GMAT Math : Arithmetic

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Example Questions

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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Example Question #442 : Arithmetic

Define an operation $\bigstar$ on the integers as follows:

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

If both |a| and |b| are even, then $a \bigstar b = a-b$ .

If |a| is odd and |b| is even, or vice versa, then $a \bigstar b = ab$ .

Add $9 \bigstar 6$ to $(-6) \bigstar (- 9)$ . What is the sum?

Possible Answers:

108

Correct answer:

108

Explanation:

Both $9 \bigstar 6$ and $(-6) \bigstar (- 9)$ can be calculated using the definition of $a \bigstar b$ for the case of exactly one of |a| and |b| being odd and one being even:

$a \bigstar b = ab$ .

$9 \bigstar 6 = 9 \cdot 6 = 54$

$(-6) \bigstar (- 9) = (- 6) \cdot (-9) = 54$

Add: 54 + 54 = 108

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Example Question #46 : Understanding Real Numbers

Define an operation $\bigstar$ on the integers as follows:

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

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

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

Multiply $6 \bigstar 1$ by $-2 \bigstar (-3)$ . What is the product?

Possible Answers:

Correct answer:

Explanation:

A prime number has exactly two factors, 1 and the number itself.

Neither 6 nor 1 is a prime number; 1 has only one factor and is not considered to be prime, and 6 has more than two factors - 1, 2, 3, and 6. Therefore, $6 \bigstar 1$ can be evaluated using the defintion of $a \bigstar b$ for two numbers whose absolute values are not prime:

$a \bigstar b = a-b$

$6 \bigstar 1 = 6-1 = 5$

2 and 3 are prime numbers, since each has exactly two factors, 1 and the number itself. Therefore, $-2 \bigstar (-3)$ can be evaluated using the defintion of $a \bigstar b$ for two numbers whose absolute values are prime:

$a \bigstar b = a+b$

$-2 \bigstar (-3) = -2 + (-3) = -5$

The product is $5 \cdot (-5) = -25$

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GMAT Math : Arithmetic

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

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

Example Question #442 : Arithmetic

Example Question #46 : Understanding Real Numbers

All GMAT Math Resources

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