Mixture and alligation for RRB NTPC: methods, patterns and practice

Mixture questions ask what happens when two things are combined: two grades of rice, milk and water, two acid solutions. They look like a separate chapter, but they are really weighted averages with a fast shortcut called alligation.

Learn the shortcut, learn why it works, and learn the two special cases (adding one component, and repeated replacement). That covers nearly every NTPC mixture question.

Two directions of the same question

The rule of alligation

A cheaper item at price c and a dearer item at price d are mixed to get a mean price m (with c < m < d). Then:

Cheaper : Dearer = (d − m) : (m − c)

Set it out as a cross:

Why it works: think of the mean price as a balance point. Every kg of the cheaper item is worth (m − c) less than the mean; every kg of the dearer item is worth (d − m) more. For the mixture to come out exactly at m, the total shortfall must equal the total excess: q₁(m − c) = q₂(d − m). Rearranging gives q₁ : q₂ = (d − m) : (m − c). Each quantity is paired with the other item's distance from the mean.

The same rule works for anything that averages: percentages of milk or acid, profit percentages, even average speeds over equal times. Water counts as a "price" of ₹0 or as 0% milk.

Adding one component

When water (or milk) is added to change a ratio, one component stays fixed. Work with that fixed amount.

• Find how much of the fixed component there is.
• Scale the new ratio so the fixed component matches.
• The change in the other component is the amount added.

Repeated replacement

From a container of V litres of milk, x litres are taken out and replaced with water, and this is repeated n times.

Milk left = V × (1 − x/V)ⁿ

Why: each time you remove x litres of a well-mixed liquid, you remove the same fraction x/V of the milk. The fraction (1 − x/V) remains. Doing it n times multiplies that fraction n times. Simple subtraction (taking 10 L of milk twice) is wrong after the first round, because the second draw is only partly milk.

Worked questions at NTPC level

Q1. Rice at ₹30 per kg is mixed with rice at ₹40 per kg to get a mixture worth ₹34 per kg. Find the ratio.

Cheaper : dearer = (40 − 34) : (34 − 30) = 6 : 4 = 3 : 2.
Check: 3 kg at 30 + 2 kg at 40 = ₹170 for 5 kg = ₹34 per kg.

Q2. 20 kg of sugar at ₹50 per kg is mixed with 30 kg at ₹60 per kg. Find the price of the mixture per kg.

Total cost = 1,000 + 1,800 = ₹2,800 for 50 kg.
Price = ₹56 per kg.

Q3. 40 litres of a mixture contain milk and water in the ratio 3 : 1. How much water must be added to make the ratio 3 : 2?

Milk = 30 L (fixed); water = 10 L.
For 3 : 2 with 30 L milk, water must be 20 L. Add 10 litres.

Q4. A container has 50 litres of milk. 10 litres are removed and replaced with water. This is done once more. How much milk is left?

Each time, 1 − 10/50 = 4/5 of the milk remains.
Milk left = 50 × 4/5 × 4/5 = 32 litres.

Q5. How many litres of a 20% acid solution and a 50% acid solution must be mixed to get 30 litres of 40% acid?

Ratio = (50 − 40) : (40 − 20) = 10 : 20 = 1 : 2.
So 10 L of 20% and 20 L of 50%.
Check: 2 + 10 = 12 L of acid in 30 L, which is 40%.

Q6. Milk costs ₹60 per litre. In what ratio should water be mixed with it so that the mixture is worth ₹50 per litre?

Water is free, so c = 0, d = 60, m = 50.
Milk : water = (50 − 0) : (60 − 50) = 5 : 1.

Common mistakes

• Writing the alligation ratio the wrong way round. The cheaper item's share is (d − m), the dearer item's distance from the mean.
• Changing the fixed component. In Q3 the milk stays 30 L; only water changes.
• Using simple subtraction for repeated replacement. Use the fraction-remaining formula.
• Using a mean outside the two prices. A mixture of ₹30 and ₹40 rice can never be worth ₹45. If your setup needs that, reread the question.
• Mixing up cost price and selling price in profit-type mixture questions. Convert to cost price first.

Practice set

1. Tea at ₹120 per kg is mixed with tea at ₹150 per kg to get ₹130 per kg. Find the ratio.
2. 60 litres of a mixture has milk and water in the ratio 2 : 1. How much water must be added to make it 1 : 1?
3. 10 kg of rice at ₹40 per kg is mixed with 15 kg at ₹50 per kg. Find the price per kg of the mixture.
4. In what ratio must items at ₹20 and ₹35 be mixed to get ₹25?
5. A container holds 80 litres of milk. 8 litres are removed and replaced with water, and this is done twice. How much milk remains?
6. Equal quantities of two mixtures, with milk and water in the ratios 3 : 1 and 5 : 3, are combined. Find the new milk-water ratio.
7. In what ratio must 25% and 45% alcohol solutions be mixed to get a 40% solution?
8. A shopkeeper mixes water with milk costing ₹48 per litre and sells the mixture at ₹48 per litre, gaining 20%. Find the ratio of milk to water.

Answers:

1. 2 : 1. (150 − 130) : (130 − 120) = 20 : 10.
2. 20 litres. Milk 40 L stays fixed; water must become 40 L.
3. ₹46. (400 + 750) ÷ 25 = 1,150 ÷ 25.
4. 2 : 1. (35 − 25) : (25 − 20) = 10 : 5.
5. 64.8 litres. 80 × 9/10 × 9/10.
6. 11 : 5. Milk fractions 3/4 = 12/16 and 5/8 = 10/16; average 11/16.
7. 1 : 3. (45 − 40) : (40 − 25) = 5 : 15.
8. 5 : 1. Cost of mixture = 48 ÷ 1.2 = ₹40; (40 − 0) : (48 − 40) = 40 : 8.

What to do next

• Draw the alligation cross for 15 questions until you place d − m under the cheaper item without thinking.
• Solve five "add water" questions by fixing the unchanged component first.
• Practise five repeated-replacement questions, including one with three rounds.
• Revise profit and loss for the "sold at cost price" variety.