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.

(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).

2. Answer Choices

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.

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\,\dfrac{1}{3} = \dfrac{10}{3}. Then x2  = \dfrac{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.

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Data Sufficiency

Probability is a common topic for Data Sufficiency questions.

A jar has 10 marbles, a mix of red and white. Two marbles are randomly chosen from the jar. If b is the probability that both will be red, is b > \dfrac{1}{3}?
(1) Less than \dfrac{1}{2} of the marbles in the jar are white.
(2) The probability that 1 white marble and 1 red marble will be chosen together is \dfrac{7}{15}.
A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D. EACH statement ALONE is sufficient.
E. Statements (1) and (2) TOGETHER are NOT sufficient.

Solution

This is a dependent probability problem. To find the probability of choosing 2 red marbles, you need to figure out the probability that the first marble will be red and that the second marble will be red. In this case, the question wants to know if that probability is larger than \dfrac{1}{3}.

Statement (1) says that less than half the marbles are white, which means that more than half the marbles are red. The best way to approach this is to systematically (but quickly) figure out what the probability of drawing two red marbles is for each scenario. Make a table where
(number of red) > (number of white) and the numbers sum to 10.

Number of
red marbles
Number of
white marbles
Probability of
2 red marbles
6 4 \dfrac{6}{10} × \dfrac{5}{9}

= \dfrac{1}{3}
7 3 \dfrac{7}{10} × \dfrac{6}{9}

= \dfrac{7}{15}
8 2 \dfrac{8}{10} × \dfrac{7}{9}

= \dfrac{28}{45}
9 1 \dfrac{9}{10} × \dfrac{8}{9}

= \dfrac{4}{5}
10 0 \dfrac{10}{10} × \dfrac{9}{9}

= 1

When less than half the marbles are white, the probability of choosing 2 red marbles can be greater than or equal to \dfrac{1}{3}. Statement (1) is not sufficient.

Statement (2) says the probability of choosing one red marble and one white marble is \dfrac{7}{15}. This is a trap. Since the probability given is exact, it may seem that only one scenario of red marbles and white marbles will work.

If you make a table of all the scenarios, you will see that when there are 7 red marbles and 3 white marbles, the probability of choosing one of each is \dfrac{7}{15}. However, it is also true in reverse: If there were 7 white marbles and 3 red marbles, the probability would also be \dfrac{7}{15}. Therefore, Statement (2) is not sufficient.

Combining Statements (1) and (2) does give enough information. From Statement (2) you know that there must be 7 of one color and 3 of the other. From Statement (1) you know that there must be more red than white. The only combination that fits this is 7 red marbles and 3 white marbles.

The correct answer is choice (C).

From a class of 12 students, two students will be randomly chosen simultaneously. If g is the number of girls in the class, what is the value of g?

(1) The probability that two girls will be chosen together is \dfrac{1}{11}.

(2) The probability that one boy and one girl will be chosen is \dfrac{16}{33}.

A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D. EACH statement ALONE is sufficient.
E. Statements (1) and (2) TOGETHER are NOT sufficient.

Solution

To answer this question, you don’t need to do the math.

Look at Statement (1). You know there is a specific number of girls (g). Since each number of girls yields a different probability of choosing 2 girls, there must be only one specific number that would yield \dfrac{1}{11}. So this is enough information to know the value of g, the number of girls.

Statement (2) requires a little more thought. Pairing a boy and a girl can be (bg) or (gb). As in many pairings, there are 2 different pairs that will give the same probability. Statement (2) is not sufficient.
(Note: If you make the table, you will see that there are two ways to get \dfrac{16}{33}, with 4 boys and 8 girls or with 4 girls and 8 boys.)

Since only Statement (1) gives enough information, the correct answer is choice (A).

In a hotel with single rooms and double rooms, what is the probability that a room chosen at random will be a double room painted red?

(1) \dfrac{1}{6} of the rooms in the hotel are painted red.

(2) \dfrac{2}{3} of the hotel’s rooms are double rooms.

A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D. EACH statement ALONE is sufficient.
E. Statements (1) and (2) TOGETHER are NOT sufficient.

Solution

Statement (1) gives the fraction of rooms painted red, but does not say anything about double rooms.

Statement (2) gives the fraction of the rooms that are double rooms, but does not say anything about red rooms.

Putting the statements together still does not give enough information. Neither statement says how many of the red rooms are double rooms. For example, if there were 12 rooms you would know that 2 were red and 8 were doubles. But you don’t know if any of the doubles are red. There is no information connecting the two categories, so you cannot find the probability.

The correct answer is choice (E).

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