ASVAB Study Guide
Fractions & Decimals · Lesson 2 of 5

Reciprocals of Fractions, Whole Numbers, and Mixed Numbers

Key rules and formulas, worked examples and 9 practice problems with step-by-step solutions.

1. Reciprocal of Fractions

Finding the inverse of a simple fraction

Formula: The reciprocal of $$\frac{a}{b} = \frac{b}{a}$$ ( with a,b ≠ 0)

Swap the positions of the numerator and the denominator.

Example

Find the reciprocal of $$\frac{4}{7}$$

Show solution

Step 1: Identify numerator and denominator:

  • Numerator = 4
  • Denominator = 7

Step 2: Swap numerator and denominator:

  • New numerator = 7
  • New denominator = 4

Result: $$\frac{7}{4}$$

Practice problems

  1. Problem 1

    Find the reciprocal of $$\frac{3}{5}$$

    Show solution

    Answer: $$\frac{5}{3}$$

    Step 1: Identify numerator (3) and denominator (5).

    Step 2: Swap numerator and denominator:

    • The top number (3) becomes the bottom.
    • The bottom number (5) becomes the top.

    => Reciprocal is $$\frac{5}{3}$$.

    => Correct Answer: B ($$\frac{5}{3}$$)

  2. Problem 2

    Find the reciprocal of $$\frac{1}{9}$$

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    Answer: 9

    Step 1: Swap numerator (1) and denominator (9) -> $$\frac{9}{1}$$.

    Step 2: Simplify fraction with denominator 1:

    • Any number divided by 1 equals itself: $$\frac{9}{1} = 9$$.

    => Reciprocal is 9.

    => Correct Answer: A (9).

  3. Problem 3

    Find the reciprocal of the result of $$(\frac{2}{3} \times \frac{9}{10})$$

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    Answer: $$\frac{5}{3}$$

    Step 1: Calculate the product inside expression:

    • $$(\frac{2}{3}) \times (\frac{9}{10}) = \frac{(2 \times 9)}{(3 \times 10)} = \frac{18}{30}$$.
    • Simplify $$\frac{18}{30}$$ by dividing by 6: $$\frac{3}{5}$$.

    Step 2: Find the reciprocal of $$\frac{3}{5}$$:

    • Swap numerator and denominator: $$\frac{3}{5}$$ becomes $$\frac{5}{3}$$.

    => Correct Answer: B ($$\frac{5}{3}$$)

2. Reciprocal of Whole Numbers

Finding the inverse of a full integer

Formula: The reciprocal of $$n = \frac{1}{n}$$ (with n ≠ 0)

Treat the whole number as a fraction with a denominator of $$1 (\frac{n}{1})$$, then flip it.

Example

Find the reciprocal of 8.

Show solution

Step 1: Express the whole number 8 as a fraction with denominator 1:

  • $$8 = \frac{8}{1}$$

Step 2: Swap the numerator and denominator:

  • $$\frac{8}{1}$$ flipped becomes $$\frac{1}{8}$$

Result: $$\frac{1}{8}$$

Practice problems

  1. Problem 4

    Find the reciprocal of 4.

    Show solution

    Answer: $$\frac{1}{4}$$

    Step 1: Write the whole number 4 as a fraction over 1: $$\frac{4}{1}$$.

    Step 2: Flip the fraction so the numerator and denominator swap positions.

    • $$\frac{4}{1}$$ becomes $$\frac{1}{4}$$.

    => Reciprocal is $$\frac{1}{4}$$.

    => Correct Answer: A ($$\frac{1}{4}$$)

  2. Problem 5

    Find the reciprocal of 15.

    Show solution

    Answer: $$\frac{1}{15}$$

    Step 1: Convert whole number 15 into fraction form: $$\frac{15}{1}$$.

    Step 2: Flip the positions of 15 and 1:

    • Top becomes 1, bottom becomes 15 -> $$\frac{1}{15}$$.

    => Reciprocal is $$\frac{1}{15}$$.

    => Correct Answer: B ($$\frac{1}{15}$$)

  3. Problem 6

    Find the reciprocal of $$(12 \div 3 + 1)$$

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    Answer: $$\frac{1}{5}$$

    Step 1: Calculate the value of the expression:

    • $$12 \div 3 = 4$$
    • $$4 + 1 = 5$$ (whole number).

    Step 2: Convert to fraction: $$5 = \frac{5}{1}$$.

    Step 3: Flip the fraction to get the reciprocal: $$\frac{1}{5}$$.

    => Correct Answer: A ($$\frac{1}{5}$$)

3. Reciprocal of Mixed Numbers

Finding the inverse of combined numbers

The reciprocal of $$a \frac{b}{c} = \frac{c}{(a \times c + b)}$$

Crucial: You must convert the mixed number to an improper fraction first, then flip it.

Example

Find the reciprocal of $$2 \frac{1}{3}$$

Show solution

Step 1: Convert mixed number $$2 \frac{1}{3}$$ to an improper fraction:

  • Numerator $$= (2 \times 3) + 1 = 6 + 1 = 7$$
  • Denominator = 3

=> Improper fraction $$= \frac{7}{3}$$

Step 2: Swap numerator and denominator to get the reciprocal:

  • $$\frac{7}{3}$$ flipped becomes $$\frac{3}{7}$$

Result: $$\frac{3}{7}$$

Practice problems

  1. Problem 7

    Find the reciprocal of $$1 \frac{1}{2}$$

    Show solution

    Answer: $$\frac{2}{3}$$

    Step 1: Convert $$1 \frac{1}{2}$$ to an improper fraction:

    • Numerator $$= (1 \times 2) + 1 = 3$$
    • Denominator = 2

    => Improper fraction $$= \frac{3}{2}$$.

    Step 2: Flip $$\frac{3}{2}$$ to find its reciprocal:

    • $$\frac{3}{2}$$ flipped becomes $$\frac{2}{3}$$.

    => Correct Answer: B ($$\frac{2}{3}$$)

  2. Problem 8

    Find the reciprocal of $$3 \frac{2}{5}$$

    Show solution

    Answer: $$\frac{5}{17}$$

    Step 1: Convert $$3 \frac{2}{5}$$ to an improper fraction:

    • Numerator $$= (3 \times 5) + 2 = 15 + 2 = 17$$
    • Denominator = 5

    => Improper fraction $$= \frac{17}{5}$$.

    Step 2: Flip $$\frac{17}{5}$$ to find its reciprocal:

    • $$\frac{17}{5}$$ flipped becomes $$\frac{5}{17}$$.

    => Correct Answer: A ($$\frac{5}{17}$$)

  3. Problem 9

    Find the reciprocal of $$5 \frac{3}{4}$$

    Show solution

    Answer: $$\frac{4}{23}$$

    Step 1: Convert $$5 \frac{3}{4}$$ to an improper fraction:

    • Numerator $$= (5 \times 4) + 3 = 20 + 3 = 23$$
    • Denominator = 4

    => Improper fraction $$= \frac{23}{4}$$.

    Step 2: Swap the numerator and denominator to get the reciprocal:

    • $$\frac{23}{4}$$ flipped becomes $$\frac{4}{23}$$.

    => Correct Answer: B ($$\frac{4}{23}$$)