Multiplying and dividing fractions, and fractions of amounts
Multiplying fractions surprises children because the answer gets smaller. Fractions of amounts and scaling are where fractions meet real problems. Bar models and area diagrams make both make sense.
Fractions of amounts
To find a fraction of an amount, divide by the denominator to find one part, then multiply by the numerator to find the parts you need.
- ¾ of 28: 28 ÷ 4 = 7, 7 × 3 = 21Divide by the denominator, multiply by the numerator.
- ¾ of 28: 28 ÷ 3 = 9.33…Dividing by the numerator.
A bar model shows why this works: draw 28 as a bar, split it into four equal parts of 7, and shade three. Children who draw it once rarely mix up the steps again. Working backwards is a common SATs question too: if ⅗ of a number is 24, then ⅕ is 8 and the whole is 40.
Scaling
Scaling problems multiply a quantity by a fraction or a whole number to make it larger or smaller, such as adjusting a recipe. A recipe for 4 people needs 300 g of flour; for 6 people, scale by 6/4 or 1½, giving 450 g. Asking “should the answer be bigger or smaller than what I started with?” before calculating catches most errors.
Multiplying fractions
To multiply two fractions, multiply the numerators and multiply the denominators. Year 6 pupils multiply simple pairs of proper fractions and write the answer in its simplest form.
- ½ × ¼ = 1/8Half of a quarter is an eighth.
- ⅔ × ¾ = 6/12 = ½Simplify the answer.
Children often expect multiplying to make things bigger, so ½ × ½ = ¼ feels wrong. Reading “×” as “of” helps: half of a half is a quarter. An area model (shade ½ of a rectangle one way, ¼ the other way, and look at the overlap) shows it visually.
Dividing fractions by whole numbers
Dividing a fraction by a whole number shares it into equal parts. ⅓ ÷ 2 means sharing a third into two pieces: each piece is ⅙. The method is to multiply the denominator by the whole number, keeping the numerator the same.
- ⅓ ÷ 2 = 1/6The denominator is multiplied by 2.
- ⅘ ÷ 2 = ⅖Or divide the numerator, when it divides exactly.
- ⅓ ÷ 2 = 1/1.5Dividing the denominator instead of multiplying it.
In the SATs
The arithmetic paper includes fractions of amounts, multiplying pairs of fractions and dividing a fraction by a whole number. Answers often need to be simplified. The reasoning papers set fractions of amounts in context, including working backwards from a part to the whole, and scaling problems with recipes, maps and measures.
Common mistakes
- Dividing by the numerator when finding a fraction of an amount.
- Forgetting to multiply after dividing.
- Expecting multiplication to make the answer bigger, and doubting a correct answer.
- Dividing the denominator instead of multiplying it.
- Leaving answers unsimplified.
Teaching ideas
Bar models for everything
For the first few weeks, every fraction-of-an-amount question gets a bar model, even easy ones. When working-backwards questions arrive, the model makes them straightforward.
Fold and shade
Children fold a paper rectangle in half one way and in quarters the other, then shade to show ½ × ¼. Counting the small rectangles makes the answer obvious.
Recipe scaling
Give a real recipe for 4 people and ask pupils to adapt it for 2, 6 and 10. Discuss which ingredients are awkward to scale and why.
Bigger or smaller?
Before any fraction calculation, pupils predict whether the answer will be bigger or smaller than the starting number. It builds number sense and catches upside-down methods.
Try it: five quick questions
SATs-style questions to use as a starter or a quick check. Tap to see each answer.
Find ⅖ of 45.
Show answer
18
⅜ of a number is 21. What is the number?
Show answer
56
Calculate ⅔ × ⅘.
Show answer
8/15
Calculate ⅗ ÷ 3.
Show answer
⅕
A recipe for 4 uses 250 ml of milk. How much for 10 people?
Show answer
625 ml
Free worksheets for this topic
Printable A4 PDFs with answers. Free to use in class or at home.
