Ratios explained: how to simplify, scale, and divide an amount in a given ratio

Mix concrete at 1:2:3, dilute squash 1:4, share the prize 3:2, scale the map at 1:25,000 — ratios are everywhere and the arithmetic is one idea applied four ways. The ratio calculator handles all of them; this guide explains the idea so the answers make sense.

What a ratio says

A ratio compares quantities: 2:3 means "two parts of this for every three of that". It is not a fraction of the whole — in a 2:3 mix, the first ingredient is 2/5 of the total, not 2/3 — which is the mistake behind most ratio errors. A ratio can compare parts to parts (2:3 orange to water) or a part to a whole (1:5 orange to drink), and the wording decides which; read carefully. Three or more terms work the same way (1:2:3 has six parts).

Simplifying and equivalent ratios

Ratios scale like fractions: multiply or divide every term by the same number and the ratio is unchanged. 6:9 simplifies to 2:3 (divide by 3, their greatest common divisor — the GCD tool); 2:3 is equivalent to 200:300. Units must match before simplifying: 20 cm to 1 m is 20:100 = 1:5, not 20:1. The unitary form — n:1 — is the handiest for comparison: a 5:3 ratio is 1.67:1, so you can compare it with 1.5:1 at a glance.

Scaling: recipes, maps, mixes

Given one term, find the others by the scale factor. A recipe at 3 eggs : 250 g flour, with 5 eggs available: factor 5/3, so 417 g flour (the recipe scaler applies it to a whole list). A map at 1:25,000: 4 cm on the map is 100,000 cm = 1 km on the ground. A 1:4 dilution to fill a 1-litre jug: 5 parts total, 200 ml concentrate and 800 ml water. The step people skip is the total number of parts — count them before dividing.

Dividing an amount in a ratio

To split £600 in the ratio 3:2: add the parts (5), find one part (£120), multiply out (£360 and £240). Three ways: 1:2:3 of £600 is six parts of £100 — £100, £200, £300. Check by adding the shares back to the total. This is exactly how the rent splitter divides rent by room size or income and how partners share profits by stake; How to split group expenses fairly, and settle up in the fewest payments covers the messier case of shared costs.

Special cases: aspect ratios and odds

An aspect ratio (16:9) is a ratio of width to height, simplified and kept in that order — 1920:1080 reduces to 16:9, and finding a missing dimension is a scaling problem (Aspect ratios explained: what 16:9, 4:3, 3:2 and 9:16 mean, and how to crop between them, the aspect ratio calculator). Odds are ratios of for-to-against: odds of 3:1 against means 1 success per 3 failures, a probability of 1/4 — not 1/3 — the classic mix-up between "3:1" and "one in three" that Is a coin flip really 50/50? unpicks. Gear ratios, exchange rates and speeds are ratios with units attached (teeth:teeth, currency:currency, km:h), which is why a rate is a ratio whose second term is usually 1.