GMAT Math : Problem-Solving Questions

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Example Questions

Example Question #1991 : Problem Solving Questions

When evaluating each of the following expressions, which one(s) require you to multiply first?

I) $12 + 45 \times 23$

II) $\left (12 + 45 \right ) \times 23$

III) $12 + \left (45 \times 23 \right )$

Possible Answers:

I and III only

I only

II and III only

III only

I and II only

Correct answer:

I and III only

Explanation:

According to the order of operations, any operations within parentheses must be performed first. In expression (II), this is the addition; in expression (III), this is the multiplication.

Expression (I) does not have any parentheses, so, by the order of operations, in the absence of grouping symbols, multiplication precedes addition.

Therefore, the correct response is I and III only.

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Example Question #36 : Real Numbers

is the multiplicative inverse of . Which of the following expressions is equivalent to

A (B + C)

for all values of the variables?

Possible Answers:

AB + 1

$AB - A^{2}$

$AB + A^{2}$

Correct answer:

AB + 1

Explanation:

By the distributive property,

A (B + C) = AB + AC

is the multiplicative inverse of , meaning that, by defintion, AC = 1 , so

AB + AC = AB + 1 .

AB + 1 is the correct choice.

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Example Question #32 : Understanding Real Numbers

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

If a < b , then $a \bigstar b = ab$ .

If $a = b \;$ , then $a \bigstar b = a+b$

If a > b , then $a \bigstar b = a-b$ .

Multiply $5 \bigstar 2$ by $2 \bigstar 5$ . What is the result?

Possible Answers:

Correct answer:

Explanation:

First, evaluate $5 \bigstar 2$ . Since 5 > 2 , use the defintion of $a \bigstar b$ for the case a > b :

$a \bigstar b = a-b$

$5 \bigstar 2 = 5 - 2 = 3$ .

Now, evaluate $2 \bigstar 5$ . Since 2 < 5 , use the defintion of $a \bigstar b$ for the case $a< b \;$ :

$a \bigstar b = ab$

$2 \bigstar 5 = 2 \cdot 5 = 10$

The product of $5 \bigstar 2$ and $2 \bigstar 5$ is $3 \cdot 10 = 30$ .

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Example Question #38 : Real Numbers

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

If a < b , then $a \bigstar b = a+b$

If $a = b \;$ , then $a \bigstar b = ab$

If a > b , then $a \bigstar b = a-b$

Divide $(-8) \bigstar (-8)$ by $8 \bigstar 8$ . What is the quotient?

Possible Answers:

Undefined

Correct answer:

Explanation:

$(-8) \bigstar (-8)$ and $8 \bigstar 8$ are both calculated by using the defintion of $a \bigstar b$ for the case $a = b \;$ :

$a \bigstar b = ab$

$(-8) \bigstar (-8) = -8 \cdot (-8) = 64$

$8 \bigstar 8 = 8 \cdot 8= 64$

Their quotient is $64 \div 64 = 1$ .

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Example Question #1992 : Problem Solving Questions

Each of $\bigcirc, \square, \bigtriangleup$ stands for a real number; if one appears more than once in a choice, it stands for the same number each time.

Which of the following diagrams demonstrates a commutative property?

Possible Answers:

$\bigcirc \times (\square \times \bigtriangleup) = ( \bigcirc \times \square) \times \bigtriangleup)$

If $\bigcirc = \bigtriangleup$ and $\bigtriangleup = \square$ , then $\bigcirc = \square$

$\bigcirc = \bigcirc$

If $\bigcirc = \bigtriangleup$ then $\bigtriangleup = \bigcirc$

$\bigcirc + \bigtriangleup = \bigtriangleup + \bigcirc$

Correct answer:

$\bigcirc + \bigtriangleup = \bigtriangleup + \bigcirc$

Explanation:

Addition and multiplication are both commutative, which means that a sum or product has the same value regardless of the order in which the addends or factors are written. The diagram

$\bigcirc + \bigtriangleup = \bigtriangleup + \bigcirc$

is the one that demonstrates this for addition.

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Example Question #33 : Understanding Real Numbers

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

If and are both negative, then $a \bigstar b = a+ b$ .

If and are not both negative, then $a \bigstar b = a- b$ .

Divide $(-5) \bigstar (-5)$ by $5 \bigstar 5$ . What is the quotient?

Possible Answers:

Undefined

Correct answer:

Undefined

Explanation:

$(-5) \bigstar (-5)$ can be evaluated using the definition of $a \bigstar b$ for the case of both and being negative:

$a \bigstar b = a+ b$

$(-5) \bigstar (-5) = -5 + (-5) = -10$

$5 \bigstar 5$ can be evaluated using the definition of $a \bigstar b$ for the case of and not both being negative:

$a \bigstar b = a- b$

$5 \bigstar 5 = 5-5 = 0$

The quotient: $-10 \div 0$ , which is undefined, as zero cannot be taken as a divisor.

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Example Question #441 : Arithmetic

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

If both and are integers, then $a \bigstar b = a+b$ .

If neither nor is an integer, then $a \bigstar b = a-b$ .

If exactly one of and is an integer, then $a \bigstar b = ab$ .

Which of the following is equal to

$\left ( \frac{3}{4} \bigstar 1 \frac{1}{3} \right ) \bigstar 2$ ?

Possible Answers:

$4\frac{1}{6}$

$-1\frac{1}{6}$

$-2\frac{7}{12}$

Correct answer:

$-1\frac{1}{6}$

Explanation:

First, evaluate $\frac{3}{4} \bigstar 1 \frac{1}{3}$ using the definition of $a \bigstar b$ for neither nor an integer:

$a \bigstar b = a-b$

$\frac{3}{4} \bigstar 1 \frac{1}{3} = \frac{3}{4} - 1 \frac{1}{3}$

$= \frac{3}{4} - \frac{4}{3}$

$= \frac{3\cdot 3 }{4 \cdot 3 } - \frac{4 \cdot 4}{4 \cdot 3}$

$= \frac{9}{12 } - \frac{16}{12}$

$= -\frac{7}{12}$

Therefore, $\left ( \frac{3}{4} \bigstar 1 \frac{1}{3} \right ) \bigstar 2 = \left ( -\frac{7}{12} \right ) \bigstar 2$ , which is evaluated using the definition of $a \bigstar b$ for exactly one of and an integer:

$a \bigstar b = ab$

$\left ( -\frac{7}{12} \right ) \bigstar 2 = -\frac{7}{12} \cdot 2 = -\frac{7}{6} = -1\frac{1}{6}$ ,

the correct response.

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Example Question #41 : Real Numbers

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

If both and are positive, then $a \bigstar b = a+b$ .

If neither nor is positive, then $a \bigstar b = a-b$ .

If exactly one of and is positive, then $a \bigstar b = ab$ .

Evaluate $(- 7) \bigstar [ \left ( -7 \right ) \bigstar \left ( -7 \right ) ]$ .

Possible Answers:

Correct answer:

Explanation:

First, evaluate $\left ( -7 \right ) \bigstar \left ( -7 \right )$ using the definition of $a \bigstar b$ for neither nor positive:

$a \bigstar b = a-b$

$\left ( -7 \right ) \bigstar \left ( -7 \right ) = -7 - (-7 ) = -7 + 7 = 0$

Therefore,

$(- 7) \bigstar [ \left ( -7 \right ) \bigstar \left ( -7 \right ) ] = (- 7) \bigstar 0$ , which is evaluated using the definition of $a \bigstar b$ for neither nor positive:

$a \bigstar b = a-b$

$(- 7) \bigstar 0 = -7 - 0 = -7$ , the correct response.

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Example Question #43 : Understanding Real Numbers

Each of $\bigcirc, \square, \bigtriangleup$ stands for a real number; if one appears more than once in a choice, it stands for the same number each time.

Which of the following diagrams demonstrates the reflexive property?

Possible Answers:

$\bigcirc = \bigcirc$

$\bigcirc + \bigtriangleup = \bigtriangleup + \bigcirc$

If $\bigcirc = \bigtriangleup$ then $\bigtriangleup = \bigcirc$

If $\bigcirc = \bigtriangleup$ and $\bigtriangleup = \square$ , then $\bigcirc = \square$

$\bigcirc +0 = \bigcirc$

Correct answer:

$\bigcirc = \bigcirc$

Explanation:

According to the reflexive property of equality, any number is equal to itself. This is demonstrated by the diagram

$\bigcirc = \bigcirc$ .

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Example Question #44 : Understanding Real Numbers

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

If both and are integers, then $a \bigstar b = a+b$ .

If neither nor is an integer, then $a \bigstar b = a-b$ .

If exactly one of and is an integer, then $a \bigstar b = ab$ .

Which of the following is equal to

$\left ( \frac{1}{2} \bigstar 2 \right ) \bigstar \left ( 1 \bigstar 1 \right )$ ?

Possible Answers:

$\frac{1}{2}$

Correct answer:

Explanation:

$\frac{1}{2} \bigstar 2$ can be evaluated using the defintion of $a \bigstar b$ for exactly one of and an integer:

$a \bigstar b = ab$

$\frac{1}{2} \bigstar 2 = \frac{1}{2} \cdot 2 = 1$

$1 \bigstar 1$ can be evaluated using the defintion of $a \bigstar b$ for and both integers:

$a \bigstar b = a+b$

$1 \bigstar 1 = 1+1 = 2$

$\left ( \frac{1}{2} \bigstar 2 \right ) \bigstar \left ( 1 \bigstar 1 \right ) = 1 \bigstar 2$ , which can be evaluated using the defintion of $a \bigstar b$ for and both integers:

$a \bigstar b = a+b$

$1 \bigstar 2 = 1 + 2= 3$ , the correct response.

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GMAT Math : Problem-Solving Questions

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #1991 : Problem Solving Questions

Example Question #36 : Real Numbers

Example Question #32 : Understanding Real Numbers

Example Question #38 : Real Numbers

Example Question #1992 : Problem Solving Questions

Example Question #33 : Understanding Real Numbers

Example Question #441 : Arithmetic

Example Question #41 : Real Numbers

Example Question #43 : Understanding Real Numbers

Example Question #44 : Understanding Real Numbers

All GMAT Math Resources

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