Dividing Fractions

The method, step by step

1. 1

   Keep the first fraction, flip the second, multiply

   3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8

   Only the second fraction is inverted. Flipping both, or flipping the first, are the two commonest errors and both produce plausible-looking answers. The first fraction is untouched throughout.

2. 2

   Convert mixed numbers before flipping

   2½ ÷ 1¼ → 5/2 ÷ 5/4 → 5/2 × 4/5

   As with multiplication, the rule applies only to proper or improper fractions. Flipping a mixed number is meaningless, and attempting it is a reliable sign that the conversion step was skipped.

3. 3

   Cancel, then multiply

   5/2 × 4/5 → cancel the 5s and 2 into 4 → 1/1 × 2/1 = 2

   Once the division has become a multiplication, everything true of multiplication applies — including cancelling across the sign. Exam numbers are usually chosen so this cancels neatly, which is a hint that the conversion was done correctly.

4. 4

   Dividing by a whole number means flipping it too

   3/5 ÷ 4 = 3/5 ÷ 4/1 = 3/5 × 1/4 = 3/20

   Any whole number is a fraction over 1, so the same rule covers it. Writing the 4 as 4/1 before flipping makes this obvious and prevents the common error of dividing only the numerator.

5. 5

   Check the size makes sense

   3/4 ÷ 2/5 = 15/8 = 1⅞ — dividing by less than 1 grew it ✓

   Dividing by a number smaller than 1 makes the answer larger, which contradicts years of whole-number intuition. Checking against that expectation catches a flipped-the-wrong-fraction error immediately.

Why 'keep, change, flip' works

Division asks how many of one thing fit into another. How many halves fit into 3? Six — so 3 ÷ ½ = 6, and that is the same as 3 × 2. The reciprocal is not a trick; it is the answer to 'how many of these fit into one', and multiplying by it scales that answer up.

Students given the mnemonic without the reason apply it confidently and cannot detect when they have flipped the wrong fraction, because they have no expectation of what the answer should look like. Thirty seconds on the 'how many fit' question buys that expectation.

• Keep the first fraction unchanged
• Change ÷ to ×
• Flip the second fraction only
• Whole numbers are n/1, so they flip to 1/n
• Dividing by less than 1 makes the answer bigger

Why the answer usually gets bigger

Every division a student meets before fractions makes numbers smaller, so an answer larger than the number you started with reads as an error. Dividing by a fraction less than 1 does exactly that, because you are asking how many small things fit inside a bigger one — and the answer to that is a large count.

This is the check that makes the topic safe. If the divisor is under 1 and the answer came out smaller, the wrong fraction was flipped. It is a one-second test and it catches the topic's main error.

Where this reappears

The same rule, unchanged, handles algebraic fractions at GCSE. Dividing (x+1)/3 by (x+1)/6 is keep-change-flip followed by cancelling, and a student secure in the numerical version meets it as familiar rather than new.

Rearranging formulae also leans on it constantly, since dividing by a fractional coefficient is how a great many equations get solved. The topic is worth more than its few lessons in the scheme of work suggest.

How we teach dividing fractions

We ask 'how many halves fit into three?' before we teach any rule. Students answer six immediately, from intuition, and then we show that 3 × 2 gives the same thing — so the reciprocal arrives as an explanation of something they already worked out, rather than as an instruction.

We also require the size check on every answer. Dividing by less than one should grow the number, and a student who checks that will never hand in an answer with the wrong fraction inverted.