Ratio & proportion tools
How to Simplify Ratios
Divide every part by their greatest common divisor. Simplest form is reached when the only divisor left is 1.
12:18 simplifies to 2:3. Both parts share a divisor of 6, and dividing both by it leaves two numbers with nothing left in common. The bar does not change shape — only the count of units used to draw it.
Why dividing is allowed and subtracting is not
A two-part ratio a:b is a statement about the quotient a/b. Divide both parts by the same non-zero k and the quotient becomes (a ÷ k)/(b ÷ k) — the k cancels between numerator and denominator, so the value is unchanged. The comparison survives; only the size of the numbers moves.
Subtraction has no such cancellation. Take 2 off both parts of 6:4 and you get 4:2, but 6/4 = 1.5 while 4/2 = 2. The numbers got smaller and the relationship changed, which is exactly what simplifying must never do.
Finding the greatest common divisor
Any common factor reduces a ratio; the greatest one finishes the job in a single step. Repeated halving works too, provided you keep going: 84:126 → 42:63 → 14:21 → 2:3. Stopping at 42:63 leaves a smaller ratio that is still not simplest, which is why the finishing condition is stated as a property of the result rather than a number of steps.
84:126 — factorised
84 = 2² × 3 × 7.
126 = 2 × 3² × 7.
Shared factors, each at the lower power: 2 × 3 × 7 = 42.
84 ÷ 42 = 2 and 126 ÷ 42 = 3, so 84:126 = 2:3.
Confirm the value did not move: 84/126 = 0.666…, and 2/3 = 0.666…
Confirm the reduction: 2 × 42 = 84 and 3 × 42 = 126, the same multiplier both times.
Decimals: clear them first
A decimal ratio is turned into an exact integer ratio before anything is reduced. Multiply every part by the power of ten that clears the longest decimal tail — multiplying all parts by one factor is the safe operation from above — then reduce.
1.5:2
- The longest tail is one decimal place, so multiply both parts by 10: 15:20.
- The greatest common divisor of 15 and 20 is 5.
- 15 ÷ 5 = 3 and 20 ÷ 5 = 4, so 1.5:2 = 3:4.
- Check: 1.5 ÷ 2 = 0.75 and 3 ÷ 4 = 0.75.
Doing this with exact integers rather than floating-point division matters for awkward inputs. 0.1:0.3 becomes 1:3 exactly; taking 0.1 ÷ 0.3 first and rounding gives 0.333, which no longer reduces cleanly.
Fractions: onto a common denominator
Put the parts over a common denominator and keep the numerators. 1/2 : 1/3 becomes 3/6 : 2/6, which is 3:2. This is the same move as clearing decimals — both parts have been multiplied by 6 — so the ratio is preserved and the result needs no further reduction here, since 3 and 2 share no divisor.
Three or more parts
The rule is unchanged, but it applies to all parts at once: take the greatest common divisor of the whole set. 4:6:10 shares a divisor of 2 and reduces to 2:3:5. Reducing pairs separately breaks the ratio, because the parts would no longer be divided by the same number.
Mixed inputs, and ratios that are already simplest
| Input | First move | Simplest form |
|---|---|---|
| 18:24 | Divide by GCD 6 | 3:4 |
| 1.5:2 | ×10, then divide by 5 | 3:4 |
| 1/2 : 1/3 | Common denominator 6 | 3:2 |
| 4:6:10 | Divide all three by 2 | 2:3:5 |
| 7:9 | Nothing shared above 1 | 7:9 — already simplest |
| 500 g : 1 kg | Convert to 500:1000 first | 1:2 |
Different-looking inputs, one procedure: make the parts exact integers in a common unit, then divide by the greatest common divisor.
7:9 is worth noticing. Neither part is prime-free of factors — 9 = 3², after all — but they share none, so the ratio cannot be reduced. “Simplest” is a property of the pair, not of either number.
Verifying the result
Two checks catch nearly everything. Multiply the simplified parts back by the divisor you removed and both original numbers should return, using the same multiplier. Then compare quotients: the original and the simplified ratio must give the same value. If the first check works but the second does not, you divided by different numbers somewhere.