Proportion

Definition

Four quantities a, b, c, d are in proportion when a : b = c : d, written a : b :: c : d. Then a and d are the extremes, b and c are the means, and the fundamental property is:

• product of extremes = product of means ⟹ a × d = b × c.
• From a : b = c : d we may also write b : a = d : c, a : c = b : d and (a + b) : b = (c + d) : d.

Standard terms

• Fourth proportional to a, b, c: the number x with a : b = c : x, so x = b c / a. For 2, 5, 8 → x = 5 × 8 / 2 = 20.
• Third proportional to a, b: the number x with a : b = b : x, so x = b²/a. For 4, 6 → x = 36/4 = 9.
• Mean proportional between a and b: the number x with a : x = x : b, so x = √(a b). Between 4 and 9 → x = 6.

Continued proportion

Three quantities are in continued proportion when a : b = b : c, i.e. b² = a c; then b is the mean proportional and a : c = a² : b².

• If a : b = 2 : 3 and b : c = 3 : 7, then a : b : c = 2 : 3 : 7 and a : c = 2 : 7.