GMAT Math : Arithmetic

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Example Questions

Example Question #11 : Sets

Number_sets

Examine the above diagram, which shows a Venn diagram representing the sets of real numbers.

If real number were to be placed in its correct region in the diagram, which one would it be - I, II, III, IV, or V?

Statement 1: If m = n+ 7 , then would be placed in Region I.

Statement 2: If p =n- 7 , then would be placed in Region I.

Possible Answers:

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 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 1 ALONE is sufficient to answer the question, but Statement 2 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:

Region I comprises the natural numbers - $\left \{ 1, 2, 3, 4,5,...\right \}$

From Statement 1 alone, m = n+ 7 is a natural number; since n = m - 7 , it follows that is the difference of a natural number and 7 - that is,

$n\in \left \{ -6, -5, -4,..., 0, 1, 2, 3,...\right \}$

could be in any of three regions - I, II, or III.

Conversely, from Statement 2 alone, is the sum of a natural number and 7 - that is,

$n \in \left \{8, 9, 10, 11, 12, 13,...\right \}$

must be a natural number and it must be in Region I.

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Example Question #161 : Arithmetic

Number_sets

Examine the above diagram, which shows a Venn diagram representing the sets of real numbers.

If real number were to be placed in its correct region in the diagram, which one would it be - I, II, III, IV, or V?

Statement 1: If $k = n^{2}$ , then would be in Region I.

Statement 2: If $h = n^{3}$ , then would be in Region III.

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:

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

Explanation:

Assume Statement 1 alone. It cannot be determined what region is in.

For example, suppose k = 4 , which is in Region I (the set of natural numbers, or positive integers). It is possible that n = 2 , putting it in Region I, or n =- 2 , putting it in Region III (the set of integers that are not whole numbers - that is, the set of negative integers).

Assume Statement 2 alone. It cannot be determined what region is in.

For example, suppose h = - 8 , which is in Region III; then n = -2 , which is also in Region III. But suppose h = - 4 ; then $n = - \sqrt[3]{4}$ , which, as an irrational number, is in Region V.

Now assume both statements. Then has an integer as a square and an integer as a cube. must either be an integer or an irrational number. But

$n = \frac{n^{3}}{n^{2}}$ , making it the quotient of integers, which is rational. Therefore, is an integer. Furthermore, its cube is negative, so is negative. The two statements together prove that is a negative integer, which belongs in Region III.

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Example Question #161 : Arithmetic

How many elements are in set ?

Statement 1: has exactly subsets.

Statement 2: has exactly proper subsets.

Possible Answers:

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.

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.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

A set with elements has exactly $2^{N}$ subsets in all, and $2^{N} - 1$ proper subsets (every subset except one - the set itself).

From Statement 1, since has $64 = 2^{6}$ subsets, it follows that it has 6 elements. From Statement 2, since has 63 proper subsets, it has 64 subsets total, and, again, 6 elements. Either statement alone is sufficient.

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Example Question #13 : Sets

Which, if either, is the greater number: or ?

Statement 1: |A + 6| = |B|

Statement 2: -A = B

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.

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 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:

Statement 1 alone gives insufficient information. For example, if A = 0 , then:

|A + 6| = |B|

|B| = |0 + 6| = |6| = 6

B = 6 or B = -6

Since -6 < 0 = A < 6 , it is unclear which of and is greater, if either.

Statement 2 gives insufficient information; if is positive, is negative, and vice versa.

Assume both to be true. The two statements form a system of equations that can be solved using substitution:

|A + 6| = |B|

-A = B

|A + 6| = |-A |

Case 1:

A + 6 = -A

2A + 6 = 0

2A = -6

A = -3

B = -A = -(-3) = 3

Case 2:

A + 6 =-( -A)

A + 6 = A

6= 0

This equation has no solution.

Therefore, the only possible solution is A = -3, B = 3 . Therefore, it can be concluded that B > A .

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Example Question #1 : Descriptive Statistics

Calculate the average of the set of numbers.

Statement 1: the range of the set is 13

Statement 2: the mode of the set is 16

Possible Answers:

Statements 1 and 2 TOGETHER are NOT sufficient.

Statement 2 ALONE is sufficient, but statement 1 is not sufficient.

EACH statement ALONE is sufficient.

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

Statement 1 ALONE is sufficient, but statement 2 is not sufficient.

Correct answer:

Statements 1 and 2 TOGETHER are NOT sufficient.

Explanation:

To answer this question, we must understand the vocabulary. The range of a set is the difference between the largest and smallest numbers. The mode of a set is the number which comes up most frequently. Neither the range nor the mode, nor both, will help us solve for the average, however.

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Example Question #2 : Dsq: Calculating Range

A high school basketball player practices 40 free throws every day. Over the past 90 days he has averaged 25 successful free throws.

What is the probability that tomorrow he will make 30 or more free throws?

Possible Answers:

$\small .08$

$\small .05$

$\small .11$

$\small .01$

$\small .02$

Correct answer:

$\small .05$

Explanation:

$\small n=40$

$\small p=.625$

$\small q=.375$

standard deviation =

$\small \sqrt{(pq)/n}=.0765$

$\small \small z=(.75-.625)/.0765=1.63$

$\small \small z=1.63$

Percentile =

$\small \small .9484$

$\small \small 1 - .9484 = .0516$

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Example Question #2 : Descriptive Statistics

We think that our average customer spends roughly $40 - $60 every time he/she visits our website. We can assume that the expenditure amounts are normally distributed, but, we do not know the standard deviation. We sample 15 web expenditures; our sample average is $52 and our sample standard deviation is $8.

We are 95% sure that our true population average is $52 plus or minus....

Possible Answers:

$11.56

$17.16

$4.05

$4.43

$15.68

Correct answer:

$4.43

Explanation:

standard deviation of the sample mean is:

$\small 8/\sqrt{15}=2.066$

95% $\small t$ for 14 d.o.f. =

$\small 2.1448$

confidence interval is

$\small 2.066(2.1448) = 4.43$

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Example Question #3 : Descriptive Statistics

The average score on a statistics test is 70%. The standard deviation is 10%.

What is the minimal test score that Judy needs in order to reach the 90^th percentile?

Possible Answers:

$\small 86.5$

$\small 88.1$

$\small 82.8$

$\small 84.1$

$\small 80.6$

Correct answer:

$\small 82.8$

Explanation:

One tailed test (Z table)

$\small p=.9$

$\small z=1.28$

$\small 1.28(10)=12.8$

$\small 70+12.8=82.8$

In English - Judy must be at least 1.28 standard deviations above the average.

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Example Question #1152 : Data Sufficiency Questions

Our final sales figures (in millions) for the year are:

$\small Jan=8$

$\small Feb=6$

$\small Mar=10$

$\small Apr=6$

$\small May=10$

$\small Jun=13$

$\small Jul=9$

$\small Aug=11$

$\small Sep=10$

$\small Oct=11$

$\small Nov=13$

$\small Dec=12$

Use Excel to calculate the correlation coefficient between the month number

$\small Jan=1$

$\small Feb=2$

etc.

and the sales figure(s).

Also, compute the $\small t$ value for testing the confidence level for the correlation coefficient.

Possible Answers:

$\small r = .6512$

$\small t = 2.81$

$\small r = .7125$

$\small t = 3.21$

$\small r = .4123$

$\small t = 2.66$

$\small r=.7732$

$\small t = 3.34$

$\small r=.5844$

$\small t = 2.22$

Correct answer:

$\small r = .7125$

$\small t = 3.21$

Explanation:

Excel indicates the correlation coefficient $\small r$ is

$\small .7125$

$\small n-2=10$

the formula for $\small t$ is -

$\small t=r/\sqrt{(1-r^{2})/(n-2)}$

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Example Question #4 : Descriptive Statistics

Our final sales figures (in millions) for the year are:

$\small Jan=8$

$\small Feb=6$

$\small Mar=10$

$\small Apr=6$

$\small May=10$

$\small Jun=13$

$\small Jul=9$

$\small Aug=11$

$\small Sep=10$

$\small Oct=11$

$\small Nov=13$

$\small Dec=12$

Using the above data, calculate the regression line

$\small Y=aX + b$

that best fits the data.

Hint – make the month number

$\small Jan=1$

$\small Feb=2$

etc.

the independent variable $\small X$ and make the sales figure the dependent variable $\small Y$ . Use Excel to calculate the slope.

Possible Answers:

$\small Y=.722X + 5.5$

$\small Y=.311X + 4.2$

$\small Y=.465X + 6.9$

$\small Y=.633X + 2.1$

$\small Y=.226X + 8.1$

Correct answer:

$\small Y=.465X + 6.9$

Explanation:

Excel indicates the slope is

$\small .465$

$\small average for X = 6.5$

$\small average for Y = 9.917$

$\small 9.917 = .465(6.5) + b$

$\small b = 6.9$

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GMAT Math : Arithmetic

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #11 : Sets

Example Question #161 : Arithmetic

Example Question #161 : Arithmetic

Example Question #13 : Sets

Example Question #1 : Descriptive Statistics

Example Question #2 : Dsq: Calculating Range

Example Question #2 : Descriptive Statistics

Example Question #3 : Descriptive Statistics

Example Question #1152 : Data Sufficiency Questions

Example Question #4 : Descriptive Statistics

All GMAT Math Resources

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