How to Multiply and Divide Fractions — KS3 Maths

To multiply fractions, multiply the numerators together and the denominators together, then simplify. To divide fractions, keep the first fraction, change the division sign to multiplication, and flip the second fraction — then multiply as normal. These two rules cover every fraction multiplication and division question you will see at KS3.

How do you multiply fractions?

Multiplying fractions is the simplest fraction operation: no need to find a common denominator.

Rule: multiply top × top, bottom × bottom, then simplify.

Worked example 1 — simple multiplication

Calculate 3/4 × 2/5.

Multiply the numerators: 3 × 2 = 6

Multiply the denominators: 4 × 5 = 20

Result: 6/20

Simplify by dividing top and bottom by 2: 3/10

Answer: 3/4 × 2/5 = 3/10

Worked example 2 — multiplying with a whole number

Calculate 5 × 2/3.

Write the whole number as a fraction: 5 = 5/1.

5/1 × 2/3

Numerators: 5 × 2 = 10

Denominators: 1 × 3 = 3

Result: 10/3, which is the mixed number 3 and 1/3

Answer: 5 × 2/3 = 10/3 = 3⅓

What is the trick for simplifying before you multiply (cross-cancelling)?

Instead of simplifying at the end, you can cancel common factors between any numerator and any denominator before you multiply. This keeps numbers smaller and reduces arithmetic errors.

Example: 4/9 × 3/8

The 4 (numerator) and 8 (denominator) share a factor of 4: 4 ÷ 4 = 1, 8 ÷ 4 = 2

The 3 (numerator) and 9 (denominator) share a factor of 3: 3 ÷ 3 = 1, 9 ÷ 3 = 3

Now multiply: 1/3 × 1/2 = 1/6

Answer: 4/9 × 3/8 = 1/6 — much easier than simplifying 12/72 at the end.

How do you divide fractions?

Division by a fraction uses the keep-change-flip rule (also called KCF or multiplying by the reciprocal).

Rule:

Keep the first fraction exactly as it is.

Change the ÷ sign to ×.

Flip the second fraction (swap numerator and denominator).

Multiply and simplify.

Worked example 3 — basic division

Calculate 3/5 ÷ 2/7.

Keep: 3/5

Change: ÷ becomes ×

Flip: 2/7 becomes 7/2

Multiply: 3/5 × 7/2 = 21/10

Simplify: 21/10 = 2 and 1/10

Answer: 3/5 ÷ 2/7 = 21/10 = 2⅒

Worked example 4 — dividing a fraction by a whole number

Calculate 4/5 ÷ 3.

Write 3 as 3/1, then flip to get 1/3.

4/5 × 1/3 = 4/15

Answer: 4/5 ÷ 3 = 4/15

How do you multiply or divide mixed numbers?

Always convert mixed numbers to improper fractions first.

Converting: multiply the whole part by the denominator, add the numerator, keep the same denominator.

Example: 2 and 1/3 = (2 × 3 + 1)/3 = 7/3

Worked example 5 — multiplying mixed numbers

Calculate 1½ × 2⅓.

Convert: 1½ = 3/2, 2⅓ = 7/3

Multiply: 3/2 × 7/3

Cross-cancel: the 3 in the numerator and 3 in the denominator cancel to 1

1/2 × 7/1 = 7/2

Convert back: 7/2 = 3½

Answer: 1½ × 2⅓ = 3½

Common mistakes to watch out for

Mistake | Wrong | Correct
Adding denominators when multiplying | 2/3 × 1/4 = 2/7 | 2/3 × 1/4 = 2/12 = 1/6
Forgetting to flip when dividing | 3/4 ÷ 1/2 = 3/8 | 3/4 × 2/1 = 6/4 = 3/2
Not converting mixed numbers first | 1½ × 2 = 2½ | 3/2 × 2/1 = 6/2 = 3
Forgetting to simplify the final answer | leaving 6/20 | simplify to 3/10

Why does flipping work?

Dividing by a number is the same as multiplying by its reciprocal. For example, dividing by 2 is the same as multiplying by 1/2. The reciprocal of a/b is b/a, because a/b × b/a = ab/ab = 1 — and multiplying by 1 leaves your fraction unchanged. This is why flipping the second fraction and switching to multiplication gives the right answer every time.

How to check your answer

Substitute your result back. If 3/5 ÷ 2/7 = 21/10, then 21/10 × 2/7 should return 3/5.

21/10 × 2/7 = 42/70 = 3/5 ✓

The check confirms the answer. Get into the habit of doing this on calculator papers where the check takes only seconds.

Frequently asked questions

How do you multiply two fractions together?

Multiply the numerators together to get the new numerator, then multiply the denominators together to get the new denominator. Simplify the result. No common denominator is needed — that is only required for adding and subtracting fractions.

What is the keep-change-flip rule for dividing fractions?

Keep-change-flip means: keep the first fraction unchanged, change the division sign to a multiplication sign, and flip the second fraction upside down (swap its numerator and denominator). Then multiply the two fractions normally and simplify.

Why do you flip the second fraction when dividing?

Because dividing by a fraction is mathematically identical to multiplying by its reciprocal. The reciprocal of a/b is b/a. Switching the operation and flipping the second fraction is a shortcut that applies this rule in one step.

How do you multiply mixed numbers?

Convert each mixed number into an improper fraction first, then multiply numerator by numerator and denominator by denominator. Simplify the result and convert back to a mixed number if required.

What is a reciprocal?

The reciprocal of a fraction is that fraction flipped — numerator and denominator swapped. The reciprocal of 3/4 is 4/3. Every non-zero number has a reciprocal: the reciprocal of 5 is 1/5.

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