Divide by fractions

Ask how many groups fit

How many half-litre portions fit in 3 litres? Each litre contains two half-litres, so three litres contain six. Thus 3÷1/2 = 6. Division can give a result larger than the starting number when each group is smaller than one unit.

For 3/4÷1/8, count eighths: 3/4 = 6/8, so six groups of one eighth fit. The quotient is 6 groups, not 6 litres unless a separate unit interpretation requires it.

Allow a fractional number of groups

For 3/4÷1/2, express both in quarters. You have three quarter-units, and a whole group needs two quarter-units. Three divided by two gives 3/2 = 11/2 groups. One full half and half of another half make three quarters.

Scaling both dividend and divisor by the same non-zero factor preserves how many divisor-sized groups fit. Multiply both by 2:

In general, dividing by a non-zero fraction c/d is multiplying by d/c. This scales the divisor to 1. The divisor must not be zero.

Equal amounts give a quotient of one: 2/3÷4/6 = 1, since both fractions name the same amount. For two positive fractions, compare the dividend with the divisor first: a smaller dividend gives a quotient below one, equal amounts give one, and a larger dividend gives a quotient above one.

Divide a fraction by a whole number

3/5÷4 = 3/5 × 1/4 = 3/20. Sharing three fifths equally among four makes a smaller part. Do not invert the first fraction: the starting amount is still three fifths.

Check your understanding 1 Share a fractional amount

Find 2/3÷5. Should the result be smaller than 2/3? Check by multiplying your answer by 5.

Your quotient, size check and multiplication check

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2/3÷5 = 2/3 × 1/5 = 2/15, which is smaller than 2/3. Five equal shares must recover the original amount: 2/15 × 5 = 2/3. Inverting the starting fraction would change the amount being shared.

Check your understanding 2 Check the quotient by multiplying

Calculate 2/3÷4/5. Show a multiplication that checks your result.

Your quotient and multiplication check

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The quotient is 2/3 × 5/4 = 10/12 = 5/6. Check 5/6 × 4/5 = 2/3. There are five sixths of a four-fifths-sized group in two thirds.

A story changes the final decision

Check your understanding 3 Decide what the quotient means

You have 3/4 m of ribbon. Each complete bow requires 2/5 m. How many complete bows can you make, and what length remains? Show the quotient before deciding the number of complete bows.

Your quotient, complete-bow count and remaining length

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3/4÷2/5 = 15/8 = 17/8 bow-sized lengths, so only one complete bow. The remaining length is 3/4-2/5 = 7/20 m. Seven eighths is a fraction of a bow-sized length, not seven eighths of a metre.