GMAT Math : GMAT Quantitative Reasoning

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Example Questions

Example Question #4 : Absolute Value

is a real number. True or false: $\left | N \right | < 5$

Statement 1: 2 N + 7 < 17

Statement 2: 4N - 17 > -37

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.

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.

Correct answer:

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

Explanation:

If $\left | N\right | < 5$ , then, by definition, -5 < N < 5 - that is, both N < 5 and N > -5 .

If Statement 1 is true, then

2 N + 7 < 17

2 N < 10

N < 5

Statement 1 alone does not answer the question, as N < 5 follows, but not necessarily N > -5 .

If Statement 2 is true, then

4N - 17 > -37

4N >-20

N > -5

Statement 2 alone does not answer the question, as N > -5 follows, but not necessarily N < 5 .

If both statements are true, then N > -5 and N < 5 both follow, and -5 < N < 5 , meaning that $\left | N \right | < 5$ .

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Example Question #11 : Dsq: Understanding Absolute Value

Of distinct integers a, b, c , which is the greatest of the three?

Statement 1: |b| <a < |c|

Statement 2: |a | + |b| = |c|

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.

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:

The two statements together are insufficient.

For example, let a = 8, b=6 . Then, from Statement 2,

|c| = |a| + |b| = |6 | + |8| = 6 + 8 = 14

Therefore, either c = - 14 or c = 14 .

In either case, Statement 2 is shown to be true, since

6 < 8 < 14

and

|b| <a < |c|

But if c = - 14 , then is the greatest of the three. If c = 14 , then is the greatest. Therefore,the two statements together are not enough.

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Example Question #11 : Dsq: Understanding Absolute Value

Of distinct integers a, b, c , which is the greatest of the three?

Statement 1: |a| < b < |c|

Statement 2: and are negative.

Possible Answers:

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.

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.

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.

Case 1: a=4 , b= 5, c =6

|a|< b < |c|

|4|< 5 < |6|

4< 5 < 6 , which is true.

Case 2: a=-4 , b= 5, c =-6

|a|< b < |c|

|-4|< 5 < |-6|

4< 5 < 6 , which is true.

But in the first case, is the greatest of the three. In the second, is the greatest.

Statement 2 gives insuffcient information, since no information is given about the sign of .

Assume both statements to be true. $|a| \ge 0$ , and from Statement 1, b > |a| ; by transitivity, b > 0 . From Statement 2, a < 0, c < 0 . This makes the greatest of the three.

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Example Question #13 : Dsq: Understanding Absolute Value

Let A,B,C be any three (not necessarily distinct) integers.

At how many points does the graph of the function

f (x)= A |x-B | + C

intersect the -axis?

Statement 1: and are nonzero integers of opposite sign.

Statement 2: and are nonzero integers of opposite sign.

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

BOTH statements TOGETHER are insufficient 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:

To determine the point(s), if any, at which the graph of a function intersects the -axis, set f(x)= 0 and solve for .

f (x)= A |x-B | + C

A |x-B | + C = 0

A |x-B | = - C

$|x-B | = - \frac{C}{A}$

At this point, we can examine the equation. Since the absolute value of a number must be nonnegative, the sign of $- \frac{C}{A}$ tells us how many solutions exist to this equation. If $- \frac{C}{A} < 0$ , there is no solution, and therefore, the graph of does not intersect the -axis. If $- \frac{C}{A} = 0$ , then there is one solution, and, therefore, the graph of intersects the -axis at exactly one point. If $- \frac{C}{A} > 0$ , then there are two solutions, and, therefore, the graph of intersects the -axis at exactly two points.

To determine the sign of $- \frac{C}{A}$ , we need to whether the signs of both and are like or unlike, or that C = 0 . Either statement alone eliminates the possibility that C = 0 , but neither alone gives the signs of both and . However, if both statements are assumed, then, since and have the opposite sign as , they have the same sign. This makes $\frac{C}{A} > 0$ and $- \frac{C}{A} < 0$ , so the graph of can be determined to not cross the -axis at all.

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Example Question #181 : Algebra

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

Statement 1: M = |N - 10|

Statement 2: N = |M + 10|

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:

Statement 1 alone gives insufficient information, as is seen in these two cases:. For example, if N = 15 , then

M = |N - 10|

M = |15 - 10| = |5| = 5

M < N

However, if N = 0 , then

M = |N - 10|

M = |0 - 10| = |-10| = 10

M > N

Therefore, it is not clear which, if either, of and is greater.

Now assume Statement 2 alone.

If is negative, then , which, being an absolute value of a number, must be nonnegative, is the greater number. If is positive, then so is M + 10 , so

N = |M + 10| = M + 10 .

Therefore,

N > M .

is the greater number in either case.

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Example Question #11 : Dsq: Understanding Absolute Value

Let A,B,C be any three (not necessarily distinct) integers.

At how many points does the graph of the function

f (x)= A |x-B | + C

intersect the -axis?

Statement 1: B < 0

Statement 2: C< 0

Possible Answers:

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.

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 insufficient to answer the question.

Explanation:

To determine the point(s), if any, at which the graph of a function intersects the -axis, set f(x)= 0 and solve for .

f (x)= A |x-B | + C

A |x-B | + C = 0

A |x-B | = - C

$|x-B | = - \frac{C}{A}$

At this point, we can examine the equation. For a solution to exist, since the absolute value of a number must be nonnegative, it must hold that $- \frac{C}{A} \geq 0$ . This happens if and are of opposite sign, or if C = 0 . However, Statement 2 tells us that C< 0 , and neither statement tells us the sign of . The two statements together provide insufficient information.

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Example Question #1 : Properties Of Integers

Is $\dpi{100} \small x+y$ odd?

(1) $\dpi{100} \small x$ is odd

(2) $\dpi{100} \small x-y$ is even

Possible Answers:

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

EACH statement ALONE is sufficient.

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

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

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

Correct answer:

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

Explanation:

For statement (1), we only know that $\dpi{100} \small x$ is odd but we have no idea about $\dpi{100} \small y$ . If $\dpi{100} \small y$ is odd, then $\dpi{100} \small x+y$ is even. If $\dpi{100} \small y$ is even, then $\dpi{100} \small x+y$ is odd. Therefore we have no clear answer to the question using this condition. For statement (2), since $\dpi{100} \small x-y$ is even, we know that $\dpi{100} \small x$ and $\dpi{100} \small y$ are either both odd or both even, therefore we know for sure that $\dpi{100} \small x+y$ is even and the answer to this question is “no”.

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Example Question #1 : Properties Of Integers

If is an integer and $\dpi{100} \small 3<k<7$ , what is the value of ?

(1) $\dpi{100} \small k$ is a factor of 20.

(2) $\dpi{100} \small k$ is a factor of 24.

Possible Answers:

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

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

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

EACH statement ALONE is sufficient.

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

Correct answer:

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

Explanation:

From statement (1), we know that the possible value of $\dpi{100} \small k$ would be 4 and 5. From statement (2), we know that the possible value of $\dpi{100} \small k$ would be 4 and 6. Putting the two statements together, we know that only $\dpi{100} \small k=4$ satisfies both conditions. Therefore both statements together are sufficient.

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Example Question #1 : Arithmetic

If is a positive integer, is divisible by 6?

1. The sum of the digits of is divisible by 6

2. is even

Possible Answers:

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

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

Correct answer:

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

Explanation:

Statement 1: Numbers whose digits sum to a number divisible by 3 are divisible by 3, but the same does not apply to sums of 6. This indicates that is divisble by 3 but is not sufficient at proving is divisible by 6.

Statement 2: Though all multiples of 6 are even, not all even numbers are multiples of 6.

Together: The fact that is a multiple of 3 and even is sufficient evidence for the conclusion that x is divisible by 6.

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Example Question #3 : Properties Of Integers

Is positive, negative, or zero?

1) $\small x^{3}$ is positive.

2) $\small x^{4}$ is positive.

Possible Answers:

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

Statements 1 and 2 TOGETHER are 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 alone is not sufficient.

Correct answer:

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

Explanation:

raised to an odd power must have the same sign as , so, if $\small x^{3}$ is positive, then is also positive. But either a positive number or a negative number raised to an even power must be positive. Therefore, $\small \small x^{4}$ being positive is inconclusive.

Therefore, the correct choice is that Statement 1, but not Statement 2, is sufficient.

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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 #4 : Absolute Value

Example Question #11 : Dsq: Understanding Absolute Value

Example Question #11 : Dsq: Understanding Absolute Value

Example Question #13 : Dsq: Understanding Absolute Value

Example Question #181 : Algebra

Example Question #11 : Dsq: Understanding Absolute Value

Example Question #1 : Properties Of Integers

Example Question #1 : Properties Of Integers

Example Question #1 : Arithmetic

Example Question #3 : Properties Of Integers

All GMAT Math Resources

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