a proportion
A proportion is a statement that two ratios are equal. If 2 apples cost 6 dollars and you want to know the cost of 5 apples at the same price, you are saying 2 : 6 is the same comparison as 5 : x, and solving for x. The everyday idea is 'keeping things in step' — doubling the apples doubles the cost, so the ratio price-per-apple never changes.
We write a proportion as a/b = c/d, or a : b = c : d, read 'a is to b as c is to d'. The power of a proportion is that you can solve for any one missing value when the other three are known: cross-multiply to get a·d = b·c, then isolate the unknown. For 2/6 = 5/x, cross-multiplying gives 2x = 30, so x = 15 dollars. The two products a·d and b·c are equal exactly when the four numbers form a true proportion — this is the test for whether two ratios agree.
Proportions are the working tool behind similar figures, scale drawings, maps, and unit conversion. Whenever a problem has the shape 'this is to that as this other thing is to what?', it is a proportion in disguise. Be careful that the two ratios are built in the same order (apples-to-dollars on both sides, not apples-to-dollars on one side and dollars-to-apples on the other), or the cross-multiplication will give the wrong answer.
A map uses 1 cm to represent 5 km. Two towns are 7 cm apart on the map. Set up 1/5 = 7/x, cross-multiply to get x = 35, so the real distance is 35 km.
Solving a proportion by cross-multiplication recovers the unknown fourth value.
Cross-multiplication is a shortcut, not a new rule — it just multiplies both sides of a/b = c/d by bd. It is only valid when b and d are nonzero, and only when both ratios are written in the same order.