GMAT Math : GMAT Quantitative Reasoning

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Example Questions

Example Question #3391 : Gmat Quantitative Reasoning

is 25% of , which is 25% of . is a positive integer.

True of false: is an integer.

Statement 1: is a prime number.

Statement 2: is an odd number.

Possible Answers:

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.

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.

Correct answer:

EITHER statement ALONE is sufficient to answer the question

Explanation:

is 25% of , so

$Y = 0.25 Z = \frac{1}{4}Z$ .

Similarly, $X = \frac{1}{4}Y$ .

Therefore,

$X = \frac{1}{4} \cdot \frac{1}{4} Z= \frac{1}{16}Z$ ,

or, equivalently,

$X = Z \div 16$

Z = 16X .

If is an integer, then 16 is a factor of . This contradicts both Statement 1, since 16 is composite, making composite, and Statement 2, since this makes a multiple of - that is, even. Either statement alone answers the question in the negative.

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Example Question #13 : Percents

is $60\%$ of , and is $60\%$ of .

True or false: 0 < N < 10 .

Statement 1: 0 < X < 5 .

Statement 2: 5 < Y < 15

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.

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

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

is 60% of , meaning that $X = 0.6N = \frac{3}{5} N$ , or, equivalently, $N =\frac{5}{3}X$ .

From Statement 1 alone, since 0 < X < 5 , it follows that

$\frac{5}{3} \cdot 0 <\frac{5}{3} X < \frac{5}{3} \cdot 5$

$0 <N < 8\frac{1}{3} < 10$

This is enough to prove that 0 < N < 10 .

is 60% of , so $N = 0.6Y = \frac{3}{5} Y$ .

From Statement 2 alone, since 5 < Y < 15 , it follows that

$\frac{3}{5} \cdot 5 <\frac{3}{5} Y < \frac{3}{5} \cdot 15$

0 < 3<N< 9 < 10 .

This is enough to prove that 0 < N < 10 .

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Example Question #1280 : Data Sufficiency Questions

Three candidates - Rodger, Stephanie, and Tina - ran for student body president. By the rules, the candidate who wins more than half the ballots cast wins the election outright; if no candidate wins more than half, there must be a runoff between the two top vote-getters. You may assume that no other names were written in.

Was there an outright winner, or will there be a runoff?

Statement 1: Rodger won 100 more votes than Stephanie and 210 more votes than Tina.

Statement 2: Tina won 25.9% of the votes.

Possible Answers:

BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.

STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.

BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.

EITHER STATEMENT ALONE provides sufficient information to answer the question.

STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 ALONE does NOT provide sufficient information to answer the question.

Correct answer:

BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.

Explanation:

For one candidate to have won the election outright, (s)he must win more than 50% of the votes.

Statement 1 alone does not prove there was an outright winner. For example, if Rodger got 211 votes, then Stephanie got 111 votes, and Tina got 1 vote; this makes Rodger's share of the votes

$211 /(211+111+1 ) \cdot 100 \% \approx 65 \%$ ,

making Rodger the outright winner. But if Rodger got 500 votes, then Stephanie got 400 votes, and Tina got 290; Rodger got the most votes, but his share is

$500 /(500+400+290) \cdot 42\%$ ,

not the required share of the vote.

Statement 2 alone does not prove this either, since 74.1% of the vote was won by either Rodger or Stephanie, but it is not specified how this is distributed; Roger or Stephanie could have won 51%, with the remaining 23.1% won by the other, resulting in an outright winner. However, it is also possible that each won half this, or about 37%, resulting in a runoff between the two.

Now assume both statements are true. Let be the number of votes won by Tina. Then Rodger won X+210 votes and Stephanie won X+210 - 100 , or X+110 , votes. The total votes are

X+210 + X+110 +X = 3X+ 320 .

Since Tina won 25.9% of the vote, we can set up an equation:

$\frac{X}{3X+ 320} = 0.259$

This can be solved for . From this, the number of votes each candidate got, the votes cast, and, finally, the percent of the vote each won can be calculated.

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Example Question #1 : Dsq: Understanding Powers And Roots

is a real number. Is positive, negative, or zero?

Statement 1: 2x - 5 > 0

Statement 2: $\frac{1}{2}x^{3} -1> 0$

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 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.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

If 2x - 5 > 0 , then 2x > 5 , and $x > \frac{5}{2} > 0$ , so must be positive.

If $\frac{1}{2}x^{3} -1> 0$ , then $\frac{1}{2}x^{3} > 1$ , $x^{3} > 2$ . and $x> \sqrt[3]{2}$ , so again, must be positive. Either statement is enough to answer the question in the affirmative.

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Example Question #1 : Powers & Roots Of Numbers

Simplify this expression as much as possible:

$\sqrt[3]{16} + \sqrt[3]{128} - \sqrt[3]{54}$

Possible Answers:

$4 + 4\sqrt[3]{2} - 3 \sqrt[3]{6}$

$3 \sqrt[3]{2}$

$9 \sqrt[3]{2}$

$\sqrt[3]{198}$

The expression is already simplified

Correct answer:

$3 \sqrt[3]{2}$

Explanation:

$\sqrt[3]{16} + \sqrt[3]{128} - \sqrt[3]{54}$

$= \sqrt[3]{8} \cdot \sqrt[3]{2} + \sqrt[3]{64} \cdot \sqrt[3]{2} - \sqrt[3]{27} \cdot \sqrt[3]{2}$

$= 2 \sqrt[3]{2} +4 \sqrt[3]{2} -3 \sqrt[3]{2}$

$=\left ( 2+4-3 \right ) \sqrt[3]{2} = 3 \sqrt[3]{2}$

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Example Question #2 : Dsq: Understanding Powers And Roots

Imagine an integer such that the units digit of is greater than 5. What is the units digit of ?

(1) The units digit of is the same as the units digit of $y^{2}$ .

(2) The units digit of is the same as the units digit of $y^{3}$ .

Possible Answers:

BOTH statements TOGETHER are not sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but the other statement alone is not sufficient.

Statement 1 ALONE is sufficient to answer the question, but the other statement alone is not sufficient.

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

EACH statement ALONE is sufficient to answer the question.

Correct answer:

Statement 1 ALONE is sufficient to answer the question, but the other statement alone is not sufficient.

Explanation:

(1) The only single-digit integer greater than 5 whose unit digit of its square term is equal to itself is 6. This statement is sufficient.

(2) There are two single-digit integers where the unit digit of the cubed term is equal to the integer itself: 6 and 9. This statement is insufficient.

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Example Question #1 : Dsq: Understanding Powers And Roots

What is the value of twelve raised to the fourth power?

Possible Answers:

1,728

144

20,736

28,380

Correct answer:

20,736

Explanation:

"Twelve raised to the fourth power" is 12⁴. If you can translate the words into their mathematical counterpart, you're done, because the actual calculation should be done by your calculator. It will tell you that $12\cdot 12\cdot 12\cdot 12=20,736$ . There is not enough time on the test for you to try to do this by hand.

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Example Question #4 : Dsq: Understanding Powers And Roots

Calculate the fifth root of :

(1) The square root of is 3125 .

(2) The tenth root of is .

Possible Answers:

Both statements TOGETHER are not sufficient.

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.

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

Correct answer:

Each statement ALONE is sufficient.

Explanation:

Using Statement (1):

$m^{\frac{1}{5}}= (m^{\frac{1}{2}})^{\frac{2}{5}}=(3125)^{\frac{2}{5}}=25$

Statement (1) ALONE is SUFFICIENT.

Using Statement (2):

$m^{\frac{1}{5}}= (m^{\frac{1}{10}})^{2}=(5)^{2}=25$

Statement (2) ALONE is SUFFICIENT.

Therefore EACH Statement ALONE is sufficient.

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Example Question #5 : Dsq: Understanding Powers And Roots

is a positive real number. True or false: is a rational number.

Statement 1: $N ^{3}$ is an irrational number.

Statement 2: $N ^{4}$ is an irrational number.

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 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:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

An integer power of a rational number, being a product of rational numbers, must itself be rational. Either statement alone asserts that such a power is irrational, so conversely, either statement alone proves irrational.

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Example Question #6 : Powers & Roots Of Numbers

N > 1 . True or false: is rational.

Statement 1: $\left (N + 1 \right )^{2}$ is rational.

Statement 2: $\left (N - 1 \right )^{2}$ is rational.

Possible Answers:

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.

EITHER 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:

Statement 1 alone is not enough to prove is or is not rational. Examples:

If N = 2 , then $\left (N+ 1 \right )^{2} = \left (2+ 1 \right )^{2} = 3^{2} = 9$

If $N = \sqrt{2} - 1$ , then $\left (N+ 1 \right )^{2} = \left ( \sqrt{2}-1 + 1 \right )^{2} =( \sqrt{2})^{2} = 2$

In both cases, $\left (N + 1 \right )^{2}$ is rational, but in one case, is rational and in the other, is irrational.

A similar argument demonstrates Statement 2 to be insufficient.

Assume both statements are true. $\left (N + 1 \right )^{2}$ and $\left (N - 1 \right )^{2}$ are rational, so their difference is as well:

$\left (N + 1 \right )^{2} - \left (N - 1 \right )^{2}$

$=\left ( N ^{2}+ 2N + 1 \right ) - \left ( N ^{2}- 2N + 1 \right )$

$= N ^{2}- N ^{2}+ 2N +2N + 1 -1$

= 4N

is rational, so by closure under division, $4N \div 4 = N$ is rational.

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GMAT Math : GMAT Quantitative Reasoning

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #3391 : Gmat Quantitative Reasoning

Example Question #13 : Percents

Example Question #1280 : Data Sufficiency Questions

Example Question #1 : Dsq: Understanding Powers And Roots

Example Question #1 : Powers & Roots Of Numbers

Example Question #2 : Dsq: Understanding Powers And Roots

Example Question #1 : Dsq: Understanding Powers And Roots

Example Question #4 : Dsq: Understanding Powers And Roots

Example Question #5 : Dsq: Understanding Powers And Roots

Example Question #6 : Powers & Roots Of Numbers

All GMAT Math Resources

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