GMAT Math : Algebra

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Example Questions

Example Question #51 : Algebra

$\left \lfloor N \right \rfloor$ is defined to be the greatest integer less than or equal to .

$\left \lceil N\right \rceil$ is defined to be the least integer greater than or equal to .

Is it true that M = N ?

(a) $\left \lfloor M \right \rfloor =\left \lfloor N \right \rfloor$

(b) $\left \lceil M\right \rceil = \left \lceil N\right \rceil$

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.

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.

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:

If M = N , then both statements are true. But both statements can also be true in some cases where $M \neq N$ .

For example, if M = 8.5, N= 8.6 , then $\left \lfloor M \right \rfloor =\left \lfloor N \right \rfloor = 8$ and $\left \lceil M\right \rceil = \left \lceil N\right \rceil = 9$ .

The two together are inconclusive.

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Example Question #51 : Algebra

$\left \lfloor N \right \rfloor$ is defined to be the greatest integer less than or equal to .

$\left \lceil N\right \rceil$ is defined to be the least integer greater than or equal to .

Is an integer?

Statement 1: $\left \lfloor N \right \rfloor = N$

Statement 2: $\left \lceil N \right \rceil = N$

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.

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.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

If is not an integer, then $\left \lfloor N \right \rfloor < N$ - for example, $\left \lfloor 1.5 \right \rfloor = 1$ , so $\left \lfloor 1.5 \right \rfloor < 1.5$ . Therefore, by Statement 1 alone, since $\left \lfloor N \right \rfloor = N$ , must be an integer. By a similar argument, Statement 2 alone proves is an integer.

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Example Question #51 : Algebra

Define $f (x) = \left\{\begin{matrix} 0\textrm{ if } x< 0\\ x \textrm{ if } x \geq 0 \end{matrix}\right.$

Is f(A) greater than, less than, or equal to f (B) ?

Statement 1: is a positive number.

Statement 2: is a negative number.

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 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.

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:

One must have information about both and to answer the question, so neither statement is sufficient by itself.

Now assume both statements to be true. Then, being positive and being negative, f(A) = A > 0 = f(B) .

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Example Question #51 : Algebra

Define $f (x) = \left\{\begin{matrix} 1\textrm{ if } x< 0\\ x \textrm{ if } x \geq 0 \end{matrix}\right.$

True or false: f(A) > f (B) .

Statement 1: is a positive number

Statement 2: is a negative number

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.

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.

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

Assume both statements. Then f(B) = 1 and f(A) = A . But this does not answer the question. For example,

If A = 2 , then

f( 2 ) = 2 > 1 = f (1)

making f(A) > f (B) true.

But if A = 0.5 , then

f( 0.5 ) = 0.5 < 1 = f (1 )

making f(A) > f (B) false.

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Example Question #52 : Algebra

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

$a \bigstar b = a ^{2} + b$

Is $M \bigstar N$ positive, negative, or zero?

Statement 1: M > N

Statement 2: N > 0

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 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:

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

Explanation:

Statement 1 alone does not provide a definite answer:

Case 1: If M = -1 and N = -2 , then M > N , and

$M \bigstar N = M^{2} + N = (-1 )^{2} + (-2) = 1-2 = -1$

Case 2: If M =1 and N = 0 , then M > N , and

$M \bigstar N = M^{2} + N = 1^{2} + 0 = 1 + 0 = 1$

However, if we assume Statement 2, then, since $M^{2} \geq 0$ , we can see:

$M \bigstar N = M^{2} + N > 0 + 0 > 0$ , and we know this is positive.

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Example Question #852 : Data Sufficiency Questions

Define an operation $\Delta$ on two real numbers as follows:

$a \Delta b = a - b ^{2}$

Is $M \Delta N$ positive, negative, or zero?

Statement 1: is negative.

Statement 2: is positive.

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:

If we know Statement 1 only - that is negative - then, since $N ^{2}$ must be nonnegative,

$M \Delta N = M - N^{2}$ must be a negative number minus a nonnegative number. This makes $M \Delta N$ negative.

If we know Statement 2 only - that is positive - then we do not have a definite answer. For example,

$2 \Delta \left ( 2 \right )= 2 - (2)^{2} = 2 - 4 = -2$

but

$6\Delta \left ( 2 \right )= 6 - (2)^{2} = 6 - 4 = 2$

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Example Question #55 : Algebra

What is the measure of $\angle 1$ ?

Statement 1: $\angle 1$ is complementary to an angle that measures $62 ^{\circ }$ .

Statement 2: $\angle 1$ is supplementary to an angle that measures $152 ^{\circ }$ .

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.

BOTH statements TOGETHER are insufficient 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.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

Complementary angles have degree measures that total $90 ^{\circ }$ , so the measure of an angle complementary to a $62 ^{\circ }$ angle would have measure $(90-62)^{\circ } = 28 ^{\circ }$ .

Supplementary angles have degree measures that total $180 ^{\circ }$ , so the measure of an angle supplementary to a $152 ^{\circ }$ angle would have measure $(180-152)^{\circ } = 28 ^{\circ }$ .

From either statement alone, we know that $m \angle 1 = 28 ^{\circ }$ .

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Example Question #52 : Algebra

$\left \lfloor x \right \rfloor$ is defined as the greatest integer less than or equal to .

You are given that M>0,N>0 .

True or false: $\left \lfloor M \right \rfloor \cdot \left \lfloor N \right \rfloor < \left \lfloor MN \right \rfloor$

Statement 1: M + 0.5 is an integer.

Statement 2: N + 0.5 is an integer.

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.

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.

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

If M>0 and M + 0.5 is an integer, then $M \in \left \{ 0.5, 1.5, 2.5, 3.5... \right \}$ ; likewise for .

We show that the two statements together are not enough to draw a conclusion by assuming both statements are true and taking two cases:

Case 1: M = 0.5,N = 0.5

$\left \lfloor M \right \rfloor \cdot \left \lfloor N \right \rfloor = \left \lfloor 0.5 \right \rfloor \cdot \left \lfloor 0.5 \right \rfloor = 0 \cdot 0 = 0$

$\left \lfloor MN \right \rfloor = \left \lfloor 0.5\cdot 0.5\right \rfloor = \left \lfloor 0.25 \right \rfloor =0$

$\left \lfloor M \right \rfloor \cdot \left \lfloor N \right \rfloor < \left \lfloor MN \right \rfloor$ is a false statement.

Case 2: M = 1.5,N = 1.5

$\left \lfloor M \right \rfloor \cdot \left \lfloor N \right \rfloor = \left \lfloor 1.5 \right \rfloor \cdot \left \lfloor 1.5 \right \rfloor =1 \cdot 1 = 1$

$\left \lfloor MN \right \rfloor = \left \lfloor 1.5\cdot 1.5\right \rfloor = \left \lfloor 2.25 \right \rfloor =2$

$\left \lfloor M \right \rfloor \cdot \left \lfloor N \right \rfloor < \left \lfloor MN \right \rfloor$ is a true statement.

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Example Question #56 : Algebra

$\left \lfloor x \right \rfloor$ is defined as the greatest integer less than or equal to .

You are given that M>0,N>0 .

True or false: $\left \lfloor M \right \rfloor \cdot \left \lfloor N \right \rfloor < \left \lfloor MN \right \rfloor$

Statement 1: is not an integer.

Statement 2: is an integer.

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.

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.

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

Assume both statements to be true. We can show at least one case where $\left \lfloor M \right \rfloor \cdot \left \lfloor N \right \rfloor < \left \lfloor MN \right \rfloor$ is true and one in which $\left \lfloor M \right \rfloor \cdot \left \lfloor N \right \rfloor < \left \lfloor MN \right \rfloor$ is false.

Case 1: M = 2.1, N = 2

Then $\left \lfloor 2.1 \right \rfloor \cdot \left \lfloor 2 \right \rfloor = 2 \cdot 2 = 4$ and $\left \lfloor 2.1 \cdot 2 \right \rfloor = \left \lfloor 4.2 \right \rfloor = 4$ .

This makes the two expressions equal and $\left \lfloor M \right \rfloor \cdot \left \lfloor N \right \rfloor < \left \lfloor MN \right \rfloor$ a false statement.

Case 2: M = 2.5, N = 2

Then $\left \lfloor 2.5 \right \rfloor \cdot \left \lfloor 2 \right \rfloor = 2 \cdot 2 = 4$ and $\left \lfloor 2.5 \cdot 2 \right \rfloor = \left \lfloor 5 \right \rfloor = 5$ .

This makes $\left \lfloor M \right \rfloor \cdot \left \lfloor N \right \rfloor < \left \lfloor MN \right \rfloor$ true.

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Example Question #57 : Algebra

Gary is writing out a geometric sequence, in order. How many terms does he need to write out before he writes a term greater than or equal to 1,000?

Statement 1: The sum of the first two terms is 5.

Statement 2: The sum of the third and fourth terms is 80.

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.

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.

Correct answer:

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

Explanation:

From Statement 1 alone, it cannot be determined what the first two terms or the common ratio are; the terms, for example, could be 1, 4 or 2,3 , giving a different common ratio in each case. By a smiliar argument Statement 2 alone is also insufficient.

If both statements are assumed, though, the following can be deduced. If is the common ratio and is the first term, then the first four terms are

$a , ar, ar^{2}, a r ^{3}$ .

a + ar = a (1 + r ) = 5

$ar^{2}+ a r ^{3} = ar^{2} (1+r)= 80$

Divide:

$\frac{ar^{2} (1+r) }{a(1+r)} =\frac{ 80}{5}$

$r ^{2} = 16$

r=4

a (1 + 4 ) = 5a = 5 , so a = 1 .

Knowing the common ratio and the first term allows us to determine all of the numbers of the sequence and to find the first one greater than or at least 1,000.

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GMAT Math : Algebra

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #51 : Algebra

Example Question #51 : Algebra

Example Question #51 : Algebra

Example Question #51 : Algebra

Example Question #52 : Algebra

Example Question #852 : Data Sufficiency Questions

Example Question #55 : Algebra

Example Question #52 : Algebra

Example Question #56 : Algebra

Example Question #57 : Algebra

All GMAT Math Resources

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