Mixtures & Alligation

Study Materials › Arithmetic › Mixtures & Alligation

Mixture questions are ratio questions with a story attached. The reliable approach is to stop thinking in percentages and start tracking absolute quantities of each component — how many litres of acid, how many of water — because those are what actually change when something is added.

The Alligation Rule

Mixing quantities with concentrations (or prices) A₁ and A₂ to give a mean Aₘ:

n₁ : n₂ = (A₂ − Aₘ) : (Aₘ − A₁)

The quantities come out in the inverse ratio of their distances from the mean.

Alligation works identically for weights, prices, speeds and interest rates. A fully worked example using group averages appears on the Averages page.

Repeated Replacement

If a vessel holds V units of liquid and x units are removed and replaced with water, repeated n times:

Final pure liquid = V × (1 − x/V)ⁿ

Example: from 100 litres of milk, 20 litres removed and replaced with water twice leaves 100 × (0.8)² = 64 litres of milk.

Adding one component changes the total too. Pouring 1 litre of water into a mixture raises both the water quantity and the overall volume. Forgetting the second change is the most common error in this topic, and it is exactly what the questions below test.

Worked Examples from Real Papers

TS Police SI 2022 — Q34

In a mixture of acid and water, adding 1 litre of water makes the mixture 20% acid. Adding 1 litre of acid to that new mixture makes it 33⅓% acid. What percentage of acid was in the original mixture?

Let the original mixture contain a litres of acid in a total of t litres.
After adding 1 litre of water the acid is unchanged but the total rises: a/(t + 1) = 1/5, so 5a = t + 1.
After adding 1 litre of acid, both the acid and the total rise: (a + 1)/(t + 2) = 1/3, so 3a + 3 = t + 2.
Substituting the first into the second: 3a + 3 = 5a − 1 + 2, giving a = 1 and t = 4.
Original acid percentage = 1/4 = 25%.
Note how the two additions differ: water changes only the denominator, acid changes both numerator and denominator. Writing that distinction down is the whole solution.

TS Police SI 2022 — Q38

Three vessels A, B and C contain syrup and water in the ratios 2:3, 3:5 and 5:7. Taking 15 litres from A, 16 litres from B and x litres from C gives a mixture with syrup to water in the ratio 2:3. Find x.

Convert each contribution into absolute quantities.
From A (15 L, ratio 2:3): syrup = 6, water = 9.
From B (16 L, ratio 3:5): syrup = 6, water = 10.
From C (x L, ratio 5:7): syrup = 5x/12, water = 7x/12.
Setting the final ratio: 3(12 + 5x/12) = 2(19 + 7x/12).
36 + 15x/12 = 38 + 14x/12 → x/12 = 2 → x = 24 litres.
Method: the quantities 15 and 16 are chosen to divide cleanly by 5 and 8. When a mixture question gives odd-looking volumes, check that they match the denominators of the given ratios — they usually do, and that confirms you have read the ratios the right way round.

Check which way the ratio runs. "Syrup and water in the ratio 2:3" means syrup is 2/5 of the total, not 2/3. Reading it as 2/3 is a standing trap, and the resulting answer is usually one of the options.

Also see: Averages · Ratio & Proportion · Percentages