the exponent should be computed before taking the negative of the value—in accordance with PEMDAS. Thus, you get –8/2 = –4.
In Quantity B:
(–2)2 =
(–2)(–2) =
4
6. (A). Do not make the mistake of thinking that 53 – 52 = 51. You cannot just subtract the exponents when you are subtracting two terms with the same base! Instead, compute the exponents and subtract:
53 – 52 =
125 – 25 =
100
Quantity A is greater. Alternatively, you could factor out 52 (this is an important technique for large numbers and exponents where pure arithmetic would be impractical):
53 – 52 =
52(51 – 1) =
52(4) =
100
7. (C). In Quantity A:
–10 – (–3)2 =
–10 – (9) =
–19
In Quantity B:
–[10 + (–3)2] =
–[10 + (9)] =
–19
8. (C). The GRE calculator will not be able to handle that many zeros. Start this calculation on paper. To make things easier, you could cancel as many zeros as you want, as long as you do the same operation to both quantities. For instance, you could divide both sides by 1,000,000,000,000 (just think of this as “1 with 12 zeros”), to get:
| Quantity A | Quantity B |
| (30)(2) | (15)(4) |
Or, just use a bit of logic: 30 million times 2 million is 60 million million, and 15 million times 4 million is also 60 million million. (A “million million” is a trillion, but this doesn’t matter as long as you’re sure that each Quantity will have the same number of zeros.)
9. 147. There are several patterns in the grid, depending on whether you look by row or by column. Within each row, there are positive and negative terms at the beginning that cancel each other. For example, in the first row, you have –2 + 2 = 0 and –1 + 1 = 0. The only terms in the first row that contribute to the sum are 3 and 4, in the far-right columns. The same is true for the other rows.
Thus, the sum of the grid is equal to the sum of only the two far-right columns. The sum in the first row in those columns is 3 + 4 = 7; the sum in the next row is 6 + 8 = 14, etc. The sum in the final row is 18 + 24 = 42. Add 7 + 14 + 21 + 28 + 35 + 42 in your calculator to get 147.
10. $2,575. At the end of the first two weeks, Molly received $160. At the end of the fourth week, she received $1, plus $160 for the total she had been paid up to that point, for a total of $161. At the end of the sixth week, she received $1, plus ($160 + $161), or $321, for the total she had been paid up to that point, making the sixth week total $322. To keep track, put these values in a table:
| Week # | Paid This Week($) | Cumulative Pay Including This Week ($) |
| 2 | 160 | 160 |
| 4 | 160 + 1 = 161 | 160 + 161 = 321 |
| 6 | 321 + 1 = 322 | 321 + 322 = 643 |
| 8 | 643 + 1 = 644 | 643 + 644 = 1,287 |
| 10 | 1,287 + 1 = 1,288 | 1,287 + 1,288 = 2,575 |
11. (B). In Quantity A, 
= 0.0003, or 0.03 cents per word. In Quantity B, 
= 0.001, or 0.1 cents per word. Quantity B is greater. Note that the calculation was not strictly necessary—it would have been more efficient to notice that the book costs 24 times the story but has 80 times the words. (Then remember to choose the greater number!)
12. (A). The ticket for the 4-year-old in Quantity B costs $0 (children under age 5 attend free).
Quantity A: The price for tickets at the Natural History Museum on a weekday for one 12-year-old and one 39-year-old = $7 + $14 = $21.
Quantity B: The price for tickets at the Natural History Museum on a weekend for one 4-year-old, two 8-year-olds, and one senior over 65 years old, after applying a coupon for $10 off the total cost is equal to ($0 + $9 + $9 + $10) – $10 = $18.
Quantity A is greater.
13. (D). Even though the range of costs ($44 to $50) is fairly small, there is still more than one possibility. A good way to work this out is to start with the simplest scenario: 3 adults and 3 children. Their tickets would cost 3(9) + 3(6) = $45. That’s in the range, so it’s one possibility.
Since children’s tickets are cheaper, you don’t want to add more children to the mix (4 children, 2 adults will give you too small a total), but try switching out 1 child for 1 adult.
For 4 adults and 2 children, tickets would cost 4(9) + 2(6) = $48. Thus, Quantity A and Quantity B could be equal, or Quantity B could be greater, so the relationship cannot be determined from the information given.
14. (C). Divide 617 by 49 with the calculator to get 12.5918…. The tens digit is 1. The tenths digit is 5. The answer is 1 + 5 = 6.
15. (B). Calculating the number of days in each quantity would be time consuming; each date range includes a lot of days! Instead, a faster approach is to compare the starting and ending dates for the two quantities.
Quantity A: The number of days between May 30, 1917 and May 15, 1996, inclusive.
Quantity B: The number of days between May 15, 1912 and May 30, 1991, inclusive.
All of the start and end dates are in May, but both the starting and ending years in Quantity B are 5 years earlier than those in Quantity A. Thus, the approximate whole number of years in both ranges is the same (about 79 years). However, the range in Quantity A starts later in the month and ends earlier in the month than the range in Quantity B. Both differences mean that Quantity B includes a greater number of days.
Alternatively, consider the following: the date range in Quantity A is about half a month less than 79 years, while the date range in Quantity B is about half a month greater than 79 years. Quantity B is greater.
16. (A). At Alfred’s, an order of 7 single cups of coffee would cost 6($2.60) = $15.60, because the 7th cup is free.
At Boris’s, an order of 7 single cups of coffee would receive the 15% discount: 7($2.60)(0.85) = $15.47.
Alternatively, because the non-discounted price of a single coffee ($2.60) and the number of single cups of coffee ordered is common to both quantities, an actual cost calculation is optional. Instead, you could compare the discounts in percent terms. At Alfred’s, “buy six drinks get one free” means that, for every seven drinks you purchase, the last one is free. That’s one in seven drinks free, or 
off, which is about 
= 14.29% off. This is smaller than the 15% discount at Boris’s, so the total cost at Alfred’s is greater. By the way, remember to pick the greater quantity (Quantity A), not the “better deal”!
17. (C). A tower that was 70 crowns tall was 70 crowns × 10 inches/crown = 700 inches tall. This same 700-inch tower, measured in scepters, would be 
= 50 scepters tall. Also, note that since the scepter is longer than the crown in absolute terms, fewer scepters will “fit” in the height of the tower, so any choices 70 or greater could be eliminated right away.
18. $99. To maximize the last employee’s contribution, minimize everyone else’s. If 15 employees could have donated a minimum of $12 and 9 employees could have donated a minimum of $19:
15(12) + 9(19) = 180 + 171 = 351.
So, the minimum that all 24 of these employees could have given is $351. Therefore, the maximum that the 25th employee could have given is 450 – 351 = 99, or $99.
19. 7.5 gallons.
First find out how many pints of the capacity is:
Now convert pints to gallons:
20. (B). To compare the values, convert the quantity on the left from pounds to kilograms and the quantity on the right from kilograms to pounds:
Quantity A |
Quantity B |
Before actually multiplying, notice that the Quantity A is divided by 2.2, while the Quantity B is multiplied by 2.2. Quantity B will be greater.
You could also solve this by noticing that the two quantities involve reverse calculations, with the same number of units (44). Since a kilogram is heavier than a pound, it takes more of the lighter pounds to equal 44 heavier kilograms than it takes of the heavier kilograms to equal 44 of the lighter pounds.
21. (C). Start by plugging 30 in for C in the equation:
30 = 
(F – 32)
Now isolate F. Begin by multiplying both sides by 
:
× 30 = F – 32
To multiply 30 by quickly, reduce before multiplying:
 × 6 |
= | F – 32 |
| 54 | = | F – 32 |
| 86 | = | F |
22. 144 kilometers. Convert miles per gallon to kilometers per liter by multiplying by the conversion ratios such that both the miles and gallons units are canceled out:
The car has 10 liters of fuel in the tank:
10 liters × 14.4 kilometers/liter = 144 kilometers
23. (E). There is a hidden trap in this question. Remember that the dimensions of this room are square feet, not feet (because 5 feet × 4 feet = 20 square feet). To avoid this trap, you should convert the dimensions to inches first, then multiply.
5 feet × 4 feet = 60 inches × 48 inches
The dimensions of the closet in inches are 60 inches by 48 inches, or 60 × 48 = 2,880 square inches. Each tile is 1 square inch, so it will take 2,880 tiles to cover the floor.
24. (C). 
of a kilogram is 600 grams. Twice 300 grams is also 600 grams. The two quantities are equal.
25. (B). One good way to keep track of large numbers (especially those that won’t fit in the GRE calculator!) is to use scientific notation (or a loose version thereof—for instance, 5.5 billion in scientific notation is 5.5 × 109, but it would be equally correct for your purposes to write it as 55 × 108).
5.5 billion = 5,500,000,000 = 5.5 × 109
75 million = 75,000,000 = 75 × 106
Since 1 in 75 million of the bacteria have the mutation, divide 5.5 billion by 75 million:
, which can also be written as 
. Only 
needs to go in the calculator, to yield 0.0733333 … Since 
is 103, move the decimal three places to the right to get 73.333 …, or answer choice (B).
Or, write one number over the other and cancel out the same number of zeros from the top and bottom before trying to use the calculator: 
26. (D). This problem is asking you to divide $4.5 billion by 1.75 million. When dealing with numbers that have many zeros, you can avoid mistakes by using scientific notation or by writing out the numbers and canceling zeros before using the calculator:
4.5 billion = 4,500,000,000 = 4.5 × 109
1.75 million = 1,750,000 = 1.75 × 106
The answer is (D). Alternatively, write one number on top of the other in fully-expanded form, and cancel zeros before using the calculator:
27. (D). This problem is asking you to divide $69.97 trillion by 6,973,738,433. When dealing with numbers that have many zeros, you can avoid mistakes by using scientific notation or by writing out the numbers and canceling zeros before using the calculator.
Before doing that, however, look at the answers—they are very far apart from one another, which gives you license to estimate. GDP is about 70 trillion. Population is about 7 billion. Thus:
28. 48 minutes, 60 minutes, and 75 minutes. Rate × Time = Distance, thus 
= Time.
The race times range from a maximum time of 
= 1.25 hours = 75 minutes for the slowest runner to a minimum time of 
= about 0.71429 hours = about 42.86 minutes for the fastest runner. All answers between (and including) 42.86 minutes and 75 minutes are correct.