Key takeaway: Dividing fractions means finding how many times one fraction fits into another, which you calculate by multiplying the first fraction by the reciprocal of the second.
Grade 66.NS.A.1Numbers and Operations
Definition of Dividing Fractions
Dividing fractions answers a “how many groups?” question. To compute \(\frac{11}{12} : \frac{1}{6}\), you ask how many sixths fit inside eleven twelfths.
The shortcut is to keep the first fraction, change the division sign to multiplication, and flip the second fraction. Why does that work? Splitting something into groups of size \(\frac{1}{6}\) is the same as counting 6 groups per whole. The result of dividing fractions can be a whole number, a mixed number or another fraction.
Formula and notation
Take any two fractions with nonzero denominators and a nonzero second numerator:
\[\frac{p}{q} : \frac{r}{s} = \frac{p}{q} \times \frac{s}{r} = \frac{p \cdot s}{q \cdot r}\]
Only the second fraction, the divisor, is turned upside down. Write whole numbers over 1 first, and simplify before or after multiplying.
Key facts
- Never flip the first fraction; only the divisor is replaced by its reciprocal.
- A proper fraction as divisor enlarges the result, because small pieces fit many times .
- A whole number becomes a fraction over 1, so \(6 : \frac{2}{3}\) turns into \(\frac{6}{1} \times \frac{3}{2}\).
- Multiply the quotient by the divisor to confirm you get the original fraction back.
Dividing Fractions example
A climbing rope section measures \(\frac{11}{12}\) meter. A coach cuts it into pieces that are \(\frac{1}{6}\) meter long. How many pieces does she get?
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Set up the division: \(\frac{11}{12} : \frac{1}{6}\).
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Multiply by the reciprocal of the divisor: \(\frac{11}{12} \times \frac{6}{1}\).
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Multiply and reduce: \(\frac{66}{12} = \frac{11}{2} = 5\frac{1}{2}\).
She gets 5 full pieces plus half of another one (\(5\frac{1}{2}\) in all).
Common mistake
Watch out
Lena computes \(\frac{11}{12} : \frac{1}{6}\) as \(\frac{12}{11} \times \frac{1}{6} = \frac{2}{11}\).
Why it is wrong
She flipped the first fraction instead of the divisor. The result \(\frac{2}{11}\) is smaller than 1, yet sixths fit into eleven twelfths more than once. Keep \(\frac{11}{12}\) and flip only \(\frac{1}{6}\) to \(\frac{6}{1}\).
Related terms and lessons
Browse the glossary: A–Z list · Grade 6 vocabulary · Numbers and Operations