Mixture and Alligation Notes for SSC Exams

Mixture and Alligation deals with mixing two or more things of different prices or concentrations. One question usually comes from it in SSC Tier 1, and the alligation rule solves it in a few seconds. These mixture and alligation notes for SSC explain the rule, the replacement formula and the milk-water problems with solved examples.

Mean Price and Alligation Rule

If two ingredients of price C (cheaper) and D (dearer) are mixed to get a mixture of mean price M, then:

• Quantity of cheaper : Quantity of dearer = (D - M) : (M - C)
• Mean price = (Q1 x P1 + Q2 x P2) / (Q1 + Q2)
• The mean price always lies between the two prices.
• The same rule works for concentration, speed, percentages and ages.

The Alligation Cross Method

Write the dearer price on the right and the cheaper price on the left. Put the mean price in the centre. Subtract along the cross: the difference of the dearer price and the mean goes against the cheaper, and the difference of the mean and the cheaper goes against the dearer. The two results are the ratio of the quantities. For example, to mix Rs 40 and Rs 60 per kg rice to get Rs 50 per kg, the ratio is (60 - 50) : (50 - 40) = 1 : 1.

Replacement Formula

Common Question Types

• Finding the ratio in which two items of different prices are mixed.
• Finding the mean price of a mixture.
• Milk and water problems where water is added or removed.
• Replacement problems where a part of the mixture is removed and replaced.
• Mixing solutions of different strengths (percentage of acid, alcohol or salt).

Solved Examples

Example 1: In what ratio must rice at Rs 40 per kg be mixed with rice at Rs 60 per kg to get a mixture worth Rs 50 per kg?
Solution: Ratio = (60 - 50) : (50 - 40) = 10 : 10 = 1 : 1.

Example 2: 30 kg of sugar at Rs 20 per kg is mixed with 20 kg of sugar at Rs 30 per kg. Find the mean price.
Solution: Mean price = (30 x 20 + 20 x 30) / 50 = 1200 / 50 = Rs 24 per kg.

Example 3: In what ratio must tea at Rs 24 per kg be mixed with tea at Rs 36 per kg to get a mixture worth Rs 28 per kg?
Solution: Cheaper : dearer = (36 - 28) : (28 - 24) = 8 : 4 = 2 : 1.

Example 4: A mixture of 40 litres has milk and water in the ratio 5 : 3. How much water must be added to make the ratio 5 : 7?
Solution: Milk = 25 L, water = 15 L. Let x be added: 25 / (15 + x) = 5 / 7, so 15 + x = 35 and x = 20 litres.

Example 5: A vessel has 40 litres of pure milk. 4 litres are taken out and replaced with water. This is repeated once more. How much milk is left?
Solution: Milk left = 40 x (1 - 4/40)^2 = 40 x (0.9)^2 = 32.4 litres.

Example 6: In what ratio must water be mixed with milk costing Rs 60 per litre to get a mixture worth Rs 48 per litre?
Solution: Water (Rs 0) : milk (Rs 60) = (60 - 48) : (48 - 0) = 12 : 48 = 1 : 4.

Quick Tips to Solve Faster

• Always write the cheaper price on the left and the dearer price on the right in the cross method.
• Use the cross method whenever you see "in what ratio".
• For repeated replacement, apply the factor (1 - x/V) once for each replacement.
• Water has a price of zero and a concentration of zero in alligation questions.
• The mean price always lies between the two prices; use this to check your answer.

Common Mistakes to Avoid

• Reversing the ratio of the cheaper and the dearer quantity.
• Using the mean price as the average of two prices when quantities are unequal.
• Forgetting to use the power n in repeated replacement.
• Not adding the amount added to the total in ratio problems.
• Taking the price of water as non-zero.

Quick Revision Points

• Cheaper : dearer = (D - M) : (M - C).
• Mean price = total cost / total quantity.
• Replacement: V x (1 - x/V)^n.
• The mean must lie between the two prices.
• Water has cost 0 and concentration 0.

Practice Questions With Answers

1. In what ratio must sugar at Rs 18 per kg be mixed with sugar at Rs 24 per kg to get a mixture worth Rs 20 per kg?
   Answer: 2 : 1 ((24 - 20) : (20 - 18)).
2. 4 kg of a variety at Rs 50 per kg is mixed with 6 kg at Rs 40 per kg. Find the mean price per kg.
   Answer: Rs 44 ((200 + 240) / 10).
3. A mixture of 20 litres has milk and water in the ratio 3 : 1. How much water should be added to make the ratio 3 : 2?
   Answer: 5 litres (15 : (5 + x) = 3 : 2).
4. A vessel holds 50 litres of milk. 10 litres are removed and replaced with water. How much milk is left?
   Answer: 40 litres (50 x 0.8).
5. In what ratio must wheat at Rs 15 per kg be mixed with wheat at Rs 20 per kg to get a mixture worth Rs 17 per kg?
   Answer: 3 : 2 ((20 - 17) : (17 - 15)).
6. A 60-litre mixture contains milk and water in the ratio 2 : 1. If 10 litres of water are added, what is the new ratio of milk to water?
   Answer: 4 : 3 (40 : 30).
7. A container has 20 litres of milk. 5 litres are replaced with water twice. How much milk is left?
   Answer: 11.25 litres (20 x 0.75 x 0.75).
8. 5 kg of an item at Rs 30 per kg is mixed with 5 kg at Rs 50 per kg. Find the mean price.
   Answer: Rs 40 ((150 + 250) / 10).
9. In what ratio must 20% and 50% acid solutions be mixed to get a 35% solution?
   Answer: 1 : 1 ((50 - 35) : (35 - 20)).
10. Two varieties of tea at Rs 200 and Rs 300 per kg are mixed in the ratio 3 : 2. Find the price of the mixture per kg.
    Answer: Rs 240 ((600 + 600) / 5).

How to Practise This Chapter

1. Learn the formulas in the table until you can write them without looking.
2. Solve the solved examples again on your own, without reading the solution.
3. Attempt the practice questions above with a timer of 10 minutes.
4. Solve the matching practice papers to test yourself in exam style.
5. Check the previous year papers to see how SSC frames this topic.

Related Notes and Practice Papers

• Ratio and Proportion Notes
• Profit and Loss Notes
• Percentage Notes
• Average Notes
• SSC Percentage Practice Paper

Explore More SSC Subjects

SSC exams test many subjects, so connect this topic with the related subjects below:

• Reasoning Notes
• Statistics Notes
• Indian Economy Notes
• English Notes
• General Awareness Notes
• All SSC subjects and notes

Frequently Asked Questions

What is alligation?

Alligation is a quick method to find the ratio in which two items of different prices or strengths are mixed to get a mixture of a given mean price or strength.

How many questions come from mixture and alligation in SSC exams?

Usually one question comes from this chapter in SSC CGL and CHSL Tier 1.

What is the alligation rule?

Cheaper quantity : dearer quantity = (Dearer price - Mean price) : (Mean price - Cheaper price).

How do I find the mean price of a mixture?

Mean price = total cost of all items / total quantity.

What is the replacement formula?

If x litres are replaced n times in V litres of pure liquid, the pure liquid left is V x (1 - x / V) raised to n.

What price do we take for water?

Water has a price of zero in milk and water problems.

Can alligation be used for percentages?

Yes. The same rule works for concentrations, percentages, speeds and ages.

How should I practise this chapter?

Learn the cross method, solve the examples here and then attempt ratio and percentage practice sets.