Why fraction division works
Count fractional groups and explain the reciprocal rule instead of memorising it alone.
Before you begin
- Equivalent fractions
- Fraction multiplication
Count using a common-sized piece
To find 3/4 ÷ 1/2, ask how many half-sized groups fit into three quarters. Rewrite one half as two quarters. Three quarter-pieces contain one group of two pieces and half of another group, giving 3/2 groups. When both fractions use the same denominator, divide their numerators because each numerator counts the same-sized piece.
Connect division with multiplication
If q groups of size c/d make a/b, then q × c/d = a/b. Multiplying both sides by d/c isolates q, provided c is not zero. This gives the reciprocal rule, but the answer still needs interpretation: it might be a number of servings, a length or an amount per person. Check by multiplying your quotient by the divisor.
Worked example
A bottle contains 5/6 L. How many 1/3 L portions does that represent?
Show the worked solution
- Rewrite 1/3 L as 2/6 L, so the question becomes five sixths divided by two sixths.
- 5 ÷ 2 = 5/2 = 2½ portions.
- Check: 2½ × 1/3 = 5/6 L. There are two full portions and half of another.
2½ portions; 2 are full portions.
Try it yourself
What is 2/3 ÷ 4/5?
Common mistakes
- Invert the divisor, not the dividend.
- A fractional number of portions differs from the number of full portions.
What can you explain now?
A rectangle has area 3/5 m² and length 3/4 m. Find its width and check it.
Compare with the explanation
4/5 m.
Width = area ÷ length = (3/5) × (4/3) = 4/5. Multiplying 4/5 m by 3/4 m gives 3/5 m².
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Original StudyForA content. AI-assisted checks completed; subject-teacher review is still pending. These lessons are not endorsed by an exam board.
Common Core mathematics standards ↗