Mixtures & Alligation

Mixtures & Alligation

Alligation is a shortcut for any weighted average — mixing two things of different "strength" (price, concentration, speed) to hit a target. It also answers ratio-of-quantities questions in one line.

The alligation rule

When two ingredients at values $c$ (cheaper) and $d$ (dearer) are blended to a mean $m$, the ratio of their quantities is

$\frac{\text{quantity of cheaper}}{\text{quantity of dearer}}=\frac{d-m}{m-c}.$

Read it off the cross: each quantity is proportional to the distance to the mean on the opposite side.

A worked example

In what ratio must a 20% salt solution be mixed with a 50% solution to obtain a 30% solution?

$\frac{\text{20\% part}}{\text{50\% part}}=\frac{50-30}{30-20}=\frac{20}{10}=\mathbf{2:1}.$

So two litres of the weak solution for every one litre of the strong.

Repeated replacement

If a vessel of volume $V$ has $x$ removed and replaced with water, $n$ times, the fraction of original liquid remaining is

$\left(1-\frac{x}{V}\right)^{n}.$

Common traps

• Putting the distances on the wrong side. The cheaper quantity pairs with $(d-m)$, the far gap — opposite sides, always.
• Mixing % with absolute amounts. Alligation gives a ratio; multiply back by the total to get litres/kg.
• Replacement vs. simple removal. The power formula assumes you refill to the same volume each time.

Checklist

• Identify the two "strengths" and the target mean
• Ratio $=\dfrac{d-m}{m-c}$ (opposite-side distances)
• Scale the ratio up to the actual quantity asked
• For repeated replacement, use $\left(1-\frac{x}{V}\right)^n$