GMAT Math : Data-Sufficiency Questions

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Example Questions

Example Question #3 : Algebra

Is $x^{2} < x$ ?

(1) 0 < x < 1

(2) x > 0

Possible Answers:

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

D: EACH statement ALONE is 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

E: Statements (1) and (2) TOGETHER are not sufficient

Correct answer:

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

Explanation:

$x^{2} < x \Leftrightarrow x^{2}-x<0 \Leftrightarrow x(x-1)<0$

From statement 1 we get that x > 0 and -1 < x - 1 < 0 .

So the first term is positive and the second term is negative, which means that x (x-1) is negative; therefore the statement 1 alone allows us to answer the question.

Statement 2 tells us that x > 0 . If x = 0.5 , we have $x^{2}=0.25$ which is less than 0.5 . Therefore in this case $x^{2}<x$ .

For x = 2 , we have $x^{2} = 4$ which is greater than . So in this case $x^{2} > x$ .

So statement 2 is insufficient.

Therefore the correct answer is A.

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Example Question #3 : Exponents

Solve the following rational expression:

$\frac{x^{m-n}10^{n-1}}{10^{(\frac{m}{2}-2)}x^{n}}$

(1) m=5

(2) m=2n

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

EACH statement ALONE is sufficient to answer the question

Both statements TOGETHER are not sufficient.

Statement (1) ALONE is sufficient, but Statement (2) ALONE is not sufficient

Correct answer:

Statement (2) ALONE is sufficient, but Statement (1) ALONE is not sufficient

Explanation:

When replacing m=5 in the expression we get:

$\frac{x^{5-n}10^{n-1}}{10^{\frac{1}{2}}x^{n}}$

Therefore statement (1) ALONE is not sufficient.

When replacing m=2n in the expression we get:

$\frac{x^{2n-n}10^{n-1}}{10^{(\frac{2n}{2}-2)}x^{n}}=\frac{x^{n}10^{n-1}}{10^{n-2}x^{n}}=\frac{10^{n-1}}{10^{n-2}}=10$

Therefore statement (2) ALONE is sufficient.

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Example Question #6 : Dsq: Understanding Exponents

Myoshi has been assigned to write one number in the circle and one number in the square in the diagram below in order to produce a number in scientifc notation.

$\bigcirc \times 10^{\square }$ .

Did Myoshi succeed?

Statement 1: Myoshi wrote in the circle.

Statement 2: Myoshi wrote in the square.

Possible Answers:

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.

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.

EITHER statement ALONE is sufficient to answer the question.

Correct answer:

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

Explanation:

A number in scientific notation takes the form

$a \times 10 ^{n}$

where $1 \le |a| < 10$ and is an integer of any sign.

Assuming Statement 1 alone, Myoshi did not succeed, since she entered an incorrect number into the circle - $|-24| = 24 \ge 10$ .

Statement 2 alone is inconclusive. Myoshi entered a correct number into the square, since is an integer. But the question is open, since it is not known whether she entered a correct number into the circle or not.

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Example Question #7 : Dsq: Understanding Exponents

is a nonzero number. Is it negative or positive?

Statement 1: $N \ge N^{2}$

Statement 2: $N >N^{3}$

Possible Answers:

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

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.

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.

Correct answer:

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

Explanation:

All negative numbers are less than their (positive) squares, as are all positive numbers greater than 1. Therefore, if Statement 1 is assumed, $N \in (0,1]$ .

can be determined to be positive.

Statement 2 alone is inconclusive. For example, if $N = \frac{1}{2}$ , then $N^{3} = \frac{1}{8}$ , and if N = -2 , $N^{3} = -8$ . In both cases, $N >N^{3}$ , but has different signs in the two cases.

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Example Question #4 : Exponents

is a nonzero number. Is it negative or positive?

Statement 1: $N < N^{2}$

Statement 2: $N < N^{3}$

Possible Answers:

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.

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.

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

Both statements together are inufficient to produce an answer. For example,

If N = 2 , then $N^{2} = 4$ and $N^{3} = 8$ .

If $N = -\frac{1}{2}$ , then $N^{2} = \frac{1}{4}$ and $N^{3} = -\frac{1}{8}$ .

In both cases, $N < N^{2}$ and $N < N^{3}$ , but the signs of differ between cases.

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Example Question #2 : Algebra

is a number not in the set $\left\{-1, 0, 1 \right\}$ .

Of the elements $\left \{ x^{4}, x^{5}, x^{6}\right \}$ , which is the greatest?

Statement 1: is a negative number.

Statement 2: $\left | x\right | > 1$

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.

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.

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 is inconclusive, as can be demonstrated by examining two negative values of other than .

Case 1: x = -2 .

Then

$x^{4} = (-2)^{4} = 16$

$x^{5} = (-2)^{5} = -32$

$x^{6} = (-2)^{6} = 64$

$x^{6}$ is the greatest of these values.

Case 2: $x = -\frac{1}{2}$

Then

$x ^{4}=\left ( -\frac{1}{2} \right )^{4} = \frac{1}{16}$

$x ^{5}=\left ( -\frac{1}{2} \right )^{5} = -\frac{1}{32}$

$x ^{6}=\left ( -\frac{1}{2} \right )^{6} = \frac{1}{64}$

$x^{4}$ is the greatest of these values.

Now assume Statement 2 alone. Either x > 1 or x < -1 .

Case 1: x > 1 .

Then $x \cdot x^{4} > 1\cdot x^{4}$ , so $x^{5} > x^{4}$ ; similarly, $x^{6} > x^{5}$ .

$x^{6}$ is the greatest of the three.

Case 2: x < -1 .

Odd power $x^{5}$ is negative, and even powers $x^{4}$ and $x^{6}$ are positive, so one of the latter two is the greatest. Since x < -1 , it follows that $x^{2} > 1$ . It then follows that $x^{2} \cdot x^{4} > 1 \cdot x^{4}$ , or $x^{6} > x^{4}$ .

Again, $x^{6}$ is the greatest of the three.

Statement 2 alone is sufficient, but not Statement 1.

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Example Question #6 : Algebra

Chord

Note: Figure NOT drawn to scale.

Examine the above diagram. True or false: $\overline{AB} \cong \overline{YZ}$ .

Statement 1: $\Delta ACB \sim \Delta ZXY$

Statement 2: $\Delta ACB$ and $\Delta ZXY$ have the same perimeter.

Possible Answers:

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

BOTH statements TOGETHER are insufficient 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.

EITHER statement ALONE is sufficient to answer the question

Correct answer:

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

Explanation:

From Statement 1 alone, it follows by the similarity of the triangles that $\angle C \cong \angle X$ . These are congruent inscribed angles of a circle, which intercept congruent arcs, so $\widehat{AXB} \cong \widehat{YCZ}$ . Since congruent arcs have congruent chords, $\overline{AB} \cong \overline{YZ}$ .

Statement 2 alone only tells us the relative perimeters of the triangles. We have no way of determining the individual sidelengths or angle measures relative to each other, so Statement 2 alone is inconclusive.

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Example Question #11 : Algebra

is a number not in the set $\left\{-1, 0, 1 \right\}$ .

Of the elements $\left \{ x^{2}, x^{3}, x^{4}\right \}$ , which is the greatest?

Statement 1: is a negative number.

Statement 2: $\left | x\right |>\frac{1}{2}$

Possible Answers:

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

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.

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

Assume both statements are known. The greatest of the three numbers must be $x^{2}$ or $x^{4}$ , since even powers of negative numbers are positive and odd powers of negative numbers are negative.

Case 1: $x = -\frac{2}{3}$

$x^{2} = \left ( -\frac{2}{3} \right )^{2} = \frac{4}{9} = \frac{36}{81}$

$x^{4} = \left ( -\frac{2}{3} \right )^{4} = \frac{16}{81}$

$x^{2} > x^{4}> x^{3}$

Case 2: x = -2 ,

then

$x^{2} = \left ( -2 \right )^{2} =4$

$x^{4} = \left (-2 \right )^{4} = 16$

$x^{4} > x^{2} > x^{3}$

In both cases, is negative and $\left | x\right |>\frac{1}{2}$ , but in one case, $x^{2}$ is the greatest number, and in the other, $x^{4}$ is. The two statements together are inconclusive.

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Example Question #801 : Data Sufficiency Questions

Philip has been assigned to write one number in the circle and one number in the square in the diagram below in order to produce a number in scientifc notation.

$\bigcirc \times 10^{\square }$ .

Did Philip succeed?

Statement 1: Philip wrote -7.5 in the circle.

Statement 2: Philip wrote in the square.

Possible Answers:

EITHER statement 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.

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 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:

A number in scientific notation takes the form

$a \times 10 ^{n}$

where $1 \le |a| < 10$ and is an integer of any sign.

Statement 1 alone proves that Philip entered a correct number into the circle, since $1 \le |-7.5| = 7.5 < 10$ . Statement 2 alone proves that he entered a correct number into the square, since is an integer. But each statement alone is insufficient, since each leaves uinclear whether the other number is valid. The two statements together, however, prove that Philip put correct numbers in both places, thereby writing a number in scientific notation.

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Example Question #801 : Data Sufficiency Questions

is an integer. Is there a real number such that $x^{A}=B$ ?

Statement 1: is negative

Statement 2: is even

Possible Answers:

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.

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.

Correct answer:

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

Explanation:

The equivalent question is "does have a real $A \textrm{th }$ root?"

If you know only that is negative, you need to know whether is even or odd; negative numbers have real odd-numbered roots, but not real even-numbered roots.

If you know only that is even, you need to know whether is negative or nonnegative; negative numbers do not have real even-numbered roots, but nonnegative numbers do.

If you know both, however, then you know that the answer is no, since as stated before, negative numbers do not have real even-numbered roots.

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

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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 #3 : Algebra

Example Question #3 : Exponents

Example Question #6 : Dsq: Understanding Exponents

Example Question #7 : Dsq: Understanding Exponents

Example Question #4 : Exponents

Example Question #2 : Algebra

Example Question #6 : Algebra

Example Question #11 : Algebra

Example Question #801 : Data Sufficiency Questions

Example Question #801 : Data Sufficiency Questions

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