The alligation shortcut, milk-water replacement problems, and weighted average price — three ideas that cover nearly every mixtures question on this exam.
Alligation is a shortcut technique for finding the ratio in which two ingredients of different values must be mixed to get a desired average value — without setting up algebraic equations.
This single formula replaces what would otherwise require solving a system of equations for every mixture problem, and is fast enough to apply mentally once practised.
Picture the cheaper price and dearer price on either side of the mean. The cheaper side's ratio number comes from subtracting across to the dearer price, and vice versa — a diagonal cross-subtraction.
A common variant asks what happens when part of a mixture is repeatedly removed and replaced with a pure ingredient — usually water replacing milk.
Each replacement removes the same fraction of whatever milk remains at that point, not a fixed amount of the original milk — this is why the formula uses a power, not simple subtraction.
After removing 8L from 40L once, 4/5 of whatever milk was present remains. The second removal takes 4/5 of THAT remaining amount, not another flat 8L worth of the original milk.
When two ingredients are mixed in a known ratio, finding the average price of the mixture is the reverse of the alligation problem — a direct weighted average instead of solving for a ratio.
When equal quantities of two ingredients are mixed, this reduces to the simple average of the two prices — but the moment quantities differ, the weighted formula must be used instead.
The simple average of two prices only works when equal quantities are mixed. The moment the ratio is anything other than 1:1, the weighted formula is required instead.
All 6 questions with a full answer key.
Full solutions are in the downloadable PDF above.
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