About Ratios
Ratios are recipes' native tongue (2:1 rice to water), maps' legend (1:50,000), engines' lifeblood (50:1 fuel mix), and the classroom bridge between fractions and proportions. Two skills cover nearly every use: reducing a ratio to its simplest honest form, and finding the missing number when two ratios must match.
Simplify mode reduces A:B and re-expresses it as 1:n and as a fraction; solve mode cross-multiplies A:B = C:x for x, with the check shown. Decimals work in solve mode; simplify keeps integers honest via the GCD.
Ratios written vertically are just fractions — arithmetic on those lives in the Fraction Calculator.
Simplifying & Cross-Multiplying
The two moves, formally:
Simplify: A : B = (A ÷ GCD) : (B ÷ GCD) Solve: A : B = C : x → x = B × C ÷ A Check: A·x should equal B·C
Worked examples: 24:36 shares GCD 12 → 2:3 (equivalently 1:1.5). Solving 3:4 = 9:x → x = 4×9÷3 = 12, and the check 3×12 = 36 = 4×9 confirms it. The scale factor (9 is 3× the 3) is often the fastest mental route.
Worked Examples
Both modes on everyday numbers — computed by this calculator:
| Problem | Answer | The move |
|---|---|---|
| Simplify 24 : 36 | 2 : 3 | ÷ GCD 12 |
| Simplify 15 : 25 | 3 : 5 | ÷ GCD 5 |
| Simplify 16 : 9 | 16 : 9 (already lowest) | GCD 1 |
| 3 : 4 = 9 : x | x = 12 | cross-multiply |
| 2 : 5 = x-solve at C=7 | x = 17.5 | 4 × 9 ÷ 3 pattern |
| 1 : 50,000 map, 3 cm | 1.5 km real | scale ratios ARE ratios |
The map row is the payoff pattern: a 1:50,000 scale means every map centimeter is 50,000 real ones — proportion solving in disguise.
Ratios in the Wild
Kitchens run on them (rice 1:2, vinaigrette 3:1, coffee brewing's famous 1:16), workshops mix by them (epoxy 1:1, two-stroke fuel 50:1, mortar 1:3), and screens name themselves after them — 16:9 is a ratio wearing a product name. Scaling a recipe or a mix is always the same operation: multiply BOTH sides.
The classic misread: a 1:4 ratio means 1 part to 4 parts — FIVE parts total, so 'dilute 1:4' fills 20% concentrate, not 25%. Cleaning products and lab protocols trip people exactly here. When instructions matter, translate the ratio to fractions-of-total first (1:4 → 1/5 and 4/5), then measure.