Mixtures and the weighted mean

Equal glasses. Mix one glass of lemonade with 10% jaggery and an equal glass with 20%: the mixture is 15%, exactly halfway, because each glass counts equally.

Unequal amounts. Mix 500 mL at 10% with 200 mL at 20%:

• jaggery: 50 + 40 = 90 mL;
• total: 700 mL;
• concentration: 90/700 ≈ 13%.

It is nearer 10%, because most of the mixture is the 10% lemonade. Each part pulls the answer towards itself, by its size.

The weighted mean. Mixing amounts w₁, w₂, …, wₙ with concentrations x₁, x₂, …, xₙ gives

(w₁x₁ + w₂x₂ + … + wₙxₙ) ÷ (w₁ + w₂ + … + wₙ),

written short as Σwx ÷ Σw. This is the weighted mean (weighted average).

'Weighted' means each value counts according to how big or important it is. The weights can be real weights (kg), volumes, days, or importance.

A built-in check: the answer always lies between the smallest and largest values, and nearer the ones with bigger weights.

An Indian formula. Brahmagupta wrote it in the Brāhmasphuṭasiddhānta (c. 628 CE) to find the mean depth of an uneven pit; Śrīdharācārya (c. 750 CE) used it for the purity of gold alloys.