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Reciprocal of a Fraction — Definition, Formula & Examples

The reciprocal of a fraction is the fraction flipped upside down — the numerator becomes the denominator and the denominator becomes the numerator. For example, the reciprocal of 34\frac{3}{4} is 43\frac{4}{3}.

Given a nonzero fraction ab\frac{a}{b}, its reciprocal (also called the multiplicative inverse) is ba\frac{b}{a}. The defining property is that a number multiplied by its reciprocal always equals 1: ab×ba=1\frac{a}{b} \times \frac{b}{a} = 1.

Key Formula

Reciprocal of ab=ba,where a≠0 and b≠0\text{Reciprocal of } \frac{a}{b} = \frac{b}{a}, \quad \text{where } a \neq 0 \text{ and } b \neq 0
Where:
  • aa = The numerator of the original fraction
  • bb = The denominator of the original fraction

How It Works

To find the reciprocal, swap the top and bottom of the fraction. If you start with a whole number like 5, write it as 51\frac{5}{1} first, then flip it to get 15\frac{1}{5}. Reciprocals are essential when dividing fractions: instead of dividing by a fraction, you multiply by its reciprocal. This technique is often called "multiply by the flip" or "keep, change, flip." Note that zero has no reciprocal, because no number multiplied by 0 can equal 1.

Worked Example

Problem: Divide 23\frac{2}{3} by 45\frac{4}{5}.
Step 1: Find the reciprocal of the second fraction by swapping its numerator and denominator.
Reciprocal of 45=54\text{Reciprocal of } \frac{4}{5} = \frac{5}{4}
Step 2: Change the division to multiplication and use the reciprocal.
23÷45=23×54\frac{2}{3} \div \frac{4}{5} = \frac{2}{3} \times \frac{5}{4}
Step 3: Multiply the numerators together and the denominators together.
2×53×4=1012\frac{2 \times 5}{3 \times 4} = \frac{10}{12}
Step 4: Simplify the result by dividing numerator and denominator by their greatest common factor, 2.
1012=56\frac{10}{12} = \frac{5}{6}
Answer: 23÷45=56\frac{2}{3} \div \frac{4}{5} = \frac{5}{6}

Another Example

Problem: Find the reciprocal of 2132\frac{1}{3} and verify it.
Step 1: Convert the mixed number to an improper fraction.
213=732\frac{1}{3} = \frac{7}{3}
Step 2: Swap the numerator and denominator to find the reciprocal.
Reciprocal of 73=37\text{Reciprocal of } \frac{7}{3} = \frac{3}{7}
Step 3: Verify by multiplying the original fraction by its reciprocal. The product should equal 1.
73×37=2121=1✓\frac{7}{3} \times \frac{3}{7} = \frac{21}{21} = 1 \checkmark
Answer: The reciprocal of 2132\frac{1}{3} is 37\frac{3}{7}.

Why It Matters

Reciprocals appear constantly in pre-algebra and algebra courses whenever you divide fractions, solve equations like 35x=9\frac{3}{5}x = 9, or work with rates and unit conversions. In science classes, you use reciprocals when converting between quantities such as speed (miles per hour) and pace (hours per mile). Mastering this skill builds a foundation for algebraic manipulation with rational expressions.

Common Mistakes

Mistake: Flipping both fractions when dividing instead of only the divisor.
Correction: In a division like ab÷cd\frac{a}{b} \div \frac{c}{d}, only the second fraction (the one you are dividing by) gets flipped. The first fraction stays the same.
Mistake: Trying to find the reciprocal of a mixed number without converting it first.
Correction: Convert the mixed number to an improper fraction before flipping. The reciprocal of 3123\frac{1}{2} is not 31\frac{3}{1} over 12\frac{1}{2} — convert to 72\frac{7}{2} first, then the reciprocal is 27\frac{2}{7}.

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