GMAT Math : Understanding real numbers

Study concepts, example questions & explanations for GMAT Math

varsity tutors app store varsity tutors android store

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:

\displaystyle B + 1

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

\displaystyle 0

\displaystyle B

\displaystyle 2 A+ B

Correct answer:

\displaystyle B

Explanation:

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

\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)

\displaystyle = B + 0

\displaystyle = B

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:

\displaystyle AB + 1

\displaystyle AB - A^{2}

\displaystyle B

\displaystyle AB - 1

\displaystyle AB + A^{2}

Correct answer:

\displaystyle AB - A^{2}

Explanation:

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

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

\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)

\displaystyle = A B- A ^{2}

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

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

\displaystyle 1

\displaystyle 0

\displaystyle AB

Correct answer:

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

Explanation:

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

\displaystyle BC = 1,

or, equivalently,

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

By way of substitution and the distributive property,

\displaystyle A (B + C)

\displaystyle = AB + AC

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

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

Example Question #34 : Real Numbers

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

Which of the following is equivalent to 

\displaystyle A\left (BC \right )

for all values of the variables?

Possible Answers:

\displaystyle 0

\displaystyle -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 

\displaystyle CA = 1

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

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

 

Example Question #431 : Arithmetic

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

I) \displaystyle 12 + 45 \times 23

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

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

Possible Answers:

III only

I and III only

II and III only

I and II only

I 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.

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

\displaystyle AB + 1

\displaystyle AB - A^{2}

\displaystyle AB + A^{2}

Correct answer:

\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, \displaystyle AC = 1, so 

\displaystyle AB + AC = AB + 1.

\displaystyle AB + 1 is the correct choice.

Example Question #32 : Understanding Real Numbers

Define an operation  on the real numbers as follows:

If \displaystyle a < b, then .

If \displaystyle a = b \;, then 

If \displaystyle a > b, then .

Multiply  by . What is the result?

Possible Answers:

\displaystyle 70

\displaystyle -30

\displaystyle -21

\displaystyle 30

\displaystyle 21

Correct answer:

\displaystyle 30

Explanation:

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

.

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

The product of  and  is \displaystyle 3 \cdot 10 = 30.

Example Question #38 : Real Numbers

Define an operation  on the real numbers as follows:

If \displaystyle a < b, then 

If \displaystyle a = b \;, then 

If \displaystyle a > b, then 

Divide  by . What is the quotient?

Possible Answers:

\displaystyle 0

\displaystyle 1

\displaystyle -1

Undefined

\displaystyle -4

Correct answer:

\displaystyle 1

Explanation:

 and  are both calculated by using the defintion of  for the case \displaystyle a = b \;:

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

Example Question #1992 : Problem Solving Questions

Each of  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 \displaystyle \bigcirc = \bigtriangleup and , then 

\displaystyle \bigcirc = \bigcirc

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

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

Correct answer:

\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

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

is the one that demonstrates this for addition.

Example Question #33 : Understanding Real Numbers

Define an operation  on the real numbers as follows:

If \displaystyle a and \displaystyle b are both negative, then .

If \displaystyle a and \displaystyle b are not both negative, then .

Divide  by . What is the quotient?

Possible Answers:

\displaystyle -10

\displaystyle 10

\displaystyle 1

Undefined

\displaystyle 0

Correct answer:

Undefined

Explanation:

 can be evaluated using the definition of  for the case of both \displaystyle a and \displaystyle b being negative:

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

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

Tired of practice problems?

Try live online GMAT prep today.

1-on-1 Tutoring
Live Online Class
1-on-1 + Class
Learning Tools by Varsity Tutors