Solves the proportion a : b = c : x by cross multiplication.
By cross-multiplying: from a/b = c/x it follows that a × x = b × c, so x = b × c / a. For 3/4 = 9/x that gives x = 4 × 9 / 3 = 12. What matters is keeping the same order on both sides — if kilometres are in the numerator on one side, they must be in the numerator on the other.
In direct proportion both quantities move the same way: twice the fabric costs twice as much. In inverse proportion they move oppositely: twice the workers finish in half the time. In the inverse case it is the product that stays constant rather than the ratio, so an ordinary proportion does not apply.
A scale of 1:25,000 means one centimetre on the map is 25,000 centimetres on the ground — 250 metres. Multiply the measured length by the second number and convert the units. Technical drawings often invert this, as in 5:1, where the part is drawn five times larger than life.
Getting the order of terms wrong, so that one fraction is inverted relative to the other. If price sits over quantity on one side, it must sit over quantity on the other too. The second common error is using a proportion where the relationship is not linear — area and volume, for instance, which scale as the square and the cube.