Alligation and Mixture

A mixture is formed by combining two or more ingredients together in a certain ratio. Alligation is the rule that works out exactly what ratio is needed — given the prices of two ingredients and the price of the desired blend, it tells you how much of each to combine.

When ingredients of different cost are mixed, the resulting cost per unit quantity of the blend is called the mean price. Of the two ingredients being combined, the one with the higher cost is the dearer ingredient, and the one with the lower cost is the cheaper ingredient.

The rule of alligation states that when ingredients are mixed to produce a mixture of a given mean price, the ratio in which they are mixed is inversely proportional to the difference between each ingredient's cost and the mean price. If $m$ is the mean price of the mixture, $c$ the cost price of the cheaper ingredient, and $d$ the cost price of the dearer ingredient, then: