Alligation and mixtures

The essentials

When quantities of different prices (or concentrations) are mixed, the mean price of the mixture is the weighted average: mean price = (total cost of all ingredients) ÷ (total quantity). For example, 3 kg at ₹50 and 2 kg at ₹75 cost ₹150 + ₹150 = ₹300 for 5 kg, so the mean price is ₹60 per kg.

The rule of alligation works the other way: given the two prices and the desired mean price, it gives the ratio in which they must be mixed. quantity of cheaper : quantity of dearer = (d − m) : (m − c), where c is the cheaper price, d the dearer price and m the mean price (c < m < d). It is often drawn as a cross diagram with c and d on top, m in the middle, and the differences written diagonally below.

Key points

• The mean price always lies between the cheapest and the dearest price.
• Alligation also works for percentages, concentrations and speeds over equal times.
• Repeated replacement: if a vessel holds x litres of liquid and y litres are taken out and replaced by water n times, the liquid left = x(1 − y/x)ⁿ litres.
• Water counts as an ingredient of price ₹0 when milk is diluted.

Exam tips

• Make sure all prices are per the same unit (per kg or per litre) before applying the rule.
• The alligation ratio is a ratio of quantities, not of costs.

Common mistakes

• Writing (m − c) : (d − m) for cheaper : dearer, which reverses the ratio.
• Using the simple average of prices when the quantities are unequal.