Proportions & Cross Multiplication

A proportion says two ratios are equal: 3/4 = x/20.

To solve, cross multiply: 3 × 20 = 4 × x, so 60 = 4x, so x = 15.

This one setup handles a huge share of real problems: scaling recipes, converting currencies, map distances, unit conversions. The key is being consistent about what goes on top. If kilometres are on top on one side, kilometres go on top on the other.

Cross multiplication is not a separate rule, it is multiplying both sides of a/b = c/d by bd, which clears both denominators and leaves ad = cb. It is valid whenever b and d are nonzero.

Setting up correctly matters more than solving. A proportion is only valid if the two ratios compare corresponding quantities in the same order, so the units in each numerator must match and the units in each denominator must match. Writing the units into the setup catches nearly every error before any arithmetic happens.

Proportions also encode direct variation: y/x = k is the same statement as y = kx, a line through the origin with slope k. When a relationship is proportional, its graph must pass through (0,0) — which is a fast test for whether a proportion is the right model at all. Situations with a fixed starting fee are not proportional, even though they are linear.