Typically one-third of all the quantitative questions are Data Sufficiency problems. Each problem consists of a question and two statements.

Data Sufficiency problems do not require you to find an exact answer. Instead, you must determine if the given statements provide enough information to answer the question.

##### Example

(question) What is the value of x + y?

(statement) (1) x = 5
(statement) (2) y = 10

### 1. Directions

These are the directions for data sufficiency problems. They don’t change; they are the same for all data sufficiency problems.

Directions: In each of the problems, a question is followed by two statements containing certain data. You are to determine whether the data provided by the statements are sufficient to answer the question. Choose the correct answer based upon the statement’s data with your knowledge of mathematics and your familiarity with everyday facts (such as the number of minutes in an hour or the meaning of die or dice).

Note: All numbers used are real numbers.

Diagrams accompanying a problem will agree with information given in the question, but may not agree with additional information given in statements (1) and (2).

Data sufficiency problems always have the same five answer choices. Memorize the choices so you don’t waste time reviewing them on test day.

(A) Statement 1 ALONE is sufficient but statement 2 alone is not sufficient to answer the question asked.

(B) Statement 2 ALONE is sufficient but statement 1 alone is not sufficient to answer the question asked.

(C) BOTH statements 1 and 2 TOGETHER are sufficient to answer the question but NEITHER statement ALONE is sufficient.

(D) EACH statement ALONE is sufficient to answer the question.

(E) Statements 1 and 2 TOGETHER are NOT sufficient to answer the question asked, and additional data is needed.

You need to understand each answer choice and how it relates to the others.
One way to do that is to rewrite the choices into simpler sentences.

(A) Statement 1 is enough.

• Notice that choice (A) requires that Statement 2 is NOT sufficient.

(B) Statement 2 is enough.

• Notice that choice (B) requires that Statement 1 is NOT sufficient.

(C) Both statements 1 and 2 are needed.

• Together the statements can answer the question, but neither statement by itself can answer the question.

(D) Each statement alone is enough.

• Either statement by itself can answer the question.

(E) The statements together are not enough.

##### Example

What is the value of x + y?

(1) x = 5
(2) y = 10

### Solution

Statement (1) tells you that x = 5. This is not enough information to answer the question, since it asks about x and y.

Statement (2) tells you that y = 10. This is not enough information to answer the question, since it asks about x and y.

If you combine the two statements, you could determine the value.
x + y = 15

Since you need both statements, the answer is option (C).

BUT remember that the problem is only asking IF the value can be found. You did not need to find the actual value (15) to answer the question.

### 3. The Basic Steps

It takes mental discipline to progress through the data sufficiency problems. The test writers deliberately build tricks into each problem. They are testing whether you can think, not whether you can calculate.

Do not think in terms of “What is the exact value?” or “Is this true or false?”

Instead, focus on one issue: “Is there enough information to answer the question?” Look at each statement and ask yourself if it provides enough information to arrive at a conclusion.

There are three basic questions that you must ask yourself on every data sufficiency problem. Consider statements (1) and (2) one at a time.

Step 1: Can you answer the question using the information from statement (1) only?

Step 2: Can you answer the question using the information from statement (2) only?

Step 3: If the answer to both of those is “no,” then ask yourself: Can you answer the question if you combine the information from both statements?

### 4. What is “Sufficient”?

Sufficient does not mean that a statement is necessarily correct or true, just that it can be used to answer the question. A statement is still sufficient if it proves the answer is “no” or that the exact value cannot be found.

##### Example

Does y + 3 = 1?

(1) y is positive.
(2) y is an odd number.

### Solution

For y + 3 to equal 1, the value of y must be negative. But Statement (1) says y is positive. So Statement (1) is sufficient because it tells you the answer is no.

For y + 3 to equal 1, y must be even. (Remember that odd + odd = even and even + odd = odd.) So Statement (2) is sufficient because it tells you the answer is no.

Since each statement alone is sufficient, the correct answer is option (D).

Is x2  ≤ 3x?

(1) x > 0

(2) x < 4

### Solution

Try plugging in a few values with x > 0.

 x x2 3x x2 ≤ 3x? 1 1 3 yes 3 9 9 yes 4 16 12 no

So Statement (1) is insufficient because the answer is “sometimes.”
Try plugging in a few values with x < 4.

 x x2 3x x2 ≤ 3x? -1 1 -3 no 0 0 0 yes 2 4 6 yes

So Statement (2) is insufficient because the answer is “sometimes.”

Combining statements (1) and (2), do all the values between 0 and 4 work? Looking at the values you tried, the answer is yes.

But be careful. The trick here is the question does not say whether x is an integer.

Try x = 3 1/3 = 10/3. Then x2  = 100/9 ≈ 11 and 3x = 10.

Since combining the statements is insufficient, the correct answer is option (E).

### 5. Process of Elimination

As you go through each statement, you can eliminate answer options. Even if you can only determine if one of the statements is sufficient (or insufficient), you can eliminate at least two answer choices.

“Statement 1 is sufficient” eliminates choices B, C and E, which require (1) to be insufficient.

“Statement 1 is insufficient” eliminates choices A and D, which require (1) to be sufficient.

“Statement 2 is sufficient” eliminates choices A, C and E, which require (2) to be insufficient.

“Statement 2 is insufficient” eliminates choices B and D, which require (2) to be sufficient.

Another way to look at which choices are eliminated is to look at what options remain available.  If (1) is sufficient, immediately cross out B, C and E. If (1) is insufficient, immediately cross out A and D.