Multiplying Fractions

La méthode, étape par étape

1. 1

   Multiply straight across

   2/3 × 4/5 = (2×4)/(3×5) = 8/15

   Tops together, bottoms together. This is genuinely all there is to the operation, which surprises students who have just spent weeks on common denominators for addition — and the surprise is worth acknowledging rather than glossing over.

2. 2

   Turn mixed numbers into improper fractions first

   1½ × 2⅓ → 3/2 × 7/3

   The multiply-across rule does not work on mixed numbers. Multiplying whole parts and fractional parts separately gives an answer that is badly wrong, because it silently drops two of the four products the multiplication actually contains.

3. 3

   Cancel common factors before multiplying

   3/2 × 7/3 → cancel the 3s → 1/2 × 7/1 = 7/2

   Any numerator can cancel with any denominator, across the multiplication sign as well as within a fraction. Cancelling first keeps the arithmetic small; cancelling afterwards means simplifying 21/6, which is more work and more risk for the same answer.

4. 4

   Read 'of' as multiply

   ¾ of 20 = ¾ × 20 = 15

   In fraction problems the word 'of' is an instruction to multiply. This one substitution converts most word problems in the topic into a calculation the student can already do, and it is the single most useful sentence in the whole subject area.

5. 5

   Convert back and check the size

   7/2 = 3½. Sensible: 1½ × 2⅓ should be a bit over 3 ✓

   Answers should be returned in the form the question used — mixed numbers in, mixed number out. Then check the size: multiplying two numbers each above 1 must give something larger than both.

Why multiplying can make a number smaller

Multiplication makes things bigger — that is the intuition built through years of whole numbers, and fractions break it. Half of a half is a quarter, so multiplying by ½ has made the number smaller. Students who never confront this treat their own correct answers as errors.

The resolution is to read multiplication by a fraction as 'a part of', not as 'lots of'. Multiplying by 3 means three lots of something and grows it; multiplying by ½ means half of something and shrinks it. Multiplying by 1 leaves it alone, which is the dividing line.

• Multiply numerators, multiply denominators
• No common denominator required
• 'of' means multiply
• Multiplying by less than 1 makes the answer smaller
• Cancel before multiplying, not after

Cancelling first is not optional at GCSE

For 2/3 × 4/5 cancelling makes no difference. For 24/35 × 21/16 it is the difference between a manageable calculation and multiplying 504 by 560. Since exam numbers are chosen to cancel, a student who does not look for it is doing arithmetic the examiner never intended.

The rule to teach is that any top can cancel with any bottom, including across the multiplication sign. Students who think cancelling only works vertically within one fraction miss most of the available simplification.

The mixed-number trap

Multiplying 1½ × 2⅓ as (1 × 2) and (½ × ⅓) gives 2⅙, and the correct answer is 3½. The method drops the two cross terms entirely, and because the wrong answer is in a plausible range it survives a casual check.

Converting to improper fractions first removes the possibility. It is one extra line of working and it is the difference between a method that always works and one that works only when no whole numbers are involved.

How we teach multiplying fractions

We teach it immediately after adding, and we say plainly that it is easier — students expect each topic to be harder than the last and will invent complications to meet that expectation. Naming multiplication as the simpler operation prevents a lot of unnecessary common-denominator work.

We drill the 'of' substitution until it is automatic, because it turns the word problems that students find hardest into calculations they find easy. Recognising the operation is most of the difficulty in this topic; performing it is not.