CAT Quant Mixture and Alligation: Method, Rules and Solved Examples

CAT Quant Mixture and Alligation questions involve combining two different values to obtain a required average or mean value. The concept can be applied to questions based on price, concentration, percentage, profit and loss, simple interest, averages and salary.

The key is to identify when the alligation method can simplify the calculation. By placing the two values and the required mean in the correct positions, you can find the mixing ratio using their differences. This guide covers the basic rules, applications and solved examples, along with simple replacement questions.

What Are CAT Quant Mixture and Alligation?

Mixture and Alligation is an important part of CAT Quant Arithmetic. One simple trick can solve questions on price, percentage, ratio, average, profit and loss and simple interest.

The trick is easy. The real skill is spotting when to use it. Use it whenever two different values are mixed to give a third average value, and that third value lies between the first two.

How Does the Alligation Method Work?

Say one sugar costs Rs. 10 per kg and another costs Rs. 70 per kg. You want a mixture that sells at Rs. 25 per kg. Mixing them 1:1 will not work. One kg of each costs Rs. 80, but two kg sold at Rs. 25 brings only Rs. 50. So you need the right ratio.

The alligation method gives it in three steps:

1. Write the two values at the corners: 10 and 70.

2. Write the required mean value in the middle: 25.

3. Take the difference along each diagonal. Always take the bigger minus the smaller, so the answer is positive.

The differences are 70 − 25 = 45 and 25 − 10 = 15. The ratio is 45:15, which is 3:1.

The table below shows how to read this ratio.

Be careful here. The difference you write under one value is the quantity of that same component, as it sits below it. Decide this clearly so that you do not reverse the ratio.

Check your answer. Take 3 kg of Rs. 10 sugar and 1 kg of Rs. 70 sugar. The cost is 30 + 70 = Rs. 100 for 4 kg. At Rs. 25 per kg, 4 kg gives Rs. 100. It matches.

How Do You Find the Quantities from the Ratio?

Questions usually give one quantity or the total. In all cases, the ratio does the work.

• First quantity given: If the ratio is 9:3 and the first is 81 units, the second is 27 units.

• Second quantity given: Use the same ratio the other way round to find the first.

• Total given: Divide the total in the ratio. If the total is 216 kg and the ratio is 3:1, the first is 162 kg and the second is 54 kg.

Can Alligation Be Used Beyond Price?

Yes. It works whenever two values combine into one overall value. This is why this topic is linked to the weighted average, which is the same idea in another form.

• Concentration: A 30% and a 70% solution are mixed to get 40%. The differences are 30 and 10, so the ratio is 3:1.

• Purity: 90% and 97% solutions give 94% purity. The differences are 3 and 4, so the ratio is 3:4. If the total mixture is 21 litres, the 90% part is 9 litres and the 97% part is 12 litres.

• Boys and girls: In a class of 400, 60% of the boys and 80% of the girls passed, and the overall result is 65%. The differences are 15 and 5, so the ratio of boys to girls is 3:1. That means 300 boys and 100 girls.

Always keep the units the same. Use percentage with percentage, rupees with rupees and litres with litres.

How Does Alligation Work in Profit and Loss?

Profit and loss questions use the same layout. Suppose a trader sells part of the goods at 8% profit and the rest at 18% profit, for an overall profit of 14%. The differences are 4 and 6, so the ratio is 2:3. If the total is 1000 kg, the part sold at 18% is 3/5 of 1000, which is 600 kg.

Now see the sign rule. A watch is sold at 16% profit and another at 12% loss, with no overall gain or loss. Write the loss as −12, not 12. The overall value is 0, and it lies between −12 and 16. The differences are 12 and 16, so the ratio is 3:4.

The table below shows why the sign matters.

When the overall value is zero, ignoring the sign gives the right answer by chance. In any other case, it fails. So always show a loss as negative, and check that the overall value lies between the two values.

If the two watches cost Rs. 840 together, the ratio 3:4 means the first watch costs 3/7 of 840, which is Rs. 360.

How Do You Use Alligation in Simple Interest?

Treat each investment rate as a value, and the overall return as the mean. Say Rs. 12,000 is invested at 10% and another amount at 20%, with an overall return of 14%. The differences are 6 and 4, so the ratio is 3:2. If Rs. 12,000 is the first part, the second is Rs. 8,000, and the total is Rs. 20,000.

There are two rules to remember:

• Convert rupees to per cent: If the question gives total interest in rupees, change it into the overall percentage first.

• Use one year: Rates are per year. If the interest is for 3 years, divide by 3 first.

For example, Rs. 18,000 is invested at 3% and 12%, and the total interest in 3 years is Rs. 2,700. The interest for one year is Rs. 900. That is, 900 ÷ 18,000 × 100 = 5%. The differences are 7 and 2, so the ratio is 7:2. The first amount is 7/9 of 18,000, which is Rs. 14,000, and the second is Rs. 4,000. The difference is Rs. 10,000. If you had used 15%, which is the 3-year rate, you would get a value outside 3 and 12. That tells you that something is wrong.

The same idea works for any "per unit" data. If 50 pets eat 210 pieces of bread in total, the average per pet is 4.2. Compare it with 3 for a cat and 5 for a dog. The ratio is 4:6, which is 2:3 for cats and dogs, so there are 20 cats and 30 dogs. Likewise, if 90 coins are worth Rs. 36.90, convert to paise (3690) and divide by 90 to get 41 paise per coin. With 20 paise and 50 paise coins, the differences are 9 and 21, giving 3:7.

How Do You Use Alligation in Averages and Salary Questions?

A company has 120 officers with an average salary of Rs. 4,000. The rest have an average of Rs. 560. The overall company average is Rs. 600. The differences are 40 and 3,400, so the ratio is 1:85. With 120 officers, the others number 85 × 120 = 10,200.

Read the question with care. The ratio gives only the number of non-officers. If the question asks for all workers, add the 120 officers back.

How Do You Add Pure Water or Use a Given Ratio?

Adding pure water. A 40-litre milk and water mixture has 10% water. Pure water is 100% water. Say the new mixture should have 20% water. The differences are 80 and 10, so the ratio is 8:1. For 40 litres of the original mixture, you add 5 litres of water.

Ratio already given. Sometimes the ratio is given, and you find the mean. Flowers at Rs. 15 and Rs. 20 per kg are mixed in the ratio 2:3. Let the mean price be x. The differences are 20 − x and x − 15, and their ratio is 2:3.

• 3 × (20 − x) = 2 × (x − 15)

• 60 − 3x = 2x − 30

• x = Rs. 18 per kg

You can also test the answer options in this kind of question. The right option gives differences in the ratio 2:3. But know the logic first.

A similar one: two honey and water mixtures have 25% and 75% honey. Two gallons of the first are mixed with three gallons of the second. Setting up (75 − x) : (x − 25) = 2:3 gives x = 55%. So honey to water is 55:45, which is 11:9. Using water instead of honey gives the same answer. Just stay on one item throughout.

What Are the Basic Replacement Questions?

In a replacement question, some mixture is taken out and replaced with another liquid. The total quantity stays the same, so the question is about what fraction remains.

Take 40 litres of pure milk. Remove 8 litres and add 8 litres of water. Repeat this three times. Each time, 4/5 of the milk stays. So milk left = 40 × (4/5)³ = 40 × 64/125 = 20.48 litres. The milk removed in total is 40 − 20.48 = 19.52 litres.

If different amounts are taken each time, find the fraction that stays each time and multiply. For 40 litres with 4, 8 and 12 litres removed, the fractions are 9/10, 4/5 and 7/10. This is only the basic level. CAT can ask harder replacement types, and difficult questions may need more than one alligation setup, but the same steps apply: find the two values, find the result and use the ratio.

Quick Summary of Mixture and Alligation Rules

The table below puts the key rules in one place. Use it for quick revision.

Alligation also helps in percentages, ratios, profit and loss, simple interest and weighted averages. For practice, solve CAT PYQs and always check that the mean lies between the two values.

Alligation becomes easy once you have seen a few types of questions, so try each one on paper. You can practice it along with the other CAT Quant Arithmetic notes and the CAT Quant Averages guide. A few CAT mock tests for Quant and the PW CAT online course can help you check how fast you spot the right setup.