Mixtures and alligation for SSC CHSL

Mixture questions describe two things combined (two kinds of rice, milk and water, a strong and a weak solution) and ask for a ratio, a price or a quantity. Most of them fall to one tool, alligation, which turns a weighted-average equation into a two-second subtraction.

If you have worked through the averages guide, you already know the idea: when two groups combine, the overall average lies between the two group averages, closer to the larger group. Alligation just reads that distance backwards to get the ratio.

The alligation rule, and why it works

If a cheaper ingredient worth c is mixed with a dearer ingredient worth d to give a mixture worth m, then

cheaper : dearer = (d − m) : (m − c)

Why: suppose x units of the cheaper and y units of the dearer are mixed. The total value is cx + dy, which must equal m(x + y). Rearranging gives x(m − c) = y(d − m), so x : y = (d − m) : (m − c). Each ingredient's share is the distance of the other ingredient from the mean.

The "values" do not have to be prices. Alligation works for anything averaged by quantity:

• price per kg or per litre
• concentration (% of acid, fraction of milk)
• average marks or speeds of two groups
• profit percentages on two parts of a stock

Changing a ratio: anchor the part that doesn't change

When only water is added to milk, the amount of milk is fixed. Set the fixed part equal to its share in the new ratio and read off the other part. The same logic works for acid in a solution: adding water changes the percentage but not the amount of acid.

Repeated replacement

A container holds V litres of a liquid. Each time, r litres are taken out and replaced with water. After n rounds,

liquid left = V × (1 − r/V)ⁿ

Why: each round removes the fraction r/V of whatever is in the container, so the original liquid is multiplied by (1 − r/V) every time, just like a repeated percentage decrease.

Seven worked questions

Q1. In what ratio must rice at ₹32/kg be mixed with rice at ₹48/kg to get a mixture worth ₹38/kg?
(48 − 38) : (38 − 32) = 10 : 6 = 5 : 3.
Check: 5 kg at 32 plus 3 kg at 48 is 160 + 144 = 304 for 8 kg, which is ₹38/kg.

Q2. 3 kg of tea at ₹200/kg is mixed with 2 kg at ₹250/kg. Find the price of the mixture.
(600 + 500) ÷ 5 = ₹220/kg. When quantities are given and the price is asked, just take the weighted average.

Q3. A 40-litre mixture has milk and water in the ratio 3 : 1. How much water must be added to make the ratio 2 : 1?
Milk = 30 and water = 10. Milk stays 30, which is now 2 parts, so water must be 15. Add 5 litres.

Q4. Container A has milk and water in the ratio 5 : 3, and container B in the ratio 3 : 1. In what ratio should they be mixed to get milk and water in the ratio 7 : 3?
Use the milk fraction as the value: A = 5/8, B = 3/4, target = 7/10. Convert to 40ths: 25, 30 and 28.
A : B = (30 − 28) : (28 − 25) = 2 : 3.
Check: 2 litres of A has 1.25 litres of milk, 3 litres of B has 2.25, total 3.5 out of 5, which is 7/10.

Q5. How much water must be added to 20 litres of a 30% acid solution to make it 20%?
Acid = 6 litres and stays fixed. At 20%, 6 litres is one-fifth of the total, so the total is 30 litres. Add 10 litres.

Q6. A milkman buys milk at ₹50/litre, adds water and sells the mixture at ₹50/litre, making a 25% profit. Find the ratio of milk to water.
The cost per litre of mixture is 50 ÷ 1.25 = ₹40. Water costs nothing.
Milk : water = (40 − 0) : (50 − 40) = 4 : 1.

Q7. A trader has 1,000 kg of sugar. Part is sold at 8% profit and the rest at 18% profit, giving 14% profit overall. How much was sold at 8%?
The values are the profit percentages. (18 − 14) : (14 − 8) = 4 : 6 = 2 : 3. So 2/5 of 1,000 = 400 kg was sold at 8%.

Common mistakes

Practice

1. In what ratio must sugar at ₹40/kg be mixed with sugar at ₹55/kg to get a mixture worth ₹45/kg?
2. A 30-litre mixture has milk and water in the ratio 4 : 1. How much water must be added to make it 2 : 1?
3. Find the price of a mixture of 4 kg of pulses at ₹90/kg and 6 kg at ₹110/kg.
4. How much water must be added to 15 litres of a 40% salt solution to make it 25%?
5. A shopkeeper mixes water with milk costing ₹60/litre and sells the mixture at ₹60/litre, gaining 20%. Find the ratio of milk to water.
6. A 60-litre mixture has milk and water in the ratio 7 : 3. How much milk must be added to make the ratio 4 : 1?
7. A vessel holds 80 litres of milk. Eight litres are taken out and replaced with water, and this is done twice. How much milk is left?
8. One alloy is 70% copper and another is 40% copper. In what ratio must they be melted together to get an alloy that is 50% copper?

Answers:

1. 2 : 1. (55 − 45) : (45 − 40) = 10 : 5.
2. 6 litres. Milk 24 is now 2 parts, so water must be 12; it was 6.
3. ₹102/kg. (360 + 660) ÷ 10.
4. 9 litres. Salt is 6 litres, which is 25% of 24 litres, so add 24 − 15.
5. 5 : 1. CP of mixture = 60 ÷ 1.2 = 50, and (50 − 0) : (60 − 50) = 50 : 10.
6. 30 litres. Water stays 18, which is 1 part, so milk must be 72; it was 42.
7. 64.8 litres. 80 × (1 − 8/80)² = 80 × 0.81.
8. 1 : 2. (50 − 40) : (70 − 50) = 10 : 20. Check: 1 kg at 70% and 2 kg at 40% give 1.5 kg of copper in 3 kg.

What to do next

• Derive the alligation rule once on paper from cx + dy = m(x + y), so you never reverse it.
• Solve 10 price-mix questions and 10 milk-and-water questions, sorting each as "alligate" or "anchor".
• Try three questions where the values are profit percentages or group averages, not prices.
• Revise averages and profit and loss if Q6 and Q7 felt slow; both feed straight into this topic.