Arithmetic Operations
Fraction Division – The Keep-Change-Flip Method
Dividing by a fraction might seem mysterious at first, but it follows one elegant rule that makes every division of fractions straightforward.
“Keep, Change, Flip” is a mnemonic for one of the oldest tricks in arithmetic: dividing by a fraction is identical to multiplying by its reciprocal, because a fraction and its reciprocal always multiply to exactly 1. This idea appears in Indian mathematical texts as early as the Bakhshali manuscript (dated to the early centuries CE) and was standard practice among medieval Arab and European arithmeticians long before symbolic algebra existed. It matters well beyond the classroom: splitting a length of material into equal pieces, working out how many fractional servings fit in a container, or scaling recipes down to a fractional batch size are all “divide by a fraction” problems in disguise.
The Rule – Keep, Change, Flip (KCF)
Keep the first fraction unchanged. Change the ÷ to ×. Flip (take the reciprocal of) the second fraction.
Why Does This Work?
Dividing by a number is the same as multiplying by its reciprocal. The reciprocal of c/d is d/c.
Worked Examples
Keep 3/4. Change ÷ to ×. Flip 1/2 to 2/1.
3/4 × 2/1 = 6/4 = 1½
5/6 × 3/2 = 15/12 = 1¼
3/5 ÷ 4/1 → 3/5 × 1/4 = 3/20
Key Takeaways
- Keep the first fraction, change to multiplication, flip the second fraction.
- Dividing by a fraction always gives a result larger than the original (when both are positive proper fractions).
- Always simplify the result.