GMAT Math : Data-Sufficiency Questions

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Example Questions

Example Question #13 : Dsq: Understanding Absolute Value

Let A,B,C be any three (not necessarily distinct) integers.

At how many points does the graph of the function

f (x)= A |x-B | + C

intersect the -axis?

Statement 1: and are nonzero integers of opposite sign.

Statement 2: and are nonzero integers of opposite sign.

Possible Answers:

EITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation:

To determine the point(s), if any, at which the graph of a function intersects the -axis, set f(x)= 0 and solve for .

f (x)= A |x-B | + C

A |x-B | + C = 0

A |x-B | = - C

$|x-B | = - \frac{C}{A}$

At this point, we can examine the equation. Since the absolute value of a number must be nonnegative, the sign of $- \frac{C}{A}$ tells us how many solutions exist to this equation. If $- \frac{C}{A} < 0$ , there is no solution, and therefore, the graph of does not intersect the -axis. If $- \frac{C}{A} = 0$ , then there is one solution, and, therefore, the graph of intersects the -axis at exactly one point. If $- \frac{C}{A} > 0$ , then there are two solutions, and, therefore, the graph of intersects the -axis at exactly two points.

To determine the sign of $- \frac{C}{A}$ , we need to whether the signs of both and are like or unlike, or that C = 0 . Either statement alone eliminates the possibility that C = 0 , but neither alone gives the signs of both and . However, if both statements are assumed, then, since and have the opposite sign as , they have the same sign. This makes $\frac{C}{A} > 0$ and $- \frac{C}{A} < 0$ , so the graph of can be determined to not cross the -axis at all.

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Example Question #181 : Algebra

Which, if either, is the greater number: or ?

Statement 1: M = |N - 10|

Statement 2: N = |M + 10|

Possible Answers:

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Correct answer:

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Explanation:

Statement 1 alone gives insufficient information, as is seen in these two cases:. For example, if N = 15 , then

M = |N - 10|

M = |15 - 10| = |5| = 5

M < N

However, if N = 0 , then

M = |N - 10|

M = |0 - 10| = |-10| = 10

M > N

Therefore, it is not clear which, if either, of and is greater.

Now assume Statement 2 alone.

If is negative, then , which, being an absolute value of a number, must be nonnegative, is the greater number. If is positive, then so is M + 10 , so

N = |M + 10| = M + 10 .

Therefore,

N > M .

is the greater number in either case.

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Example Question #11 : Dsq: Understanding Absolute Value

Let A,B,C be any three (not necessarily distinct) integers.

At how many points does the graph of the function

f (x)= A |x-B | + C

intersect the -axis?

Statement 1: B < 0

Statement 2: C< 0

Possible Answers:

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

To determine the point(s), if any, at which the graph of a function intersects the -axis, set f(x)= 0 and solve for .

f (x)= A |x-B | + C

A |x-B | + C = 0

A |x-B | = - C

$|x-B | = - \frac{C}{A}$

At this point, we can examine the equation. For a solution to exist, since the absolute value of a number must be nonnegative, it must hold that $- \frac{C}{A} \geq 0$ . This happens if and are of opposite sign, or if C = 0 . However, Statement 2 tells us that C< 0 , and neither statement tells us the sign of . The two statements together provide insufficient information.

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Example Question #1 : Arithmetic

Is $\dpi{100} \small x+y$ odd?

(1) $\dpi{100} \small x$ is odd

(2) $\dpi{100} \small x-y$ is even

Possible Answers:

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

EACH statement ALONE is sufficient.

Correct answer:

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

Explanation:

For statement (1), we only know that $\dpi{100} \small x$ is odd but we have no idea about $\dpi{100} \small y$ . If $\dpi{100} \small y$ is odd, then $\dpi{100} \small x+y$ is even. If $\dpi{100} \small y$ is even, then $\dpi{100} \small x+y$ is odd. Therefore we have no clear answer to the question using this condition. For statement (2), since $\dpi{100} \small x-y$ is even, we know that $\dpi{100} \small x$ and $\dpi{100} \small y$ are either both odd or both even, therefore we know for sure that $\dpi{100} \small x+y$ is even and the answer to this question is “no”.

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Example Question #2 : Arithmetic

If is an integer and $\dpi{100} \small 3<k<7$ , what is the value of ?

(1) $\dpi{100} \small k$ is a factor of 20.

(2) $\dpi{100} \small k$ is a factor of 24.

Possible Answers:

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

EACH statement ALONE is sufficient.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Correct answer:

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Explanation:

From statement (1), we know that the possible value of $\dpi{100} \small k$ would be 4 and 5. From statement (2), we know that the possible value of $\dpi{100} \small k$ would be 4 and 6. Putting the two statements together, we know that only $\dpi{100} \small k=4$ satisfies both conditions. Therefore both statements together are sufficient.

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Example Question #983 : Data Sufficiency Questions

If is a positive integer, is divisible by 6?

1. The sum of the digits of is divisible by 6

2. is even

Possible Answers:

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

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

EACH statement ALONE is sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

Correct answer:

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Explanation:

Statement 1: Numbers whose digits sum to a number divisible by 3 are divisible by 3, but the same does not apply to sums of 6. This indicates that is divisble by 3 but is not sufficient at proving is divisible by 6.

Statement 2: Though all multiples of 6 are even, not all even numbers are multiples of 6.

Together: The fact that is a multiple of 3 and even is sufficient evidence for the conclusion that x is divisible by 6.

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Example Question #984 : Data Sufficiency Questions

Is positive, negative, or zero?

1) $\small x^{3}$ is positive.

2) $\small x^{4}$ is positive.

Possible Answers:

Statement 2 ALONE is sufficient, but Statement 1 alone is not sufficient.

EACH statement ALONE is sufficient.

Statement 1 ALONE is sufficient, but Statement 2 alone is not sufficient.

Statements 1 and 2 TOGETHER are not sufficient.

BOTH statements TOGETHER are sufficient, but neither statement ALONE is sufficient.

Correct answer:

Statement 1 ALONE is sufficient, but Statement 2 alone is not sufficient.

Explanation:

raised to an odd power must have the same sign as , so, if $\small x^{3}$ is positive, then is also positive. But either a positive number or a negative number raised to an even power must be positive. Therefore, $\small \small x^{4}$ being positive is inconclusive.

Therefore, the correct choice is that Statement 1, but not Statement 2, is sufficient.

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Example Question #3 : Arithmetic

The greatest common factor of 32 and a number is 16. What is ?

1) 3 is also a factor of .

2) 5 is also a factor of .

Possible Answers:

EITHER Statement 1 or Statement 2 ALONE is sufficient to answer the question.

BOTH statements TOGETHER are NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is not sufficient.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is not sufficient.

Correct answer:

BOTH statements TOGETHER are NOT sufficient to answer the question.

Explanation:

That cannot be determined, even if both statements are known to be true, can be proved by demonstrating two examples of that fit these conditions. We can do this by comparing the prime factorizations of 32 and .

Example: B = 240

To find GCF (32, 240) :

$32 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2$

$240 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 5$

$GCF (32, 240) = 2 \cdot 2 \cdot 2 \cdot 2 = 16$

Example: B = 720

To find GCF (32, 720) :

$32 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2$

$720 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 5$

$GCF (32, 720) = 2 \cdot 2 \cdot 2 \cdot 2 = 16$

In each case, 3 and 5 are factors of , and in each case, $\small GCF (32, B) = 16$ .

The answer is that both statements together are insufficient.

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Example Question #6 : Dsq: Understanding The Properties Of Integers

What is the area of a rhombus in square inches?

1) One of its angles measures $30^\circ$

2) One of its sides measures 10 inches

Possible Answers:

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is not sufficient.

BOTH statements TOGETHER are NOT sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is not sufficient.

EITHER Statement 1 or Statement 2 ALONE is sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation:

As is true of any other parallelogram, the area of the rhombus is the base multiplied by the height. The common sidelength alone can be used to determine the base, but without the angles, the height cannot be determined. Using trigonometry, the angle can be used to determine the height relative to the base, but without the base, the height is unknown.

If we know both of the given statements, then part of one base, an altitude from an endpoint of the opposite base, and one adjacent side form a 30-60-90 triangle. The hypotenuse of that triangle is 10 inches, and the altitude is half that, or 5 inches. This makes the area 50 square inches.

The answer is that both statements together are sufficient to answer the question, but neither statement alone.

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Example Question #4 : Arithmetic

Data Sufficiency Question

Out of 100 students, 60 took French and 25 took German. How many students took neither?

1. 15 students took Spanish

2. 7 students took both French and German

Possible Answers:

both statements taken together are sufficient to answer the question, but neither statement alone is sufficient

statement 1 alone is sufficient, but statement 2 alone is not sufficient to answer the question

each statement alone is sufficient

statement 2 alone is sufficient, but statement 1 alone is not sufficient to answer the question

statements 1 and 2 together are not sufficient, and additional data is needed to answer the question

Correct answer:

statement 2 alone is sufficient, but statement 1 alone is not sufficient to answer the question

Explanation:

Statement 1 does not tell us anything about the number of students taking French or German. The information from statement 2 is sufficient, if 60 took French, 25 took German, and 7 took both, we can calculate the number that took neither.

100-(60+25-7)=23

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GMAT Math : Data-Sufficiency Questions

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #13 : Dsq: Understanding Absolute Value

Example Question #181 : Algebra

Example Question #11 : Dsq: Understanding Absolute Value

Example Question #1 : Arithmetic

Example Question #2 : Arithmetic

Example Question #983 : Data Sufficiency Questions

Example Question #984 : Data Sufficiency Questions

Example Question #3 : Arithmetic

Example Question #6 : Dsq: Understanding The Properties Of Integers

Example Question #4 : Arithmetic

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