GMAT Math : Arithmetic

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Example Questions

Example Question #21 : Real Numbers

True or false: is an integer.

Statement 1: The multiplicative inverse of is not an integer.

Statement 2: The additive inverse of is an integer.

Possible Answers:

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

BOTH STATEMENTS TOGETHER do NOT provide 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.

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

Correct answer:

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

Explanation:

Assume Statement 1 alone. The multiplicative inverse of a number is the number which, when multiplied by that number, yields a product of 1. If the multiplicative inverse of is not an integer, it is possible for to be an integer or to not be one, as is shown in these examples:

If N = 2 , which is an integer, then, since $2 \cdot \frac{1}{2} = 1$ , the multiplicative inverse of is $\frac{1}{2}$ , which is not an integer, If $N = \frac{2}{3}$ , which is not an integer, then, since $\frac{2}{3} \cdot \frac{3} {2} = 1$ , the multiplicative inverse of is $\frac{3} {2}$ , which is not an integer.

Assume Statement 2 alone. The additive inverse of a number is the number which, when added to that number, yields a sum of 0. If is the additive inverse of the number , then

M+N = 0 , or

$N = -M = -1 \cdot M$

by Statement 2, is an integer; ., the product of integers, is itself an integer.

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Example Question #26 : Real Numbers

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

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

If and are not both whole numbers, then $a \bigstar b = 0$ .

Evaluate $M \bigstar N$ .

Statement 1: is not an integer.

Statement 2: N = 0 .

Possible Answers:

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.

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.

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

Assume Statement 1 alone. Every whole number (0, 1, 2, 3,...) is an integer, so if is not an integer, it cannot be a whole number. Therefore, since and are not whole numbers, the second defintion of $\bigstar$ is used, and $M \bigstar N= 0$ .

Assume Statement 2 alone. N = 0 , which is a whole number, so it is not clear what definition of $\bigstar$ is used. If is not a whole number, the second defintion of $\bigstar$ is used, and $M \bigstar N= 0$ . If is also a whole number, then the first defintion is used:

$M \bigstar N = M +N = M + 0 = M$ .

However, we do not know the value of .

Overall, Statement 2 provides insufficient information to answer the question.

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Example Question #27 : Real Numbers

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

If and are both prime integers, then $a \bigstar b = 1$ .

If and are not both prime integers, then $a \bigstar b = 2$ .

Evaluate $M \bigstar N$ .

Statement 1: M= 1 .

Statement 2: is a factor of .

Possible Answers:

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.

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

Assume Statement 1 alone. A prime number is an integer with exaclty two factors - 1 and the number itself. 1 is considered to not be prime, since it has one factor. Therefore, if M = 1 , and are not both prime, and the second defintion of $a \bigstar b$ is used. $M \bigstar N = 1$ .

Assume Statement 2 alone.

Case 1: M = 1 .

In this case, Statement 1 is true, and as demonstrated before, $M \bigstar N = 1$ .

Case 2: M = 2, N = 2 .

In this case, is a factor of , since any integer is a factor of itself. Both integers are prime, so the first defintion of $a \bigstar b$ is used. $M \bigstar N = 1$ .

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Example Question #28 : 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 = 0 .

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.

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:

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

Explanation:

Assume Statement 1 alone. M = -N , 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

$M \bigstar N = M+ N = M + (-M) = 0$ .

Assume Statement 2 alone. Again, the definition of $a \bigstar b$ for and not both positive is used, and

$0 \bigstar N = 0+ N =N$ .

However, we do not know the value of .

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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 #271 : Arithmetic

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:

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

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

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

Statement (2) ALONE is sufficient, but statement (1) 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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GMAT Math : Arithmetic

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #21 : Real Numbers

Example Question #26 : Real Numbers

Example Question #27 : Real Numbers

Example Question #28 : Real Numbers

Example Question #29 : Real Numbers

Example Question #3371 : Gmat Quantitative Reasoning

Example Question #31 : Dsq: Understanding Real Numbers

Example Question #271 : Arithmetic

Example Question #2 : Dsq: Calculating Ratio And Proportion

Example Question #3 : Ratio & Proportions

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