Ratios and Proportions
A ratio compares two quantities of the same kind; a proportion states that two ratios are equal. The distinction matters because it determines which operation you need — simplification for a single ratio, cross-multiplication for a pair.
Formulas
Proportion: a : b = c : x → x = bc ÷ a
Share of total: part ÷ (a + b) × total
As a percentage: a ÷ (a + b) × 100
Multiplying or dividing both terms of a ratio by the same non-zero number leaves the relationship unchanged, which is why 3:4, 6:8, and 75:100 all describe the same split. Adding to both terms does not preserve it — a common error when someone tries to adjust a mixture by adding a fixed amount of each ingredient.
Part-to-Part and Part-to-Whole
Most confusion with ratios comes from mixing up these two readings of the same numbers.
Part to part — 3 : 2
Three of one thing for every two of another. There are five shares in total, so the first quantity is three fifths of the whole, not three quarters.
Part to whole — 3 in 5
A fraction of the total, which converts directly to 60%. Concentrations, dilutions, and odds are usually quoted one way or the other, and getting the reading wrong changes the answer substantially.
Dilution ratios — 1 : 10
In cleaning and chemistry this normally means one part concentrate to ten parts water, giving eleven parts total, not one part in ten. Check which convention a product uses before mixing.
Scaling Recipes, Plans, and Mixtures
Scaling a recipe or a drawing is proportion work: set the known ratio equal to the target and solve for the missing quantity. If a recipe for 4 serves 500g of flour and you need 7 servings, then 4:500 = 7:x gives x = 875g.
Two cautions apply. Not everything scales linearly — baking times, pan depths, and seasoning generally do not — and rounding each ingredient separately can drift the mixture off its intended ratio. Where the proportion is critical, such as mortar, resin, or fuel mix, scale from the total and derive each part rather than rounding as you go.