GMAT Math : GMAT Quantitative Reasoning

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Example Questions

Example Question #29 : Real Numbers

Define an operation $\bigstar$ on the positive integers as follows:

If and are both prime integers, then $a \bigstar b =|a- b|$ .

If and are not both prime integers, then $a \bigstar b = 0$ .

Evaluate $M \bigstar N$ .

Statement 1: M= 2 .

Statement 2: is a factor of .

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.

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.

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:

Assume Statement 1 alone. A prime number is an integer with exaclty two factors - 1 and the number itself. 2 has only two factors, 1, and 2, and is therefore prime. However, we do not know . If is not prime, then $M \bigstar N =2 \bigstar N = 0$ . If is prime, then $M \bigstar N =2 \bigstar N = |2 -N|$ , which cannot be calculated without knowing .

Assume Statement 2 alone.

Unless both numbers are primes, the second definition of $a \bigstar b$ is used, and $M \bigstar N = 0$ .

The only way for both numbers to be primes and to be a factor of is for both and to be the same prime number. If this is the case, the first definition is used:

$M \bigstar N =|M -N| = |N -N| = |0| = 0$ .

Therefore, $M \bigstar N = 0$ regardless.

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Example Question #3371 : Gmat Quantitative Reasoning

Define an operation $\bigstar$ on the real numbers as follows:

If both and are integers, then $a \bigstar b = a+b$ .

If and are not both integers, then $a \bigstar b = a-b$ .

Evaluate $M \bigstar N$ .

Statement 1: M = N

Statement 2: 1< M < 2 and 1< N < 2

Possible Answers:

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.

EITHER 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 sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation:

Assume Statement 1 alone. The defintion that will be used to calculate $M \bigstar N$ is not clear, since it is not known whether the numbers are integers. For example, if

M = N = 2 ,

since both are integers, the definition $a \bigstar b = a+b$ will be used, and

$M \bigstar N = 2 \bigstar 2 = 2+2 = 4$ .

If $M = N = \frac{1}{2}$ , the defintion $a \bigstar b = a-b$ . will be used, and

$M \bigstar N = \frac{1}{2} \bigstar \frac{1}{2} =\frac{1}{2} - \frac{1}{2} =0$ .

Assume Statement 2 alone. Both numbers fall between two consecutive integers, so neither is an integer, and the definition $a \bigstar b = a-b$ will be used. However, the value of $M \bigstar N$ cannot be calculated, since , , and their difference are unknown.

Now assume both statements to be true. From Statement 2, the definition $a \bigstar b = a-b$ will be used. Since M = N ,

$M \bigstar N = M - N= M- M = 0$

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Example Question #31 : Dsq: Understanding Real Numbers

Define an operation $\bigstar$ on the real numbers as follows:

If both and are both positive, then $a \bigstar b = a+b$ .

If both and are not both positive, then $a \bigstar b = a-b$ .

Evaluate $M \bigstar N$ .

Statement 1: M = -N .

Statement 2: M < N .

Possible Answers:

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.

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

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

Assume both statements to be true. From Statement 1, N = -M , so and are each other's opposites. Either one is positive and one is negative, or both are equal to 0; either way, the definition of $a \bigstar b$ for and not both positive is used, and

$M \bigstar N = M - N$

$M \bigstar (-M) = M - (-M) = 2M$

Statement 2 tells us nothing except that M < N , so it can only be deduced that is negative, and so is .

With no further information, $M \bigstar N = M - N$ cannot be evaluated.

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Example Question #1 : Ratio & Proportions

At Branchwood Middle School, there are 4 sixth graders for every 5 seventh graders and 6 seventh graders for every 5 eighth graders. How many sixth graders are there?

1. The ratio of sixth graders to eighth graders is 24:25

2. There are 75 eighth graders at the middle school.

Possible Answers:

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

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.

Correct answer:

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

Explanation:

Statement 1: This information can already be determined from the original information.

Statement 2: The two ratios can be connected by the common element of seventh graders. Convert the two ratios so that the seventh grade value is the same in both. The ratio of sixth graders: seventh graders:eighth graders = 24:30:25.

Therefore, 24x + 30x + 25x = total students

25x = 75

x=3

Sixth graders =24x = 24(3)=72

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Example Question #2 : Dsq: Calculating Ratio And Proportion

Fred is looking at a map and is wondering how far it is from Washington City to Bush Corner. He sees that on the map, they are three and a half inches apart. From the map distance between the two, how far is it in actual miles?

Statement 1: The distance from Adamsville to Clinton Ridge on the map is four and a half inches.

Statement 2: In actuality, it is 85 miles from Adamsville to Clinton Ridge.

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:

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

Explanation:

Fred needs to know the ratio of actual miles to map inches to solve this problem; once he knows this, he can multiply this ratio by the map distance from Washington City to Bush Corner. Neither of the facts alone about the other two cities are helpful, but if he knows both, he can determine the ratio he needs to achieve his goal.

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Example Question #3 : Ratio & Proportions

Last year, a computer shop had an average of 75 computers in stock at the start of each day, and it sold an average of 300 computers each week.

This year, the shop is expecting to sell an average of 732 computers each week. The computer store wants to use last year's ratio of computers in stock to computers sold to decide how many computers to have in stock each day. What should this target number be?

Possible Answers:

2,928

30.5

132

198

183

Correct answer:

183

Explanation:

We can set up a proportion to solve for the number of computers the store should keep in stock each day:

$\frac{75}{300}=\frac{x}{732}$

Cross-multiply:

$300x=75\times 732$

300x=54,900

Divide both sides by 300:

x=183

The store should aim to have 183 computers in stock at the start of each day.

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Example Question #1 : Ratio & Proportions

Eleanor has a a scale model of the Voyager I probe. What is the radius of the dish in the model?

I) The length of the antennae on the model is $\frac{1}{60}$ the length of the actual antennae.

II) The radius of the dish on the actual probe is feet.

Possible Answers:

Statement I is sufficient to answer the question, but statement II is not sufficient to answer the question.

Neither statement is sufficient to answer the question. More information is needed.

Either statement is sufficient to answer the question.

Both statements are needed to answer the question.

Statement II is sufficient to answer the question, but statement I is not sufficient to answer the question.

Correct answer:

Both statements are needed to answer the question.

Explanation:

The secret to this problem is to set up a proportion.

We are given the ratio of one part of the model from Statement I, and asked to find the length of another part.

We need both I) and II) to set up the following proportion.

$\frac{1}{60}=\frac{x}{5}$

$\frac{5}{60}=\frac{1}{12}=x$

So both statements are needed.

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Example Question #5 : Ratio & Proportions

In a car dealership there are SUVs to every sedans, and sedans to every sports car. How many sedans are there?

There are SUVs.
There are sport cars.

Possible Answers:

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

Statement 1 and 2 are not sufficient, and additional data is needed to answer the question.

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

Each statement alone is sufficient.

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

Correct answer:

Each statement alone is sufficient.

Explanation:

We can combine the given ratios into one ratio by using the least common mulitiple of 8 and 6, which is 24.

In order to arrive to 24, we need to multiply the first ratio by 3 and the second ratio by 4. We can then arrive to a single ratio of 21:24:20

Using our ratio, we have an equation: 21x + 24x + 20x =total

Statement 1: 21x=84

x=4

then 24(4)=96 sedans

so Statement 1 is sufficient to answer the question

Statement 2: 20x=80

x=4

then 24(4)=96 sedans

so Statement 2 is also sufficient to answer the question

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Example Question #1 : Percents

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.

Possible Answers:

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.

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

EACH statement ALONE is sufficient to answer the question asked.

BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.

Correct answer:

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

Explanation:

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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Example Question #1261 : Data Sufficiency Questions

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.

Possible Answers:

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

D. EACH statement ALONE is sufficient.

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

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

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

Correct answer:

D. EACH statement ALONE is sufficient.

Explanation:

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 $x = 87.5\%$ .

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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 #29 : Real Numbers

Example Question #3371 : Gmat Quantitative Reasoning

Example Question #31 : Dsq: Understanding Real Numbers

Example Question #1 : Ratio & Proportions

Example Question #2 : Dsq: Calculating Ratio And Proportion

Example Question #3 : Ratio & Proportions

Example Question #1 : Ratio & Proportions

Example Question #5 : Ratio & Proportions

Example Question #1 : Percents

Example Question #1261 : Data Sufficiency Questions

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