GRE Arithmetic Practice Questions Practice Questions with Answers

Arithmetic constitutes approximately 25% to 30% of the Quantitative Reasoning section, making it a cornerstone for achieving a high score. Success on this portion of the exam requires more than just basic calculation skills; it demands a deep understanding of number properties, fractions, percentages, and ratios. By engaging with high-quality GRE arithmetic practice questions, you can sharpen your mental math and identify the logical traps frequently set by the test makers at ETS. This guide provides a structured approach to these fundamental topics to ensure you are fully prepared for test day.

Concept Explanation

GRE arithmetic encompasses the study of integers, fractions, decimals, real numbers, and their various properties and operations. At its core, this subject involves understanding how numbers interact through addition, subtraction, multiplication, and division, as well as more complex concepts like prime factorization, remainders, and exponents. To excel in this section, you must be comfortable with the GRE Prep requirements for number theory, such as the difference between rational and irrational numbers and the rules governing even and odd integers. For instance, knowing that the product of two odd integers is always odd can save precious seconds during a timed section. Proficiency in these areas allows you to manipulate expressions efficiently and solve Word problems that translate real-world scenarios into mathematical equations.

Solved Examples

1. Number Properties: If $n$ is an even integer and $m$ is an odd integer, which of the following must be an odd integer?
   1. $n + m$ is the correct approach. Let $n = 2$ and $m = 3$.
   2. Substitute values: $2 + 3 = 5$. Since 5 is odd, the property holds.
   3. General rule: $\text{Even} + \text{Odd} = \text{Odd}$. Therefore, $n + m$ is always odd.
2. Percentages: A laptop originally priced at $800 is discounted by 20%, and then an additional 10% discount is applied to the sale price. What is the final price?
   1. Calculate the first discount: $20\% \text{ of } 800 = 0.20 \times 800 = 160$.
   2. Find the new price: $800 - 160 = 640$.
   3. Calculate the second discount: $10\% \text{ of } 640 = 0.10 \times 640 = 64$.
   4. Final price: $640 - 64 = 576$. Note: You cannot simply add the percentages to get a 30% discount.
3. Ratios: The ratio of apples to oranges in a basket is 3:5. If there are 40 oranges, how many apples are there?
   1. Set up a proportion: $\frac{3}{5} = \frac{x}{40}$.
   2. Cross-multiply to solve for $x$: $5x = 3 \times 40$.
   3. $5x = 120$.
   4. Divide by 5: $x = 24$. There are 24 apples.

Practice Questions

1. Which of the following is a prime factor of the sum $2^5 + 2^6$?

2. A store clerk increased the price of a jacket by 25%. After the increase, the new price is $150. What was the original price of the jacket?

3. If $x$ and $y$ are positive integers such that $x$ is a multiple of 6 and $y$ is a multiple of 9, then $xy$ must be a multiple of which of the following? (Select all that apply: 12, 18, 54, 72)

4. What is the remainder when $7^{15}$ is divided by 5?

5. Quantity A: The number of prime numbers between 10 and 20. Quantity B: The number of prime numbers between 20 and 30.

6. A mixture is composed of liquids A, B, and C in the ratio 2:3:5. If the total volume of the mixture is 500 ml, how many more milliliters of liquid C are there than liquid A?

7. If $\frac{3}{x} = \frac{y}{4}$, what is the value of $xy$?

8. What is the greatest common divisor (GCD) of 48 and 72?

9. If the average (arithmetic mean) of five consecutive integers is 12, what is the sum of the least and greatest of these integers?

10. Solve for $z$: $0.4z + 1.2 = 3.6$.

Answers & Explanations

1. 3: Factor out the smallest power: $2^5(1 + 2^1) = 2^5(3)$. The prime factors are 2 and 3.
2. $120: Let $P$ be the original price. $1.25P = 150$. Dividing 150 by 1.25 gives $P = 120$.
3. 18 and 54: $x = 6a$ and $y = 9b$, so $xy = 54ab$. Since 54 is a multiple of 18 and 54, both are correct. 12 and 72 are not guaranteed.
4. 3: Look for a pattern in units digits or remainders: $7^1 \div 5$ rem 2; $7^2 \div 5$ rem 4; $7^3 \div 5$ rem 3; $7^4 \div 5$ rem 1. The cycle is 2, 4, 3, 1. For $7^{15}$, $15 \div 4$ has a remainder of 3, so the remainder is the 3rd in the cycle, which is 3.
5. Quantity A is greater: Primes between 10-20 are {11, 13, 17, 19} (Total: 4). Primes between 20-30 are {23, 29} (Total: 2).
6. 150 ml: Total parts = $2+3+5 = 10$. One part = $500/10 = 50$ ml. Liquid C = $5 \times 50 = 250$. Liquid A = $2 \times 50 = 100$. Difference = $250 - 100 = 150$.
7. 12: Cross-multiplying the equation $\frac{3}{x} = \frac{y}{4}$ gives $3 \times 4 = x \times y$, so $xy = 12$.
8. 24: Prime factorization of $48 = 2^4 \times 3$. Prime factorization of $72 = 2^3 \times 3^2$. The GCD is the product of the lowest powers of common factors: $2^3 \times 3 = 8 \times 3 = 24$.
9. 24: In a set of consecutive integers, the mean is the middle number. The integers are 10, 11, 12, 13, 14. Sum of least and greatest = $10 + 14 = 24$.
10. 6: Subtract 1.2 from both sides: $0.4z = 2.4$. Divide by 0.4: $z = 6$.

To improve your efficiency, you might use an AI Question Generator to create variations of these problems, which helps in recognizing underlying patterns rather than just memorizing steps. For students who also study for other standardized exams, the logic used here is quite similar to the quantitative sections found in ACT Prep materials.

1. If the ratio of x to y is 4:7 and the sum of x and y is 77, what is the value of x?

• A28
• B33
• C44
• D49

Frequently Asked Questions

What arithmetic topics are most common on the GRE?

The most frequent topics include percent changes, ratios, prime factorization, and number properties like even/odd rules. You should also be prepared for operations with fractions and decimals.

Can I use a calculator for GRE arithmetic questions?

Yes, an on-screen calculator is provided during the GRE Quantitative sections. However, it is basic, so relying on mental math for simple steps is often faster.

How is number theory tested in the GRE arithmetic section?

Number theory is tested through questions on divisibility, remainders, and prime numbers. These often appear in Quantitative Comparison formats where you must determine the relationship between two numerical expressions.

What is the best way to avoid silly mistakes in arithmetic?

Always double-check your signs (positive vs. negative) and ensure you are answering the specific question asked, such as finding "x" versus "x + 5". Using an AI Exam Simulator can help you practice under timed pressure to minimize errors.

Are square roots and exponents considered arithmetic on the GRE?

Yes, the properties of exponents and square roots are core arithmetic concepts. You must know how to simplify radical expressions and apply the laws of exponents to solve equations.