GMAT Math : GMAT Quantitative Reasoning

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Example Questions

Example Question #51 : Dsq: Understanding Functions

Evaluate $\displaystyle (f+g) (0)$.

Statement 1: $\displaystyle f$ is an odd function.

Statement 2: $\displaystyle g$ is an odd function.

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:

$\displaystyle (f+g) (0) = f(0) + g(0)$, so we need to find the values of both f(0) $\displaystyle f(0)$ and g(0) $\displaystyle g(0)$ in order to answer this question.

Assume Statement 1 alone. By defintion of an odd function, from Statement 2, for every $\displaystyle a$ in the domain of $\displaystyle g$, $\displaystyle f(a) = -f(a)$. In specific, setting a = 0 $\displaystyle a = 0$,

$\displaystyle f(0) = -f(0)$.

The only number whose opposite is itself is 0, so

$\displaystyle f(0) = 0$,

and it follows that

$\displaystyle (f+g) (0) = 0 + g(0) = g(0)$.

However, we have no way of knowing the value of g(0) $\displaystyle g(0)$, so the expression cannot be evaluated.

By a similar argument, if Statement 2 alone is assumed, $\displaystyle (f+g) (0) = f(0) + 0 = f(0)$, but, since f(0) $\displaystyle f(0)$ is unknown, the expression cannot be evaluated.

Now assume both statements. It follows that $\displaystyle f(0) = g(0) = 0$, and

$\displaystyle (f+g) (0) = f(0) + g(0) = 0+0 = 0$.

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Example Question #91 : Algebra

Given a function $\displaystyle f$, it is known that:

$\displaystyle f(1) = 7$

$\displaystyle f(2) = 10$

$\displaystyle f(3) = 6$

$\displaystyle f(4) = 9$

$\displaystyle f(5) = 8$

Given a function $\displaystyle g$, evaluate $\displaystyle (f + g) (4)$.

Statement 1: $\displaystyle g(4) = g(-4)$

Statement 2: $\displaystyle g$ is an odd function.

Possible Answers:

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.

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.

Correct answer:

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

Explanation:

$\displaystyle (f + g) (4) = f(4)+ g(4) = 9 + g(4)$, so to answer the question, it is necssary and sufficient to evaluate g(4) $\displaystyle g(4)$. Neither statement alone gives us this value.

However, assume both statements to be true. By defintion of an odd function, from Statement 2, for every $\displaystyle a$ in the domain of $\displaystyle g$, $\displaystyle g(a) = -g(-a)$, so, in specific, $\displaystyle g(4) = -g(-4)$. From Statement 1, however, $\displaystyle g(4) = g(-4)$. This means that $\displaystyle g(4) = -g( 4)$, and g(4) $\displaystyle g(4)$ , being equal to its own opposite, must be equal to 0. Therefore,

$\displaystyle (f + g) (4) = 9 + g(4) = 9 + 0 = 9$.

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Example Question #61 : Dsq: Understanding Functions

Given a function $\displaystyle f$, it is known that:

$\displaystyle f(0) = 11$

$\displaystyle f(1) = 7$

$\displaystyle f(2) = 10$

$\displaystyle f(3) = 6$

$\displaystyle f(4) = 9$

$\displaystyle f(5) = 8$

Given a function $\displaystyle g$, evaluate $\displaystyle (f g) (0)$.

Statement 1: $\displaystyle g$ is an odd function.

Statement 2: $\displaystyle g (c) = 0$ for every positive integer $\displaystyle c$.

Possible Answers:

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.

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:

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

Explanation:

$(f g) (0) = f(0) \cdot g (0) = 11 \cdot g(0)$ $\displaystyle (f g) (0) = f(0) \cdot g (0) = 11 \cdot g(0)$, so to answer the question, it is necessary and sufficient to evaluate g(0) $\displaystyle g(0)$.

Assume Statement 1 alone. By defintion of an odd function, from Statement 2, for every $\displaystyle a$ in the domain of $\displaystyle g$, $\displaystyle g(-a) = -g(a)$. In specific, setting a = 0 $\displaystyle a = 0$,

$\displaystyle g(0) = -g(0)$.

The only number whose opposite is itself is 0, so

$\displaystyle g(0) = 0$,

and it follows that

$(f g) (0) = 11 \cdot 0 = 0$ $\displaystyle (f g) (0) = 11 \cdot 0 = 0$.

Statement 2 only gives the values of $\displaystyle g$ for positive integers; this information is irrelevant.

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Example Question #61 : Dsq: Understanding Functions

Let $\displaystyle f$ be a function with the set of all real numbers as its domain, and let the function have an inverse $\displaystyle f^{-1}$.

What is the $\displaystyle y$-intercept of the graph of $f^{-1}$ $\displaystyle f^{-1}$?

Statement 1: $\displaystyle f(0) = 6$.

Statement 2: The graphs of $\displaystyle f$ and $f^{-1}$ $\displaystyle f^{-1}$ intersect only at the point (4, 4) $\displaystyle (4, 4)$.

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:

The $\displaystyle y$-intercept of the graph of $f^{-1}$ $\displaystyle f^{-1}$ is the point (0, b) $\displaystyle (0, b)$ at which the graph intersects the $\displaystyle y$-axis. At that point, $f^{-1}(0) = b$ $\displaystyle f^{-1}(0) = b$, or, equivalently, $\displaystyle f(b) = 0$.

Therefore, we need to find the value $\displaystyle b$ for which $\displaystyle f(b) = 0$.

Between the two statements, we only know that $\displaystyle f(0) = 6$ and $\displaystyle f(4) = 4$. The value of $\displaystyle b$ for which $\displaystyle f(b) = 0$ cannot be determined.

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Example Question #61 : Functions/Series

True or false: $\left \{ a_{n} \right \}$ $\displaystyle \left \{ a_{n} \right \}$, $\displaystyle n = 1, 2, 3,...$ is an arithmetic sequence.

Statement 1: $a_{1}+ a_{4} \ne a_{2} + a_{3}$ $\displaystyle a_{1}+ a_{4} \ne a_{2} + a_{3}$

Statement 2: $a_{6}- a_{5} = a_{8} - a_{7}$ $\displaystyle a_{6}- a_{5} = a_{8} - a_{7}$

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:

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

Explanation:

An arithmetic sequence is one in which the difference of each term in the sequence and the one preceding is constant.

Assume Statement 1 alone.

$a_{1}+ a_{4} \ne a_{2} + a_{3}$ $\displaystyle a_{1}+ a_{4} \ne a_{2} + a_{3}$

$a_{1}+ a_{4} - a_{1} - a_{3} \ne a_{2} + a_{3} - a_{1} - a_{3}$ $\displaystyle a_{1}+ a_{4} - a_{1} - a_{3} \ne a_{2} + a_{3} - a_{1} - a_{3}$

$a_{4} - a_{3} \ne a_{2} - a_{1}$ $\displaystyle a_{4} - a_{3} \ne a_{2} - a_{1}$,

meaning that at least two such differences are unequal, and proving the sequence is not arithmetic.

Statement 2 alone only proves that two such differences are equal, but says nothing about any of the other (infinitely many) such differences. Therefore, it leaves the question unresolved.

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Example Question #63 : Dsq: Understanding Functions

Define $\displaystyle f(x)=4x+A$ and $\displaystyle g(x) =Bx+5$.

Is it true that $f = g^{-1}$ $\displaystyle f = g^{-1}$ ?

Statement 1: A = -10 $\displaystyle A = -10$

Statement 2: $B=\frac{1}{2}$ $\displaystyle B=\frac{1}{2}$

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.

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.

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

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

For $f = g^{-1}$ $\displaystyle f = g^{-1}$ to be each other's inverse, it must be true that

$(f\circ g)(x) =x$ $\displaystyle (f\circ g)(x) =x$ and $(g\circ f)(x) =x$ $\displaystyle (g\circ f)(x) =x$

We can look at the first condition.

$(f\circ g)(x) =x$ $\displaystyle (f\circ g)(x) =x$

$(f\circ g)(x) =f(g(x)) = f(Bx+5)=4(Bx+5)+A=4Bx+20+A= x$ $\displaystyle (f\circ g)(x) =f(g(x)) = f(Bx+5)=4(Bx+5)+A=4Bx+20+A= x$

For this to be true, it must hold that:

4B=1 $\displaystyle 4B=1$

$B=\frac{1}{4}$ $\displaystyle B=\frac{1}{4}$

and

20+A=0 $\displaystyle 20+A=0$

A = -20 $\displaystyle A = -20$

Since both statements violate these conditions, it is impossible for $f = g^{-1}$ $\displaystyle f = g^{-1}$, even if you are only given one of them.

The answer is that either statement alone is sufficient to answer the question.

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Example Question #1 : Simplifying Algebraic Expressions

Evaluate the expression $(x+y)^{2}-x(x+3y)+xy$ $\displaystyle (x+y)^{2}-x(x+3y)+xy$

1) x = 3 $\displaystyle x = 3$

2) y = 2 $\displaystyle y = 2$

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.

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.

Correct answer:

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

Explanation:

Simplify the expression:

$(x+y)^{2}-x(x+3y)+xy$ $\displaystyle (x+y)^{2}-x(x+3y)+xy$

$= x^{2}+2xy+y^{2}-x^{2}-3xy+xy$ $\displaystyle = x^{2}+2xy+y^{2}-x^{2}-3xy+xy$

$= x^{2}-x^{2} +xy+2xy-3xy+y^{2}$ $\displaystyle = x^{2}-x^{2} +xy+2xy-3xy+y^{2}$

$= y^{2}$ $\displaystyle = y^{2}$

Therefore, we only need to know $\displaystyle y$ - If we know y = 2 $\displaystyle y = 2$, we calculate that $(x+y)^{2}-x(x+3y)+xy = y^{2} = 2^{2} = 4$ $\displaystyle (x+y)^{2}-x(x+3y)+xy = y^{2} = 2^{2} = 4$

The answer is that Statement 2 alone is sufficient to answer the question, but Statement 1 is not.

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Example Question #2 : Simplifying Algebraic Expressions

Evaluate: $\frac{(x+y)^{2}-(x-y)^{2}}{x}$ $\displaystyle \frac{(x+y)^{2}-(x-y)^{2}}{x}$

Statement 1: x = 5 $\displaystyle x = 5$

Statement 2: y=3 $\displaystyle y=3$

Possible Answers:

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.

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:

$\frac{(x+y)^{2}-(x-y)^{2}}{x} = \frac{(x^{2}+2xy+y^{2})-(x^{2}-2xy+y^{2})}{x} = \frac{4xy}{x}= 4y$ $\displaystyle \frac{(x+y)^{2}-(x-y)^{2}}{x} = \frac{(x^{2}+2xy+y^{2})-(x^{2}-2xy+y^{2})}{x} = \frac{4xy}{x}= 4y$

Therefore, you only need to know the value of $\displaystyle y$ to evaluate this; knowing the value of $\displaystyle x$ is neither necessary nor helpful.

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Example Question #91 : Algebra

Evaluate the expression for positive x,y $\displaystyle x,y$:

$\frac{4x^{2}}{y^{-4}} \cdot \frac{3x^{2}y^{2}}{8} \cdot \frac{5}{(2x^{2}y^{2})^{2}}$ $\displaystyle \frac{4x^{2}}{y^{-4}} \cdot \frac{3x^{2}y^{2}}{8} \cdot \frac{5}{(2x^{2}y^{2})^{2}}$

Statement 1: x = 3 $\displaystyle x = 3$

Statement 2: y = 4 $\displaystyle y = 4$

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.

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:

$\frac{4x^{2}}{y^{-4}} \cdot \frac{3x^{2}y^{2}}{8} \cdot \frac{5}{(2x^{2}y^{2})^{2}}$ $\displaystyle \frac{4x^{2}}{y^{-4}} \cdot \frac{3x^{2}y^{2}}{8} \cdot \frac{5}{(2x^{2}y^{2})^{2}}$

$= \frac{4x^{2}}{y^{-4}} \cdot \frac{3x^{2}y^{2}}{8} \cdot \frac{5}{2^{2} (x^{2})^{2}(y^{2})^{2}}$ $\displaystyle = \frac{4x^{2}}{y^{-4}} \cdot \frac{3x^{2}y^{2}}{8} \cdot \frac{5}{2^{2} (x^{2})^{2}(y^{2})^{2}}$

$= \frac{4x^{2}}{y^{-4}} \cdot \frac{3x^{2}y^{2}}{8} \cdot \frac{5}{4 x^{2\cdot 2}y^{2\cdot 2}}$ $\displaystyle = \frac{4x^{2}}{y^{-4}} \cdot \frac{3x^{2}y^{2}}{8} \cdot \frac{5}{4 x^{2\cdot 2}y^{2\cdot 2}}$

$= \frac{4x^{2}}{y^{-4}} \cdot \frac{3x^{2}y^{2}}{8} \cdot \frac{5}{4 x^{4}y^{4}}$ $\displaystyle = \frac{4x^{2}}{y^{-4}} \cdot \frac{3x^{2}y^{2}}{8} \cdot \frac{5}{4 x^{4}y^{4}}$

$= \frac{4x^{2}\cdot 3x^{2}y^{2}\cdot 5}{y^{-4}\cdot 8\cdot 4 x^{4}y^{4}}$ $\displaystyle = \frac{4x^{2}\cdot 3x^{2}y^{2}\cdot 5}{y^{-4}\cdot 8\cdot 4 x^{4}y^{4}}$

$= \frac{4 \cdot 3\cdot 5 \cdot x^{2}\cdot x^{2}\cdot y^{2}}{8\cdot 4 \cdot x^{4} \cdot y^{4}\cdot y^{-4}}$ $\displaystyle = \frac{4 \cdot 3\cdot 5 \cdot x^{2}\cdot x^{2}\cdot y^{2}}{8\cdot 4 \cdot x^{4} \cdot y^{4}\cdot y^{-4}}$

$= \frac{60 \cdot x^{2+2}\cdot y^{2}}{32 \cdot x^{4} \cdot y^{4-4}}$ $\displaystyle = \frac{60 \cdot x^{2+2}\cdot y^{2}}{32 \cdot x^{4} \cdot y^{4-4}}$

$= \frac{60 \cdot x^{4}\cdot y^{2}}{32 \cdot x^{4}}$ $\displaystyle = \frac{60 \cdot x^{4}\cdot y^{2}}{32 \cdot x^{4}}$

Cancel the $x^{4}$ $\displaystyle x^{4}$ from both halves:

$= \frac{60 y^{2}}{32 } = \frac{15 y^{2}}{8 }$ $\displaystyle = \frac{60 y^{2}}{32 } = \frac{15 y^{2}}{8 }$

As can be seen by the simplification, it turns out that only the value of $\displaystyle y$, which is given only in Statement 2, affects the value of the expression.

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Example Question #3003 : Gmat Quantitative Reasoning

What is the value of x+y $\displaystyle x+y$?

(1) 7x=7-7y $\displaystyle 7x=7-7y$

(2) $\displaystyle 13x+13y=13$

Possible Answers:

EACH statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but the other statement alone is not sufficient

BOTH statements TOGETHER are not sufficient to answer the question.

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

Statement 1 ALONE is sufficient to answer the question, but the other statement alone is not sufficient

Correct answer:

EACH statement ALONE is sufficient to answer the question.

Explanation:

(1) Add $\displaystyle 7y$ to both sides to make 7x+7y=7 $\displaystyle 7x+7y=7$. Then divide through by 7 to get

x+y=1 $\displaystyle x+y=1$. This statement is sufficient.

(2) Divide both sides by 13. The equation becomes x+y=1 $\displaystyle x+y=1$. This statement 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 #51 : Dsq: Understanding Functions

Example Question #91 : Algebra

Example Question #61 : Dsq: Understanding Functions

Example Question #61 : Dsq: Understanding Functions

Example Question #61 : Functions/Series

Example Question #63 : Dsq: Understanding Functions

Example Question #1 : Simplifying Algebraic Expressions

Example Question #2 : Simplifying Algebraic Expressions

Example Question #91 : Algebra

Example Question #3003 : Gmat Quantitative Reasoning

All GMAT Math Resources

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