Mixtures & Alligations • Topic 1 of 4

Mixing Two Substances

When you blend two quantities of different per-unit values, the value of the blend is their weighted average: M = (q₁c₁ + q₂c₂)/(q₁ + q₂), where the c’s are the strengths (price, purity, percentage) and the q’s are the amounts. The key CAT insight is that "strength" can be anything per unit — ₹/kg of two teas, the alcohol % of two solutions, the milk % of two cans, even the marks-per-student of two sections. The mean M must always lie strictly between c₁ and c₂; if your answer falls outside that band you have made a sign or arithmetic slip. Set the problem up by fixing the total or the ratio first, then apply the formula. The slow students plug in litres and grind; the fast ones spot that only the RATIO of quantities matters and jump straight to alligation in the next topic.

✅ Solved examples

1. Tea worth ₹150/kg is mixed with tea worth ₹200/kg in the ratio 3 : 2. Find the price per kg of the mixture.
M = (3×150 + 2×200)/(3+2) = (450 + 400)/5 = 850/5 = ₹170/kg.
2. 5 litres of a 20% acid solution is mixed with 3 litres of a 60% acid solution. Find the acid % of the blend.
Acid = 5×0.20 + 3×0.60 = 1.0 + 1.8 = 2.8 L in 8 L ⇒ 2.8/8 = 35%.
3. A 40% milk mixture (10 L) is combined with a 70% milk mixture (20 L). Milk % of the result?
Milk = 10×0.40 + 20×0.70 = 4 + 14 = 18 L in 30 L ⇒ 18/30 = 60%.
4. Two classes of 30 and 20 students score average marks 60 and 80. Find the combined average.
M = (30×60 + 20×80)/(30+20) = (1800 + 1600)/50 = 3400/50 = 68 marks.

✏️ Practice — try these, take hints as needed

1. Sugar at ₹40/kg is mixed with sugar at ₹60/kg in ratio 1 : 3. Mixture price?
Weighted average with weights 1 and 3.
(1×40 + 3×60)/4.
(40 + 180)/4.
₹55/kg
2. 4 L of 25% syrup is mixed with 6 L of 50% syrup. Concentration?
Find total sugar first.
4×0.25 + 6×0.50 = 1 + 3.
4 L sugar in 10 L.
40%
3. Average age of 8 men is 30; of 12 women is 25. Combined average age?
Weighted average, weights 8 and 12.
(8×30 + 12×25)/20.
(240 + 300)/20.
27 years
4. Milk (10 L, 80% pure) is mixed with milk (15 L, 60% pure). Purity of blend?
Pure milk = 10×0.8 + 15×0.6.
8 + 9 = 17 L in 25 L.
17/25 × 100.
68%
5. Two alloys with 30% and 50% copper are mixed in ratio 2 : 3. Copper % in new alloy?
(2×30 + 3×50)/5.
(60 + 150)/5.
210/5.
42%

📝 Topic test — 8 questions

Auto-graded with full solutions; saved to your dashboard. Use the calculator and formula sheet (top-right) any time.

Loading questions…
Alligation Rule →