Complex Fraction
In short: Complex fraction describes a quotient written with fractions inside its numerator, its denominator, or both, and it equals the top value divided by the bottom value.
7th GradeNumbers and Operations7.RP.A.1
What is Complex Fraction?
Picture a long division bar with smaller bars hiding above and below it: that layered look is a complex fraction. Halves, fifths or ninths can occupy the top slot, the bottom slot, or both.
In symbols, \( rac{a}{b}\) qualifies when \(a\) or \(b\) is itself a ratio of integers and \(b
eq 0\). The long bar means division, so \( rac{ rac{p}{q}}{ rac{r}{s}} = rac{p}{q} \div rac{r}{s}\). Rates measured in fractional units, such as miles per half hour, produce this layered shape naturally.
How to spot it
- A speed or price compares two measurements and at least one is fractional, like 3/4 mile covered in 1/3 hour.
- Several fraction bars appear in one expression, with a longer main bar separating the upper part from the lower part.
- A question asks for a unit rate, meaning the amount per one unit, and the given quantities are not whole numbers.
- A mixed number divides another mixed number and the result is written as a single stacked fraction.
The method
- Replace the long main bar with a division sign placed between the upper fraction and the lower fraction.
- Change any mixed number into an improper fraction before you divide.
- Multiply the top fraction by the reciprocal of the bottom fraction, then cancel common factors.
- Simplify the product and attach the correct unit if the problem comes from a context.
Complex Fraction: solved example
Problem
A Maine lobster boat crew hauls \( rac{5}{8}\) of a trap line in \( rac{15}{16}\) of an hour. Write that pace as a layered quotient and simplify it to get the part of the line hauled each hour.
Solution
Pace \(= \frac{\frac{5}{8}}{\frac{15}{16}} = \frac{5}{8} \div \frac{15}{16}\).
Multiply by the reciprocal: \(\frac{5}{8} \times \frac{16}{15} = \frac{80}{120} = \frac{2}{3}\). The crew hauls \(\frac{2}{3}\) of the line per hour.
Try it
Question 1
Simplify \(\frac{\frac{7}{9}}{\frac{14}{27}}\).
Show the answer
\(\frac{7}{9} \times \frac{27}{14} = \frac{189}{126} = \frac{3}{2}\).
Question 2
A hiker walks \(1\frac{1}{2}\) miles in \(\frac{3}{8}\) hour. How many miles per hour is that?
Show the answer
\(\frac{3}{2} \div \frac{3}{8} = \frac{3}{2} \times \frac{8}{3} = 4\) miles per hour.
Complex Fraction vs. Rate of Change
A complex fraction is a way of writing a quotient whose parts are fractions; it is only a form. A rate of change is a meaning: how one quantity changes per unit of another. A rate with fractional measurements can be written as a complex fraction, but not every complex fraction describes a rate.
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Whole dictionary: A–Z · 7th Grade · Numbers and Operations