Ratios and proportions
A ratio compares two quantities; a proportion states that two ratios are equal. Almost every scaling problem — recipes, maps, model kits, screen resolutions, mixing paint, currency conversion — is a proportion with one unknown term, solved by cross-multiplication.
Worked example
A 16:9 screen is 1920 pixels wide. How tall? Set up 16 : 9 = 1920 : d, cross-multiply to get 16d = 9 × 1920 = 17,280, so d = 1080.
Part-to-part vs. part-to-whole
A source of real confusion. If a class has a 3:2 ratio of girls to boys, that is part to part — there are 5 parts in total, so girls are 3/5 (60%) of the class, not 3/4. Ratios are not fractions of the whole unless you add the parts up first.
To split a total in a given ratio, add the parts, divide the total by that sum to get the value of one part, then multiply. Splitting $450 in a 4:5 ratio: 9 parts, $50 each, giving $200 and $250. Enter a total above and the calculator does it.
Using ratios in everyday problems
Almost any scaling question is a ratio with one term missing. Once you see the shape, the arithmetic is always the same.
- Recipes. Rice at 2:1 water to grain means 1.5 cups of rice needs 3 cups of water — regardless of how many people you are feeding.
- Maps and models. A 1:25,000 map means 1 cm on paper is 25,000 cm, or 250 m, on the ground.
- Mixing. Two-stroke fuel at 50:1 means 100 ml of oil in 5 litres of petrol. Paint, concrete and fertiliser all work the same way.
- Currency and unit rates. Any “price per” comparison is a ratio, which is why unit pricing on a supermarket shelf is the only honest way to compare pack sizes.
- Gears and pulleys. A 3:1 gear ratio trades three turns of input for one of output, gaining torque.
Scaling a recipe up
A dough uses flour and water in a 5:3 ratio and you have 750 g of flour. Set up 5 : 3 = 750 : d, cross-multiply to get 5d = 2,250, so d = 450 g of water. The same three lines solve every problem in this section.
Common ratios worth recognising
- 16:9 — the standard widescreen aspect ratio (1920×1080, 3840×2160).
- 4:3 — older monitors and most tablet cameras.
- 3:2 — 35 mm photography and many laptop screens.
- 1.618:1 — the golden ratio, φ, which shows up in design more as folklore than as science but is genuinely elegant maths.
- 1:√2 — the A-series paper ratio, which is why folding A4 in half gives A5 with identical proportions.