Simplifying Ratios

A recipe for a crowd might say 6 cups of flour to 4 of sugar, but you're only baking for a few friends. Simplifying a ratio lets you describe the very same mix with the smallest, tidiest numbers — the easiest form to scale up or down.

A ratio compares two amounts, and the same comparison can be written many ways. If you divide both parts by the same number, the comparison doesn't change — you just describe it with smaller, tidier numbers. That gives an equivalent ratio:

6 : 4 \;=\; 3 : 2 \qquad (\text{both divided by } 2)

This is exactly like simplifying a fraction: just as \tfrac{6}{4} = \tfrac{3}{2}, the ratio 6 : 4 is the very same mix as 3 : 2 — six reds to four blues is the same recipe as three reds to two blues, only with bigger piles.

To reach the simplest form, keep dividing both parts by a common factor until they share no common factor left. The quickest route is to divide once by the highest common factor (HCF) — the biggest number that goes into both. The HCF of 6 and 4 is 2, so 6 : 4 drops straight to 3 : 2, which can't be reduced any further.

Three worked examples

The recipe is always the same: spot a number that divides both parts, then divide both.

10 : 15 — both share the factor 5, so 10 : 15 = 2 : 3. The 2 and 3 share nothing, so that's simplest.
12 : 8 — you can go in steps. Halve once: 12 : 8 = 6 : 4. Halve again: 6 : 4 = 3 : 2. Keep going until nothing divides both. (Dividing once by the HCF, 4, gets there in one jump.)
4 : 12 — divide both by the first part to get the handy "1 : n" form: 4 : 12 = 1 : 3, which tells you the second amount is three times the first.

The traps when simplifying:

Divide both parts by the same number. Halving 6 : 4 gives 3 : 2 — not 3 : 4 (you must halve the 4 too).
Divide, don't subtract. 6 : 4 is not 2 : 0; taking away 4 from each changes the mix.
Keep going until no common factor is left. 6 : 4 = 3 : 2 is finished, but 12 : 8 = 6 : 4 is not — it still halves.
Same units first. 1\text{ m} : 50\text{ cm} is really 100 : 50 = 2 : 1, not 1 : 50.

A rescue centre has 4 cats and 6 dogs.

cat cat cat cat : dog dog dog dog dog dog

Bundle them into 2 equal groups and each group is 2 cats and 3 dogs. So 4 : 6 = 2 : 3 — two cats for every three dogs.

A jug is made with 6 oranges and 4 lemons.

orange orange orange orange orange orange : lemon lemon lemon lemon

Both numbers divide by 2, so the recipe is 6 : 4 = 3 : 2. A smaller jug with 3 oranges and 2 lemons tastes exactly the same — same mix, fewer fruits.

See it shrink

Both bars are cut by the same factor, so they keep the same proportion — only the numbers get smaller.

Group the counters

Here is a pile of counters in some ratio — reds to blues. Bundle them into equal groups and each group shows the simplest form: the same mix, written with the smallest numbers. Press Refresh for a new pile.

Three worked examples

See it shrink

Group the counters

See it explained