Alligation & Mixtures

Alligation answers one question: in what ratio must two things be mixed to reach a given average?

quantity of cheaper : quantity of dearer = (dearer − mean) : (mean − cheaper)

Notice the crossing: the cheaper quantity is paired with the difference belonging to the dearer side. That crossing is the whole rule, and getting it backwards is the single most common mistake in the topic. A one-second check: the mixture is always closer in price to whichever ingredient you used more of.

Three types follow from it:

1. Two priced goods. Apply the rule directly.
2. Adding water. Water is free, so its "price" is 0. Milk : water = (target − 0) : (cost − target).
3. Changing a milk–water ratio by adding water. The milk never changes — compute the milk, work out how much water the new ratio demands, and subtract the water already present.

A fourth type is repeated replacement: if x litres are drawn from a full vessel of V litres and replaced with water, n times over, the milk left is V(1 − x/V)^n. It is a power, not a subtraction — the later draws also remove water that was added earlier.

cheaper : dearer = (dearer − mean) : (mean − cheaper) · water is priced 0 · repeated replacement: milk left = V(1 − x/V)^n

In what ratio must rice at ₹30/kg be mixed with rice at ₹45/kg so that the mixture is worth ₹35/kg? → cheaper : dearer = (45 − 35) : (35 − 30) = 10 : 5 = 2 : 1. Sanity check: 2 kg at ₹30 costs ₹60 and 1 kg at ₹45 costs ₹45, so 3 kg costs ₹105, i.e. ₹35 per kg. Note also that ₹35 is nearer ₹30 than ₹45, and we did use more of the cheaper rice.