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) $12 + 45 \times 23$ $\displaystyle 12 + 45 \times 23$

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

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

Possible Answers:

III only

I and III only

I only

I and II only

II and III 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

$\displaystyle A$ is the multiplicative inverse of $\displaystyle C$. Which of the following expressions is equivalent to

$\displaystyle A (B + C)$

for all values of the variables?

Possible Answers:

$\displaystyle B$

$AB - A^{2}$ $\displaystyle AB - A^{2}$

$AB + A^{2}$ $\displaystyle AB + A^{2}$

AB + 1 $\displaystyle AB + 1$

$\displaystyle AB$

Correct answer:

AB + 1 $\displaystyle AB + 1$

Explanation:

By the distributive property,

$\displaystyle A (B + C) = AB + AC$

$\displaystyle A$ is the multiplicative inverse of $\displaystyle C$, meaning that, by defintion, AC = 1 $\displaystyle AC = 1$, so

$\displaystyle AB + AC = AB + 1$.

AB + 1 $\displaystyle AB + 1$ is the correct choice.

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Example Question #441 : Arithmetic

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

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

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

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

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

Possible Answers:

$\displaystyle 70$

$\displaystyle -30$

$\displaystyle 21$

$\displaystyle 30$

$\displaystyle -21$

Correct answer:

$\displaystyle 30$

Explanation:

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

$a \bigstar b = a-b$ $\displaystyle a \bigstar b = a-b$

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

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

$a \bigstar b = ab$ $\displaystyle a \bigstar b = ab$

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

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

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Example Question #442 : Arithmetic

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

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

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

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

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

Possible Answers:

$\displaystyle -4$

Undefined

$\displaystyle 1$

$\displaystyle 0$

$\displaystyle -1$

Correct answer:

$\displaystyle 1$

Explanation:

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

$a \bigstar b = ab$ $\displaystyle a \bigstar b = ab$

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

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

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

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Example Question #443 : Arithmetic

Each of $\bigcirc, \square, \bigtriangleup$ $\displaystyle \bigcirc, \square, \bigtriangleup$ 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:

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

$\bigcirc = \bigcirc$ $\displaystyle \bigcirc = \bigcirc$

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

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

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

Correct answer:

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

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

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

is the one that demonstrates this for addition.

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Example Question #444 : Arithmetic

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

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

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

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

Possible Answers:

$\displaystyle -10$

$\displaystyle 1$

Undefined

$\displaystyle 10$

$\displaystyle 0$

Correct answer:

Undefined

Explanation:

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

$a \bigstar b = a+ b$ $\displaystyle a \bigstar b = a+ b$

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

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

$a \bigstar b = a- b$ $\displaystyle a \bigstar b = a- b$

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

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

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Example Question #41 : Understanding Real Numbers

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

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

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

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

Which of the following is equal to

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

Possible Answers:

$\displaystyle 3$

$4\frac{1}{6}$ $\displaystyle 4\frac{1}{6}$

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

$\displaystyle 2$

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

Correct answer:

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

Explanation:

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

$a \bigstar b = a-b$ $\displaystyle a \bigstar b = a-b$

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

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

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

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

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

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

$a \bigstar b = ab$ $\displaystyle a \bigstar b = ab$

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

the correct response.

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Example Question #42 : Understanding Real Numbers

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

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

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

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

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

Possible Answers:

$\displaystyle 0$

$\displaystyle -7$

$\displaystyle -98$

$\displaystyle -343$

$\displaystyle 7$

Correct answer:

$\displaystyle -7$

Explanation:

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

$a \bigstar b = a-b$ $\displaystyle a \bigstar b = a-b$

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

Therefore,

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

$a \bigstar b = a-b$ $\displaystyle a \bigstar b = a-b$

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

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Example Question #41 : Understanding Real Numbers

Which of the following diagrams demonstrates the reflexive property?

Possible Answers:

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

$\bigcirc +0 = \bigcirc$ $\displaystyle \bigcirc +0 = \bigcirc$

$\bigcirc = \bigcirc$ $\displaystyle \bigcirc = \bigcirc$

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

Correct answer:

$\bigcirc = \bigcirc$ $\displaystyle \bigcirc = \bigcirc$

Explanation:

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

$\bigcirc = \bigcirc$ $\displaystyle \bigcirc = \bigcirc$.

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Example Question #44 : Understanding Real Numbers

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

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

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

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

Which of the following is equal to

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

Possible Answers:

$\frac{1}{2}$ $\displaystyle \frac{1}{2}$

$\displaystyle 0$

$\displaystyle 3$

$\displaystyle 1$

$\displaystyle 2$

Correct answer:

$\displaystyle 3$

Explanation:

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

$a \bigstar b = ab$ $\displaystyle a \bigstar b = ab$

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

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

$a \bigstar b = a+b$ $\displaystyle a \bigstar b = a+b$

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

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

$a \bigstar b = a+b$ $\displaystyle a \bigstar b = a+b$

$1 \bigstar 2 = 1 + 2= 3$ $\displaystyle 1 \bigstar 2 = 1 + 2= 3$, 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 #441 : Arithmetic

Example Question #442 : Arithmetic

Example Question #443 : Arithmetic

Example Question #444 : Arithmetic

Example Question #41 : Understanding Real Numbers

Example Question #42 : Understanding Real Numbers

Example Question #41 : Understanding Real Numbers

Example Question #44 : Understanding Real Numbers

All GMAT Math Resources

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