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You already know about halves and quarters. In Key Stage 2 you go much further — working with any fraction, comparing them, adding and subtracting them, multiplying them, and converting between mixed numbers and improper fractions.
Equivalent fractions have the same value, even though they look different.
1/2 = 2/4 = 3/6 = 4/8 = 5/10 ...
To find an equivalent fraction, multiply (or divide) both the numerator and denominator by the same number.
Equivalent fractions form a "family" — multiplying top and bottom by the same number scales up; dividing scales down. Here is the family of 1/2:
flowchart TD
A[1/2]
A -->|x 2/2| B[2/4]
A -->|x 3/3| C[3/6]
A -->|x 4/4| D[4/8]
A -->|x 5/5| E[5/10]
A -->|x 6/6| F[6/12]
B -->|/ 2/2| A
C -->|/ 3/3| A
D -->|/ 4/4| A
E -->|/ 5/5| A
F -->|/ 6/6| A
All these fractions have the same value. Always simplify back to 1/2 — the "head" of the family.
A fraction is in its simplest form when the numerator and denominator share no common factor other than 1.
To simplify, divide both numbers by their highest common factor (HCF).
Example: Simplify 12/18
Example: Simplify 15/20
To compare fractions with different denominators, convert them to a common denominator first.
Example: Which is larger, 3/4 or 5/7?
Example: Order 1/2, 2/5, 3/10 from smallest to largest.
Same denominator: simply add or subtract the numerators.
Different denominators: find a common denominator first.
Example: 1/3 + 1/4
Example: 3/4 - 1/6
A mixed number has a whole number part and a fraction part: 2 and 3/4.
An improper fraction has a numerator larger than (or equal to) its denominator: 11/4.
Converting mixed to improper:
Converting improper to mixed:
Multiplying a fraction by a whole number:
Multiplying a fraction by a fraction:
To divide a fraction by a whole number, multiply the denominator by that whole number (and keep the numerator the same).
Why it works: dividing by 2 is the same as halving — it makes each piece smaller, so there are more pieces per whole.
Examples:
Check: 1/6 x 3 = 3/6 = 1/2. ✓
To find a fraction of an amount, divide by the denominator then multiply by the numerator.
Three families share two large pizzas at a party. Each family has 4 children. The pizzas are cut into 12 equal slices each, so there are 24 slices in total. Every child wants the same amount of pizza. How much does each child get, expressed as a fraction?
Step 1 — Total number of children.
3 families x 4 children = **12 children**
Step 2 — Total number of slices.
2 pizzas x 12 slices = **24 slices**
Step 3 — Slices per child.
24 / 12 = **2 slices each**
Step 4 — Express as a fraction of one pizza. Each pizza has 12 slices, so each slice is 1/12 of a pizza. Two slices = 2/12 of a pizza, which simplifies to 1/6 of a pizza.
Divide top and bottom by 2: 2/12 = 1/6 ✓
Step 5 — Express as a fraction of all the food. Each child receives 2 slices out of 24 total slices = 2/24 = 1/12 of all the pizza.
Step 6 — Add a fourth family with 5 children. Now there are 12 + 5 = 17 children, but only 24 slices. Each child cannot receive 2 whole slices.
24 / 17 ≈ 1.41 slices per child — not a whole number.
Cut each slice into smaller equal pieces. Cut every slice into 17 tiny pieces? That gives 24 x 17 = 408 tiny pieces, which can be shared 24 each — impractical, but mathematically valid.
Step 7 — A more practical approach. Cut a third pizza into 12 slices (now 36 slices total). Each child receives 36 / 17 = 2.12 slices ≈ 2 slices and a small extra. Or simpler: order four pizzas (48 slices). Each child gets 48 / 17 ≈ 2.82 slices ≈ 2 large slices plus most of a third.
Step 8 — Compare two children's shares using a bar model.
Child A: |1/12|1/12| = 2/12 = 1/6
Child B: |1/12|1/12|1/12| = 3/12 = 1/4
Child B has more pizza. Compare: 1/6 vs 1/4. Common denominator is 12 → 2/12 vs 3/12 → 1/4 > 1/6.
Step 9 — Combine three children's shares.
Child A: 1/6 = 2/12
Child B: 1/4 = 3/12
Child C: 1/3 = 4/12
Total = 2/12 + 3/12 + 4/12 = **9/12 = 3/4 of one pizza**.
So three children together have eaten three-quarters of a pizza.
Step 10 — Find a fraction of an amount. A pizza box contains 36 small pizzas. What is 3/4 of 36? Divide by the denominator (4): 36 / 4 = 9. Multiply by the numerator (3): 9 x 3 = 27 small pizzas.
Step 11 — Multiply two fractions. A child eats 1/2 of their 1/6 share of a pizza. How much of the whole pizza is that?
1/2 x 1/6 = (1 x 1) / (2 x 6) = **1/12** of a pizza
So the child ate just one-twelfth of one pizza — a small slice indeed.
Step 12 — Divide a fraction by a whole number. Three children share 3/4 of a remaining pizza equally between them. How much does each get?
3/4 / 3 = 3 / (4 x 3) = 3/12 = **1/4** of a pizza per child
This is a satisfying answer: three children sharing three-quarters means each gets a quarter, which makes sense.
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