DSQ: Calculating percents - GMAT Quantitative Reasoning

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Question

What percent of students in a school are freshmen with glasses?

(1) Of the freshmen in the school, 10% wear glasses.

(2) Of the non-freshmen in the school, 20% wear glasses.

Answer

To answer this question, it is necessary to know the total number of students in the school and the number of freshmen with glasses.

(1) This indicates that 10% of the freshmen have glasses; neither the total number of freshmen nor the total number of students in the school is provided $\dpi{100} \small \rightarrow$ NOT sufficient.

(2) Provides the percent of non-freshmen who wear glasses (irrelevant to the question at hand). It does not provide the total number of students in the school or the number of freshmen with glasses $\dpi{100} \small \rightarrow$ NOT sufficient.

From (1) and (2), the percent of non-freshmen with glasses is known and the percent of the freshmen with glasses is known, but not the percent of the school who are freshmen with glasses.

For example:

If there are 100 freshmen, including 10 freshmen with glasses, and 100 non-freshmen, including 20 non-freshmen with glasses, then $\dpi{100} \small \frac{10}{200}=5%$ of the school are freshmen with glasses.

If there are 300 freshmen, including 30 freshmen with glasses and 100 non-freshmen, including 20 non-freshmen with glasses, then $\dpi{100} \small \frac{30}{400}=7.5%$ of the school are freshmen with glasses.

Both statements together are not sufficient.

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Question

In 2013 there are 300 employees at Company ABC. If the number of employees at Company ABC increased by 200% from 1993 to 2013, by what percent did the number of employees at Company ABC increase from 2003 to 2013?

(1) In 2003 there were 160 employees at Company ABC.

(2) From 1993 to 2003 the number of employees increased by 60% at Company ABC.

Answer

For statement (1), since we know the numbers of employees in 2003 and 2013, we can directly calculate the percentage change: $\frac{(300 - 160)}{160}= 87.5%$ .

For statement (2), since we know the percentage change from 1993 to 2013 and the percentage change from 1993 to 2003, we can set the percentage change from 2003 to 2013 to be and then calculate from the following:

$(1 + 60%) \cdot (1 + x) = 1+200%$ .

Solve and we can get .

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Question

Data sufficiency question- do not actually solve the question

How many male students are in a class?

1. There are 42 students in the class.

2. 55 percent of the students are female.

Answer

In order to calculate the number of males in the class, you need to know the total number of students and the number of females (which can be calculated using the percentage).

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Question

Data sufficiency question- do not actually solve the question

There are 20 cats in an animal shelter. How many black, female cats are in the shelter?

1. 14 of the cats are male

2. 25 percent of the cats are black

Answer

The information provided will allow you to calculate the number of females and the number of black cats, but there is not enough information to quantify the number of black, female cats

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Question

A certain pet store sells cats and dogs. The number of dogs is 250% greater than the number of cats. How many cats are in the store?

1. There are 40 dogs in the store.

2. There are 56 cats and dogs in the store.

Answer

$dogs = 2.5 \times cats$

Statement 1: sufficient

$40: dogs = 2.5 \times cats$

$cats = \frac{40}{2.5}=16$

Statement 2: sufficient

$cats = x=\frac{56}{3.5}=16$

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Question

Mrs. Smith purchased groceries whose price, before tax, was $147.64. What was the tax rate on those groceries (nearest hundredth of a percent)?

The tax she paid was $9.23.

The total amount she paid was $156.87

Answer

To determine the tax rate, you need to know the purchase price, which is given, and the amount of tax paid. The amount of tax is given in Statement 1. Statement 2 alone, however, also allows you to find the amount of tax; just subtract:

Either way, you can now find the tax rate:

$\frac{9.23}{147.64} \cdot 100 \approx 0.0625 = 6.25%$

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Question

Jared, a jewelry salesman, makes a 10% commission on the selling price of all the jewelry he sells. Last month, he earned $4,000 in commission. What is the total selling price of the jewelry he sold this month?

1. His sales this month were 25% more than his sales last month.

2. He earned $5,000 in commission this month.

Answer

Each statement has enough information to calculate the answer.

Using statement 1, we know that he sold 25% more than last month. Given in the question, last month he earned $4,000. If $4,000 is 10% of the total price of his jewelry sold last month, we do some math (below) to find he sold $40,000 worth of jewelry. Let be the total price of jewelry sold last month. Then,

so

Increasing that by 25% we see that

$40,000\cdot 1.25=50,000$

NOTE: when multiplying the percents, make sure you use the decimal values.

Using statement 2, we know that we can just calculate the total selling price of the jewelry he sold this month based on the commission percentage. So, if we let equal the total price of jewelry sold this month, we get

or

Therefore we see we can use either statement individually to find what we are looking for.

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Question

A distributor ordered 3,250 chairs. 2,315 chairs were delivered. What percentage of the chairs have not yet been delivered?

Answer

First let's find the number of chairs not delivered:

$3,250-2,315=935\ \textup{undelivered chairs}$

Therefore the fraction of chairs not delivered is $\frac{935}{3,250}$ .

Divide the numerator by the denominator to convert this fraction into a decimal:

$935\div 3,250=0.287$

Multiply by 100 to turn this decimal into a percentage:

$0.287\cdot 100=28.7%$

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Question

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

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

(2) Of the men employed in the company, have a college degree.

Answer

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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Question

is of , and is of .

True or false: .

Statement 1: .

Statement 2:

Answer

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

From Statement 1 alone, since , 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 .

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

From Statement 2 alone, since , it follows that

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

.

This is enough to prove that .

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Question

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.

Answer

Statement 1 alone is inconclusive. For example, if , then is 30% of 10 - that is,

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

an integer.

But if , 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 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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Question

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.

Answer

Assume Statement 1 alone.

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

and

or

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

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 , . 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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Question

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.

Answer

Assume both statements are true.

Case 1: .

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:

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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Question

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.

Answer

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,

$X = Z \div 16$

or

.

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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Question

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.

Answer

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 votes and Stephanie won , or , votes. The total votes are

.

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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