GMAT Math : Understanding real numbers

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Example Questions

Example Question #431 : Arithmetic

$\displaystyle A$ is the additive inverse of $\displaystyle C$.

Which of the following is equivalent to

$\displaystyle A+ (B + C)$

for all values of the variables?

Possible Answers:

B + 1 $\displaystyle B + 1$

$A+ \frac{1}{A} + B$ $\displaystyle A+ \frac{1}{A} + B$

$\displaystyle 0$

$\displaystyle B$

2 A+ B $\displaystyle 2 A+ B$

Correct answer:

$\displaystyle B$

Explanation:

If $\displaystyle A$ is the additive inverse of $\displaystyle C$, then

C+A = 0 $\displaystyle C+A = 0$.

It follows by way of the commutative and associative properties that

$\displaystyle A+ (B + C)$

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

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

= B + 0 $\displaystyle = B + 0$

= B $\displaystyle = B$

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Example Question #31 : Understanding Real Numbers

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

$\displaystyle A (B + C)$

for all values of the variables?

Possible Answers:

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

$\displaystyle B$

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

AB + 1 $\displaystyle AB + 1$

AB - 1 $\displaystyle AB - 1$

Correct answer:

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

Explanation:

If $\displaystyle A$ is the additive inverse of $\displaystyle C$, then

C+A = 0 $\displaystyle C+A = 0$, or, equivalently,

C = -A $\displaystyle C = -A$

By way of the distributive property and substitution,

$\displaystyle A (B + C)$

$\displaystyle = A B + A C$

$\displaystyle = A B + A (-A)$

$= A B- A ^{2}$ $\displaystyle = A B- A ^{2}$

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Example Question #1982 : Problem Solving Questions

$\displaystyle B$ 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:

$\displaystyle A$

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

$\displaystyle 1$

$\displaystyle 0$

$\displaystyle AB$

Correct answer:

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

Explanation:

If $\displaystyle B$ is the multiplicative inverse of $\displaystyle C$, then

BC = 1 $\displaystyle BC = 1$,

or, equivalently,

$C = \frac{1}{B}$ $\displaystyle C = \frac{1}{B}$.

By way of substitution and the distributive property,

$\displaystyle A (B + C)$

$\displaystyle = AB + AC$

$= AB + A \cdot \frac{1}{B}$ $\displaystyle = AB + A \cdot \frac{1}{B}$

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

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Example Question #34 : Real Numbers

$\displaystyle A$ is the multiplicative inverse of $\displaystyle C$.

Which of the following is equivalent to

$A\left (BC \right )$ $\displaystyle A\left (BC \right )$

for all values of the variables?

Possible Answers:

$\displaystyle 0$

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

$A^{2}B$ $\displaystyle A^{2}B$

$\displaystyle 1$

$\displaystyle B$

Correct answer:

$\displaystyle B$

Explanation:

If $\displaystyle A$ is the multiplicative inverse of $\displaystyle C$, then

CA = 1 $\displaystyle CA = 1$.

By way of the commutative and associative properties, substitution, and the identity property of multiplication:

$A\left (BC \right ) = \left (BC \right ) A = B (CA) = B \cdot 1 = B$ $\displaystyle A\left (BC \right ) = \left (BC \right ) A = B (CA) = B \cdot 1 = B$

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Example Question #431 : Arithmetic

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:

II and III only

I only

III only

I and 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

$\displaystyle A$ 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:

$\displaystyle AB$

$\displaystyle B$

AB + 1 $\displaystyle AB + 1$

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

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

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 #31 : Understanding Real Numbers

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:

$\displaystyle 30$

$\displaystyle 21$

$\displaystyle 70$

$\displaystyle -21$

$\displaystyle -30$

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 #38 : Real Numbers

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:

$\displaystyle 0$

$\displaystyle 1$

$\displaystyle -1$

Undefined

$\displaystyle -4$

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 #1992 : Problem Solving Questions

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:

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

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

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

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 #33 : Understanding Real Numbers

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:

$\displaystyle -10$

$\displaystyle 10$

$\displaystyle 1$

Undefined

$\displaystyle 0$

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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GMAT Math : Understanding real numbers

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #431 : Arithmetic

Example Question #31 : Understanding Real Numbers

Example Question #1982 : Problem Solving Questions

Example Question #34 : Real Numbers

Example Question #431 : Arithmetic

Example Question #36 : Real Numbers

Example Question #31 : Understanding Real Numbers

Example Question #38 : Real Numbers

Example Question #1992 : Problem Solving Questions

Example Question #33 : Understanding Real Numbers

All GMAT Math Resources

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