## Overview

While problem solving questions may be familiar to you, data sufficiency questions are—within the standardized test world—unique to the GMAT. You’ll be given a question followed by two statements that provide information that might be useful in answering the question. The true task is not necessarily to answer the actual question (though that may happen in the process), but to determine what information you’ll need in order to answer it in the first place. You may need one of the statements, maybe the other, maybe both together, maybe either would work individually, or maybe neither helps at all!

Data sufficiency questions are designed to measure the test-taker’s ability to:

- Analyze a quantitative problem
- Recognize which information is relevant
- Determine at what point there is sufficient information to solve a problem

This section requires knowledge of the following:

- Arithmetic
- Elementary algebra
- Basic concepts of geometry

Click through our practice questions to test your data sufficiency problem-solving skills. Each question is followed by the answer and a detailed explanation of how to solve the problem.

## Question 1

If a is an integer, is b an integer?

- The average (arithmetic mean) of a and b is not an integer.
- The average (arithmetic mean) of 3a, 2b, and a + 12, is 2b.

## Answer & Explanation

We need to determine whether or not b is an integer.

Statement 1 tells us that the average of a and b is not an integer. Recall that the average of a group is equal to the sum of the group’s terms divided by the number of terms. Thus, the average of a and b is . The average will __not__ be an integer whenever a + b is __not__ an even number.

If a = 1 and b = 2, the average of the two numbers is , which is not an integer. If a = 1 and b = 1.5, the average of a and b is , which is also not an integer. Since we can pick both integer and non-integer values for b that satisfy Statement 1, Statement 1 alone is insufficient to answer the question. The answer must be either B, C, or E.

Statement 2 tells us that the average of 3a, 2b, and a + 12 is 2b. Using the average formula, we can write this statement as the equation . If we combine the a terms in the numerator of the fraction on the left side, we get . Multiplying both sides of the equation by 3 gives us: 4a + 2b + 12 = 6b. Subtracting 2b from both sides, we get: 4a + 12 = 4b. Finally, dividing both sides of the equation by 4, we get a + 3 = b. Since a is an integer, a + 3 must also be an integer; therefore, b is an integer. Statement 2 alone is sufficient to answer the question.

The correct answer choice is **B**.

## Question 2

Is the positive integer k a multiple of 54?

- k is a multiple of 6.
- k is a multiple of 9.

## Answer & Explanation

We must determine whether or not k is a multiple of 54.

Statement 1 tells us that k is a multiple of 6. Let’s look at a few multiples of 6 to see if any/all of them are multiples of 54. 6 and 12 are multiples of 6, but neither is a multiple of 54. On the other hand, 54 is a multiple of both 6 and 54. Since there are multiples of 6 that both are and are not multiples of 54, Statement 1 alone is insufficient to answer the question. The answer must be B, C, or E.

Statement 1 tells us that k is a multiple of 9. We can use the same logic we applied above: 9 and 18 are both multiples of 9, but neither is a multiple of 54. However, 54 is a multiple of both 9 and 54. Therefore, we can also conclude that Statement 2 alone is insufficient to answer the question. The answer must be C or E.

Taken together, Statements 1 and 2 tell us that *k* is a multiple of both 6 and 9. We cannot simply multiply together 6 × 9 = 54 and conclude that *k* is a multiple of 54. For example, *k* could be 18 or 36, since 18 and 36 are multiples of both 6 and 9. (The reason for this is that the Least Common Multiple of 6 and 9 is 18, not 54).

Since we can still find values of*k* that both are and are not multiples of 54, Statements 1 and 2 together do not provide us with enough information to answer the question. The correct answer choice is **E**.

## Answer & Explanation

The question asks whether *n* is negative. Remember that when a negative number is raised to an even exponent, the result is always positive.

Statement 1 tells us that (1 – *n*^{2}) < 0. We can add *n*^{2} to both sides of the inequality to get 1 < *n*^{2}, or *n*^{2} > 1. If the square of a number is greater than 1, the number itself must either be greater than 1, or less than –1. For example, (–2)^{2} = 4. Since we do not know if *n* > 1 or *n* < –1, Statement 1 is insufficient. The answer must be B, C, or E.

Statement 2 tells us that *n*^{2} – *n* – 2 < 0. Since the expression on the right is quadratic, we should try to factor it. In this case, *n*^{2} – *n* – 2 = (*n* – 2)(*n* + 1), so we can rewrite the inequality as (*n* – 2)(*n* + 1) < 0. This tells us that the product of two expressions is less than zero. This can only be true if one of the expressions is positive and the other is negative. (*n* – 2) is positive if *n* > 2, zero if *n* = 2, and negative if *n* < 2. Similarly, (*n* + 1) is positive if *n* > –1, zero if *n* = –1, and negative if *n* < –1. We can figure out when the product of these two terms is negative by using a diagram:

By representing visually where each of the expressions is positive and negative, we can see more clearly that (*n* + 1)(*n* – 2) is negative when –1 < *n* < 2. In this region, (*n* + 1) is positive and (*n* – 2) is negative. Since we do not know if *n* is positive or negative, Statement 2 is also insufficient. The answer must be C or E.

Taken together, Statement 1 tells us that *n* > 1 or *n* < –1, and Statement 2 tells us that –1 < *n* < 2. The only overlap between these two regions is 1 < *n* < 2. Since 1 < *n* < 2, *n* must be positive. The answer to the question in the prompt is No. Since both statements together are sufficient to answer the question, answer choice **C** is correct.