Fractions
Dividing Fractions – Keep, Change, Flip
Dividing fractions uses the Keep, Change, Flip method (KCF). It turns division into multiplication.
“Keep, Change, Flip” is a memory device for a much older mathematical fact: dividing by a number is the same as multiplying by its reciprocal. This equivalence was already understood by medieval Islamic and Indian mathematicians working with ratios, and it is the same principle that lets you turn “divide by 2” into “multiply by 0.5” with whole numbers. It shows up constantly in real measuring problems: cutting a length of ribbon into equal pieces, working out how many servings fit in a container, or converting between units, are all division-by-a-fraction problems in disguise.
Dividing Fractions as Measuring
Dividing by a fraction is a measuring question: “how many of these fit into that?” 3/4 ÷ 1/4 asks: how many 1/4-sized pieces fit inside a length of 3/4? You can answer this just by marking a length of 3/4 and counting off 1/4-long segments along it, no Keep-Change-Flip required.
Picture a bar cut into quarters. Shade 3 of the 4 quarters to mark out a length of 3/4, then measure that shaded length using a ruler that's exactly 1/4 long — count how many times it fits.
The shaded length of 3/4 is measured out and fits exactly 3 blocks of size 1/4, with nothing left over. So 3/4 ÷ 1/4 = 3 — the same answer the Keep-Change-Flip method gives (3/4 × 4/1 = 12/4 = 3), just seen directly as a measurement instead of calculated.
The KCF Rule
a/b divided by c/d = a/b times d/c. Keep the first fraction. Change division to multiplication. Flip the second fraction.
Why It Works
Dividing by c/d is the same as multiplying by d/c because (c/d) times (d/c) = 1. You multiply by the reciprocal.
Examples
Keep 3/4. Change to multiply. Flip 2/5 to 5/2. 3/4 times 5/2 = 15/8 = 1 and 7/8.
5/6 times 3/10 = 15/60 = 1/4.
Dividing by a Whole Number
Write 6 as 6/1. 3/5 times 1/6 = 3/30 = 1/10.
Dividing Mixed Numbers
5/2 divided by 4/3 = 5/2 times 3/4 = 15/8 = 1 and 7/8.
15/4 divided by 5/2 = 15/4 times 2/5 = 30/20 = 3/2 = 1 and 1/2.
Real-Life Example
Key Takeaways
- Keep the first fraction, change to multiply, flip the second.
- Always convert mixed numbers to improper fractions before dividing.
- The reciprocal of a/b is b/a.
- Never divide with a common denominator - that is for addition only.