GMAT Math : Arithmetic

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Example Questions

Example Question #15 : Dsq: Understanding Real Numbers

Evaluate (A+B)+ C .

Statement 1: B = 0

Statement 2: is the additive inverse of .

Possible Answers:

BOTH STATEMENTS TOGETHER do 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.

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

EITHER STATEMENT ALONE provides sufficient information to answer the question.

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

Correct answer:

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

Explanation:

Assume Statement 1 alone. Since B = 0 , then by substitution,

(A+B)+ C = (A+0)+ C = A + C .

But without further information, the expression cannot be evaluated.

Assume Statement 2 alone. is the additive inverse of , so, by definition, C+A= 0 . Applying the commutative, then associative, properties,

(A+B)+ C = C + (A+B) = (C+A)+ B = 0 + B = B

But without further information, the expression cannot be evaluated.

Now assume both statements to be true. From Statement 2 and Statement 1 combined,

(A+B)+ C = B = 0 .

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Example Question #16 : Dsq: Understanding Real Numbers

Evaluate A (B + C) .

Statement 1: AB =128

Statement 2: AC = - 37

Possible Answers:

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.

BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.

EITHER STATEMENT ALONE provides sufficient information to answer the question.

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

Correct answer:

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

Explanation:

By the distributive property of multiplication over addition,

$A (B + C) = A \cdot B + A \cdot C$

Assume Statement 1 alone. $A \cdot B = 128$ , so, by substitution,

$A (B + C) = 128+ A \cdot C$ .

But, without knowing , , or their product, it is impossible to determine the value of the expression. Statement 2 provides insufficient information, for similar reasons.

Now assume both statements to be true. Then

$A (B + C) = A \cdot B + A \cdot C$

Since $A \cdot B = 128$ and $A \cdot C = - 37$ , by substitution,

A (B + C) = 128 + (-37 ) = 91 .

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Example Question #17 : Dsq: Understanding Real Numbers

is a real number. True or false: is a rational number.

Statement 1: $N^{3}-2N^{2}-2N + 4 = 0$

Statement 2: $N^{4} - 4 = 0$

Possible Answers:

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

EITHER STATEMENT ALONE provides 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.

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

BOTH STATEMENTS TOGETHER do 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 polynomial expression in

$N^{3}-2N^{2}-2N + 4 = 0$

can be factored as follows:

$N^{2} (N-2)-2 (N-2) = 0$

$(N^{2}-2 ) (N-2) = 0$

Either N - 2 = 0 , in which case N = 2 , which, as an integer is also rational, or

$N ^{2}- 2 = 0$ , in which case:

$N^{2} = 2$

or either $N = - \sqrt{2}$ or $N = \sqrt{2}$ , both irrational.

Therefore, Statement 1 provides insufficient information.

Assume Statement 2 alone. If $N^{4} - 4 = 0$ , then $N^{4} = 4$ , making a fourth root of 4. A rational root of an integer must itself be an integer, and there is no integer which, when taken to the fourth power, is equal to 4. Therefore, must be irrational.

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Example Question #261 : Arithmetic

Evaluate $A^{B}$ .

Statement 1: A = 1

Statement 2: B = 0

Possible Answers:

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

BOTH STATEMENTS TOGETHER do NOT provide 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.

EITHER STATEMENT ALONE provides sufficient information to answer the question.

Correct answer:

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

Explanation:

From Statement 1 alone, since 1 raised to the power of any real number is equal to 1, we know that

$A^{B} = 1 ^{B} = 1$ .

However, we cannot determine the value of $A^{B}$ knowing only that B = 0 . If is a nonzero number, then $A^{B} = A^{0} = 1$ . However, if A = 0 , then

$A^{B} = 0^{0}$ ,

which is an undefined expression.

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Example Question #19 : Dsq: Understanding Real Numbers

is a real number. True or false: is a rational number.

Statement 1: A square whose side has length has area $\frac{1}{4}$ .

Statement 2: $N \div 6$ is a rational number.

Possible Answers:

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

BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.

EITHER STATEMENT ALONE provides sufficient information to answer the question.

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

Correct answer:

EITHER STATEMENT ALONE provides sufficient information to answer the question.

Explanation:

From Statement 1 alone, since a side of a square with area $\frac{1}{4}$ has as its sidelength

$\sqrt{\frac{1}{4}} = \frac{\sqrt{1}}{\sqrt{4}} = \frac{1}{2}$ ,

$N= \frac{1}{2} = 1 \div 2$ .

This is the quotient of two integers; by definition, this is rational.

From Statement 2 alone, if we let $M = N \div 6$ , then $N = 6 \cdot M$ . is rational, and so is 6, so their product, , is also rational.

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Example Question #11 : Real Numbers

Evaluate $A \cdot (B + C)$ .

Statement 1: A = 0

Statement 2: B = 0

Possible Answers:

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

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

Correct answer:

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

Explanation:

Assume Statement 1 alone.

$A \cdot (B + C) = 0 \cdot (B + C) = 0$ , since 0 multiplied by any number yields a product of 0.

Assume Statement 2 alone.

$A \cdot (B + C) = A \cdot (0 + C) = A \cdot C$ , since 0 added to any number yields the sum 0. However, without knowing and , or their product, it is impossible to determine the result.

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Example Question #21 : Real Numbers

Evaluate AB + AC .

Statement 1: is the multiplicative inverse of .

Statement 2: is the additive inverse of .

Possible Answers:

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

BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.

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

EITHER STATEMENT ALONE provides 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 product of a number and its multiplicative inverse is 1, so AB = 1 , and AB + AC = 1+ AC . But without knowing anything else, the expression cannot be evaluated.

Assume Statement 2 alone. The sum of a number and its additive inverse is 0, so B + C = 0 . By the distributive property,

$AB + AC = A (B + C ) = A \cdot 0 = 0$ .

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Example Question #22 : Real Numbers

is a real number. True or false: is a rational number.

Statement 1: $N^{2} - 1$ is a rational number.

Statement 2: N + 1 is a rational number.

Possible Answers:

BOTH STATEMENTS TOGETHER do 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.

EITHER STATEMENT ALONE provides sufficient information to answer the question.

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

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:

Statement 1 provides insufficient information to determine whether is rational or not.

If N = 2 , which, being an integer, is rational, then $N^{2} - 1 = 2^{2} - 1 = 4 - 1 = 3$ , which, being an integer, is rational.

If $N = \sqrt{2}$ . which is irrational, then $N^{2} - 1 =\left ( \sqrt{2} \right )^{2} - 1 = 2 - 1 = 1$ , which, being an integer, is rational.

Statement 2 is, however, sufficient. If N + 1 = M for some rational number , then N = M - 1 . 1, being an integer, is rational, and as the difference of rational numbers, is rational.

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Example Question #23 : Real Numbers

is a real number. True or false: is a rational number.

Statement 1: $N + \frac{6}{11}$ is an irrational number.

Statement 2: $\frac{5}{7}N$ is an irrational number.

Possible Answers:

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

EITHER STATEMENT ALONE provides sufficient information to answer the question.

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

BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.

Correct answer:

EITHER STATEMENT ALONE provides sufficient information to answer the question.

Explanation:

Assume Statement 1 alone. $\frac{6}{11}$ , the quotient of integers, is rational. The sum of any two rational numbers is rational, so if is rational, then $N + \frac{6}{11}$ is a rational number. However, $N + \frac{6}{11}$ is irrational, so, contrapositively, is irrational.

Assume Statement 2 alone. $\frac{5}{7}$ , the quotient of integers, is rational. The product of any two rational numbers is rational, so if is rational, then $\frac{5}{7}N$ is a rational number. However, $\frac{5}{7}N$ is irrational, so, contrapositively, is irrational.

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Example Question #24 : Real Numbers

Evaluate AB + CD .

Statement 1: is the additive inverse of .

Statement 2: is the additive inverse of .

Possible Answers:

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

BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.

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

EITHER STATEMENT ALONE provides sufficient information to answer the question.

Correct answer:

BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.

Explanation:

Assume both statements to be true. The additive inverse of a number is the number that can be added to that number to yield sum 0. We show that the value of AB + CD cannot be determined from these two statements by examining two cases:

Case 1: A = 1, B = -1, C = 2, D = -2

A + B = 1 + (-1) = 0 and C+ D = 2 + (-2) = 0 , so is the additive inverse of and is the additive inverse of .

AB + CD = 1(-1)+ 2 (-2) = -1 +(-4) = -5 .

Case 2: A = 1, B = -1, C = 3, D = -3

A + B = 1 + (-1) = 0 and C+ D = 3 + (-3) = 0 , so is the additive inverse of and is the additive inverse of .

AB + CD = 1(-1)+ 3(-3) = -1 +(-9) = -10

In both scenarios, the conditions of both statements are met, but AB + CD assumes different values. The two statements together are insufficient to answer the question.

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GMAT Math : Arithmetic

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #15 : Dsq: Understanding Real Numbers

Example Question #16 : Dsq: Understanding Real Numbers

Example Question #17 : Dsq: Understanding Real Numbers

Example Question #261 : Arithmetic

Example Question #19 : Dsq: Understanding Real Numbers

Example Question #11 : Real Numbers

Example Question #21 : Real Numbers

Example Question #22 : Real Numbers

Example Question #23 : Real Numbers

Example Question #24 : Real Numbers

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