Mixed numbers and division stories
Translate a situation into fraction division and explain any unused amount.
Before you begin
- Improper fractions and mixed numbers
- Dividing fractions
Convert the whole quantity first
A mixed number combines whole units and a proper fraction. Convert it into one improper fraction before multiplication or division: 2⅓ = 7/3 because two wholes contain six thirds. Treating 2⅓ as 2 × 1/3 would change its value. Draw complete units and the remaining piece if you are unsure of the conversion.
Distinguish groups from leftover material
Division may produce a mixed-number answer even when the context only allows complete objects. For 2½ metres cut into 3/4-metre lengths, the quotient is a count of lengths. Its fractional part is a fraction of one cut length, not a length in metres. Multiply that fraction by the divisor, or subtract the used material from the original amount, to find the leftover.
Worked example
A 3½ m ribbon is cut into 3/4 m pieces. How many complete pieces and how much ribbon remain?
Show the worked solution
- Convert 3½ to 7/2, then calculate (7/2) ÷ (3/4) = 14/3 = 4⅔.
- Four complete pieces use 4 × 3/4 = 3 m.
- The remaining ribbon is 3½ − 3 = 1/2 m; it is two-thirds of one 3/4 m piece.
4 complete pieces; 1/2 m remains.
Try it yourself
What is 2¼ ÷ 1½?
Common mistakes
- A fractional quotient has units of groups; it is not automatically the leftover amount.
- Convert mixed numbers before applying the reciprocal rule.
What can you explain now?
Share 4½ kg of rice equally among 6 families. How much does each family receive?
Compare with the explanation
3/4 kg.
Convert 4½ to 9/2. Dividing by six gives 9/12 = 3/4. Six shares of 3/4 kg total 4½ kg.
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Bring the ideas together
Dividing fractions
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Common Core mathematics standards ↗