GMAT Math : Understanding real numbers

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Example Question #21 : Understanding Real Numbers

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

I) $14 + 50 \div 5 \cdot 6$

II) $14 \div 50\cdot 5+ 6$

III) $14 \cdot 50+ 5\div 6$

Possible Answers:

None

I and III only

I, II, and III

III only

II and III only

Correct answer:

III only

Explanation:

Since there are no grouping symbols in any of the expressions, the order of operations dictates that the multiplication and division must be worked before the addition in each. The operations must be worked from left to right. Only in expression (III) does the multiplication occur before the division when reafing left to right, so this is the only expession evaluated by multiplying first.

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Example Question #22 : Real Numbers

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

$\textup{I. 12 + 3 - 14 - 19}$

$\textup{II. 12 - 3 + 14 - 19}$

$\textup{II. 12 - 3 - 14 + 19}$

Possible Answers:

$\textup{I and II}$

$\textup{I only}$

$\textup{II and III}$

$\textup{I and III}$

$\textup{I, II , and III}$

Correct answer:

$\textup{I only}$

Explanation:

In the absence of grouping symbols, additions and subtractions are performed from left to right. Only the expression in (I) has an addition as its leftmost operation, so this is the only expression that is evaluated by adding first.

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Example Question #22 : Understanding Real Numbers

When evaluating the expression

$123 \div 45 \times 32 \div 10 \times 12 \div 77$ ,

which of the following operations should be performed second?

Possible Answers:

The leftmost division

The rightmost multiplication

The rightmost division

The middle division

The leftmost multiplication

Correct answer:

The leftmost multiplication

Explanation:

In the absence of grouping symbols, multiplications and divisions are performed with precedence moving from left to right. Therefore, the five operations, each a multiplication or a division, are performed left to right, and the second operation performed is the leftmost multiplication.

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Example Question #21 : Understanding Real Numbers

Boxes

The above figure shows the first six of a sequence of boxes. If the sequence extends indefinitely, if is the lower right entry in the twelfth box, which of the following is a true statement?

Possible Answers:

$1,000<N \le 2,000$

$3, 000<N \le 4,000$

$2,000<N \le 3,000$

$4,000<N \le 5,000$

$0<N \le 1,000$

Correct answer:

$2,000<N \le 3,000$

Explanation:

The sequence of numbers in the upper left entries of the boxes is simply the sequence of natural numbers. The upper left number in the twelfth box will be .

The sequence of numbers in the upper right entries of the boxes is the sequence of powers of 2, beginning with $2 ^{0}= 1$ . The upper right number in the twelfth box will be $2 ^{11} = 2,048$ .

The sequence of numbers in the lower left entries of the boxes is the Fibonacci sequence, whose terms after the second are each the sum of the preceding two. To get the twelfth term requires a little work, but we can do it as follows:

$F_{7} = F_{5}+ F_{6} = 5+8 = 13$

$F_{8} = F_{6}+ F_{7} = 8 + 13= 21$

$F_{9} = F_{7}+ F_{8} = 13+21 = 34$

$F_{10} = F_{8} + F_{9}= 21+34 = 55$

$F_{11} = F_{9}+ F_{10} = 34 + 55 = 89$

$F_{12} = F_{10} +F_{11} = 55 + 89 = 144$

The entry in the lower right is the sum of the other three entries, so the lower right entry in the twelfth box will be

12+ 2,048 + 144 = 2,204 .

The correct choice is that $2,000<N \le 3,000$ .

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Example Question #25 : 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 distributive property?

Possible Answers:

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

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

$\bigcirc +0 = \bigcirc$

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

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

Correct answer:

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

Explanation:

The distributive property of multiplication over addition is the idea that, if a number is multiplied by a sum, the result will be the same as if the products of that number and each individual addend are added. The diagram that demonstrates this is

$\bigcirc \times (\square + \bigtriangleup) = \bigcirc \times \square + \bigcirc \times \bigtriangleup$ .

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Example Question #26 : 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 an associative property?

Possible Answers:

$\bigcirc = \bigcirc$

$\bigcirc \times 1 = \bigcirc$

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

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

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

Correct answer:

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

Explanation:

Addition and multiplication are both associative, which means that a sum or product of three numbers will result in the same value regardless of which two are added or multiplied first. This is demonstrated in multiplication by the diagram

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

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Example Question #27 : 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 transitive property?

Possible Answers:

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

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

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

$\bigcirc = \bigcirc$

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

Correct answer:

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

Explanation:

According to the transitive property of equality, if two numbers are equal to the same third number, they are equal to each other. This is demonstrated by this diagram:

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

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Example Question #28 : 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 symmetric property?

Possible Answers:

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

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

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

$\bigcirc +0 = \bigcirc$

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

Correct answer:

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

Explanation:

According to the symmetric property of equality, if one number is equal to another, the second is equal to the first. This is demonstrated by the diagram

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

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Example Question #29 : 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 an identity property?

Possible Answers:

$\bigcirc = \bigcirc$

$\bigcirc +0 = \bigcirc$

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

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

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

Correct answer:

$\bigcirc +0 = \bigcirc$

Explanation:

0 is called the additive identity, since it can be added to any number to yield, as a sum, the latter number. This property is demonstrated in the diagram

$\bigcirc +0 = \bigcirc$

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Example Question #30 : Real Numbers

is the additive inverse of .

Which of the following is equivalent to

ABC

for all values of the variables?

Possible Answers:

$-A^{2}B$

$A^{2}B$

Correct answer:

$-A^{2}B$

Explanation:

If is the additive inverse of , then

C+A = 0 , or, equivalently,

C = -A

Substituting,

$ABC = AB (-A) = A \cdot (-A) \cdot B = -A^{2}B$ .

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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 #21 : Understanding Real Numbers

Example Question #22 : Real Numbers

Example Question #22 : Understanding Real Numbers

Example Question #21 : Understanding Real Numbers

Example Question #25 : Real Numbers

Example Question #26 : Real Numbers

Example Question #27 : Real Numbers

Example Question #28 : Real Numbers

Example Question #29 : Real Numbers

Example Question #30 : Real Numbers

All GMAT Math Resources

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