GMAT Math : Arithmetic

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Example Question #291 : Arithmetic

What percent of a company's employees are women with a college degree?

(1) Of the women employed in the company, $40\%$ do not have a college degree.

(2) Of the men employed in the company, $80\%$ have a college degree.

Possible Answers:

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

Each statement ALONE is sufficient.

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

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

Both statements TOGETHER are not sufficient.

Correct answer:

Both statements TOGETHER are not sufficient.

Explanation:

Statement (1) indicates the percentage of women who do not have a college degree. From that statement, we know that 60% of the women in the company have a college degree. However, we do not know the total number of employees or the total number of women in the company. Therefore, statement (1) alone is not sufficient.

Statement (2) indicates the percentage of men who have a college degree but from that statement we cannot find the total number of employees or the total number of women with a college degree.

We need the total number of women and the total number of employees in order to calculate the percentage of employees who are women with a college degree. We show it with the following example:

If we assume that there are 100 women and 100 men working in the company, we can attempt to find the percentage of employees who are women with a college degree in the following way:

$\frac{(100-40)}{100+100}=\frac{60}{200}=\frac{30}{100}$

So in that example, 30% of the employees are women with a college degree.

However if we change the number of women to 500 and the number of men to 600, the percentage of employees who are women with a college degree is:

$\frac{(0.6\times 500)}{500+700}=\frac{300}{1200}=\frac{25}{100}$

The percentage of employees who are women with a college degree becomes 25%.

Therefore both statements together are not sufficient.

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Example Question #9 : Percents

is 30% of , and 60% of . and are positive integers.

True or false: is a positive integer.

Statement 1: is a multiple of 5.

Statement 2: is a multiple of 5.

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.

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.

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. For example, if N = 10 , then is 30% of 10 - that is,

$X = 0.30N = 0.30\cdot 10 = 3$ ,

an integer.

But if N = 15 , then is 30% of 15 - that is,

$X = 0.30N = 0.30\cdot 15 = 4.5$ ,

not an integer.

If Statement 2 alone is assumed, then Y = 5M for some integer . 60% of this is

$X = 0.60 \cdot 5M = 3M$ .

is three times an integer and is itself an integer.

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Example Question #292 : Arithmetic

is 30% of , and is 30% of . is an integer. True or false: is an integer.

Statement 1: is an integer.

Statement 2: is divisible by 100.

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.

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

Assume Statement 1 alone.

If is 30% of and is 30% of , then:

B = 0.30 C

and

A = 0.30 B = 0.30 (0.30 C) = 0.09 C

$A = \frac{9}{100}C=9\cdot \frac{C}{100}$

is an integer; for to be an integer, must be divisible by 100. This makes Statement 2 a consequence of Statement 1, so we can now test Statement 2 alone.

If is divisible by 100, then for some integer , C= 100M . 30% of this is $B = 0.30 \cdot 100M = 30M$ . is 30 times some integer, and is therefore an integer itself.

Either statement alone answers the question.

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Example Question #12 : Percents

is 20% of , and is 20% of . is an integer. True or false: is an integer.

Statement 1: is not an integer.

Statement 2: is divisible by 72.

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.

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

Assume both statements are true.

Case 1: C= 72 .

Then is 20% of this, so

$B = 0.20 \cdot 72 = 14.4$

is 20% of , so

$A = 0.20 \cdot 14.4= 2.88$

Case 2: C= 360

Then is 20% of this, so

$B = 0.20 \cdot 360 = 72$

is 20% of , so

$A = 0.20\cdot 72 = 14.4$

In both situations, the conditions of the problem, as well as both statements, are true, but in one case, is not an integer and in the other case, is an integer. The statements together are inconclusive.

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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:

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.

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.

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:

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:

EACH statement ALONE is sufficient to answer the question.

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

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

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

BOTH statements TOGETHER are not 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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GMAT Math : Arithmetic

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #291 : Arithmetic

Example Question #9 : Percents

Example Question #292 : Arithmetic

Example Question #12 : Percents

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

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