Key takeaway: A complex fraction is a fraction whose numerator, denominator, or both contain fractions, and it is simplified by dividing the top fraction by the bottom fraction.
Grade 77.RP.A.1Numbers and Operations
Definition of Complex Fraction
A complex fraction is a fraction built from other fractions, such as \(\frac{3/4}{1/6}\). The long bar means divide, so this reads “three fourths divided by one sixth”.
Complex fractions appear naturally in unit rates with fractional amounts. If Rina walks \(\frac{3}{4}\) mile in \(\frac{1}{6}\) hour, her speed in miles per hour is the complex fraction \(\frac{3/4}{1/6}\). To simplify, rewrite the main fraction bar as a division sign and multiply by the reciprocal of the bottom fraction.
Formula and notation
The main fraction bar is drawn longer than the others:
\[\frac{\;\frac{a}{b}\;}{\frac{c}{d}} = \frac{a}{b} : \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}\]
Here \(b\), \(c\) and \(d\) are not zero. A whole number or mixed number can sit in the top or bottom as well.
Key facts
- The main bar means division of the top expression by the bottom one.
- Only the bottom fraction is replaced by its reciprocal.
- The result is the same as multiplying the top and bottom by a common denominator.
- Unit rates with fractional amounts, like miles per hour, are written this way.
Complex Fraction example
Rina walks \(\frac{3}{4}\) mile in \(\frac{1}{6}\) hour. Write her speed as a complex fraction and simplify it.
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Speed is distance over time: \(\frac{3/4}{1/6}\) miles per hour.
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Rewrite as division and flip the divisor: \(\frac{3}{4} \times \frac{6}{1}\).
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Multiply: \(\frac{18}{4} = \frac{9}{2} = 4.5\).
Rina walks at 4.5 miles per hour.
Common mistake
Watch out
Sol flips the top fraction instead: \(\frac{4}{3} \times \frac{1}{6} = \frac{4}{18} = \frac{2}{9}\).
Why it is wrong
Only the bottom fraction (the divisor) is turned upside down. Keeping \(\frac{3}{4}\) and flipping \(\frac{1}{6}\) to \(\frac{6}{1}\) gives 4.5. The wrong answer \(\frac{2}{9}\) mile per hour is too slow: walking \(\frac{3}{4}\) mile in a sixth of an hour is a brisk pace.
Related terms and lessons
Browse the glossary: A–Z list · Grade 7 vocabulary · Numbers and Operations