Averages & Mixtures • Topic 3 of 4

Mixtures & Alligation

A mixture problem blends two ingredients of known "strength" (price, concentration, purity) and asks about the result — or, run in reverse, asks in what ratio to blend them to hit a target. Because the blend’s strength is the weighted average of the parts, alligation gives the ratio in one line: the amounts are in the ratio of the cross differences, n₁ : n₂ = (a₂ − M) : (M − a₁), where M is the target and a₁ < M < a₂. To make a 40%-milk blend from 20% and 50% milks, the ratio is (50 − 40) : (40 − 20) = 10 : 20 = 1 : 2, so use twice as much of the weaker one. Alligation is the GRE-fast route for "how many litres of water to add", "what ratio of two coffees", and profit questions dressed as tea blends. A close cousin is repeated replacement — draw off some mixture and top up with pure solvent — where the remaining original follows original × (1 − r/V)ⁿ after n rounds.

✅ Solved examples

1. In what ratio should 20% and 50% salt solutions be mixed to get 30%?
Alligation: (50 − 30) : (30 − 20) = 20 : 10 = 2 : 1 (weaker : stronger).
2. How much water must be added to 10 litres of 40% acid to dilute it to 25%?
Acid stays fixed at 4 L. Need 4 = 25% of total ⇒ total = 16 L, so add 16 − 10 = 6 litres.
3. Coffee at $8/kg is mixed with coffee at $14/kg to sell at $10/kg. Find the mixing ratio.
(14 − 10) : (10 − 8) = 4 : 2 = 2 : 1 (cheaper : dearer).
4. From 20 litres of pure juice, 4 litres are removed and replaced with water; this is done twice. How much pure juice remains?
Remaining = 20 × (1 − 4/20)² = 20 × (0.8)² = 20 × 0.64 = 12.8 litres.

✏️ Practice — try these, take hints as needed

1. In what ratio should 15% and 40% solutions be mixed to get 25%?
Alligation cross differences.
(40 − 25) : (25 − 15).
15 : 10.
3 : 2
2. How much water must be added to 12 litres of 30% acid to make it 20%?
Acid amount stays 3.6 L.
3.6 = 20% of new total ⇒ total = 18.
Added = 18 − 12.
6 litres
3. Rice at $2/kg and rice at $3.50/kg are mixed to sell at $3/kg. Ratio of cheaper to dearer?
(3.50 − 3) : (3 − 2).
0.5 : 1.
Scale up.
1 : 2
4. A 25% alcohol solution is mixed with a 55% one in the ratio 2 : 3. Resulting strength?
Weighted average.
(2·25 + 3·55)/5.
(50 + 165)/5.
43%
5. From 50 litres of pure milk, 5 litres are removed and replaced with water, once. How much pure milk remains?
Fraction kept = 1 − 5/50.
= 0.9.
50 × 0.9.
45 litres

📝 Topic test — 8 questions

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