GMAT Math : Understanding real numbers

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Example Question #41 : 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

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

Possible Answers:

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

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

$4\frac{1}{6}$

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 #42 : Understanding 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 #45 : Understanding Real Numbers

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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Example Question #47 : 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 $a \bigstar b = ab$ .

Subtract $2 \bigstar 50$ from $4 \bigstar 25$ . What is the result?

Possible Answers:

148

Correct answer:

Explanation:

2 is a prime number, since 2 has only two factors, 1 and 2 itself. 50 is not a prime number, since 50 has other factors, such as 2. $2 \bigstar 50$ can be evaluated using the definition of $a \bigstar b$ for exactly one of |a| and |b| prime:

$a \bigstar b = ab$

$2 \bigstar 50 = 2 \cdot 50 = 100$

Neither 4 nor 25 are prime, since each has factors other than 1 and itself; for example, $4 = 2 \cdot 2$ and $25 = 5 \cdot 5$ . $4 \bigstar 25$ can be evaluated using the definition of $a \bigstar b$ for neither |a| nor |b| prime:

$a \bigstar b = a-b$

$4 \bigstar 25 = 4 - 25 = -21$

The difference:

$(4 \bigstar 25 ) -(2 \bigstar 50 ) = -21 - 100 = -121$

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Example Question #48 : Understanding Real Numbers

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

A (B + C)

for all values of the variables?

Possible Answers:

$AB + \frac{A}{B}$

Correct answer:

Explanation:

If is the additive inverse of , then

B+C= 0 , or, equivalently,

By way of substitution and the identity property of addition,

A (B + C)

$= A \cdot 0$

= 0

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Example Question #49 : 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 $a \bigstar b = ab$ .

Evaluate $\left ( 17 \bigstar 13 \right ) \bigstar 7$ .

Possible Answers:

210

1,547

Correct answer:

210

Explanation:

17 and 13 are both prime numbers, since each has exactly two factors - 1 and the number itself. Therefore, we first evaluate $17 \bigstar 13$ using the definition of $a \bigstar b$ for |a| and |b| both prime:

$a \bigstar b = a+b$

$17 \bigstar 13 = 17+13 = 30$

Therefore, $\left ( 17 \bigstar 13 \right ) \bigstar 7 = 30 \bigstar 7$ . 7 is also prime, since its only two factors are 1 and 7 itself. 30, however, is not prime, since 30 has factors other than 1 and itself - for example, $2 \cdot 15 = 30$ . Therefore, $30 \bigstar 7$ is evaluated using the definition of $a \bigstar b$ for exactly one of |a| and |b| prime:

$a \bigstar b = ab$

$30 \bigstar 7 = 30 \cdot 7 = 210$ , the correct response.

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Example Question #50 : 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 $a \bigstar b = ab$ .

Which of the following expressions is the greatest of the five?

Possible Answers:

$4 \bigstar 25$

$10 \bigstar 10$

$5 \bigstar 20$

$2 \bigstar 50$

$1 \bigstar 100$

Correct answer:

$1 \bigstar 100$

Explanation:

Of the integers shown in the five choices, the following are primes, since they have exactly two factors, 1 and the number itself: 2, 5.

1 is not consdered to be a prime, having exactly one factor (1). Also, 4, 10, 20, 25, 50, and 100 are not primes, since each has at least one factor other than 1 and itself.

$2 \bigstar 50$ and $5 \bigstar 20$ can both be evaluated using the definition of $a \bigstar b$ for exactly one of and prime - that is, by multiplying the numbers:

$2 \bigstar 50 = 2 \times 50 = 100$

$5 \bigstar 20 = 5 \times 20 = 100$

Each of $1 \bigstar 100$ , $4 \bigstar 25$ , and $10 \bigstar 10$ can be evaluated using the definition of $a \bigstar b$ for neither of and prime - that is, by adding the numbers:

$1 \bigstar 100 = 1 + 100 = 101$

$4 \bigstar 25 = 4+25 = 29$

$10 \bigstar 10 = 10 + 10 = 20$

The greatest of the five expressions is $1 \bigstar 100$ .

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GMAT Math : Understanding real numbers

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #41 : Understanding Real Numbers

Example Question #42 : Understanding Real Numbers

Example Question #43 : Understanding Real Numbers

Example Question #44 : Understanding Real Numbers

Example Question #45 : Understanding Real Numbers

Example Question #46 : Understanding Real Numbers

Example Question #47 : Understanding Real Numbers

Example Question #48 : Understanding Real Numbers

Example Question #49 : Understanding Real Numbers

Example Question #50 : Understanding Real Numbers

All GMAT Math Resources

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