Average, Mixture and Alligation Notes

Average, mixture and alligation are closely related chapters of arithmetic. All three deal with the idea of a "mean value" and how it changes when quantities are added, removed or mixed. They appear in every competitive examination, such as SSC, banking, railways, CSAT and state-level tests, and the questions can usually be solved in less than a minute if you know the right shortcut. This note explains the formulas, the alligation method, the replacement formula, and many solved examples in simple language, so that you can solve these questions quickly and accurately.

Average: The Basic Idea

The average (arithmetic mean) of a set of numbers is the sum of the numbers divided by how many there are. The key relation is Sum = Average × Number of items. Almost every problem becomes easy if you convert the average into a sum. For example, if the average of 5 numbers is 20, their total is 100.

Useful Results

• The average of the first n natural numbers is (n + 1)/2; of the first n even numbers is (n + 1); of the first n odd numbers is n.
• For numbers in arithmetic progression (equal gaps), the average is the average of the first and last terms: the average of 11, 13, 15, ..., 29 is (11 + 29)/2 = 20.
• If every number is increased by k, the average also increases by k; if every number is multiplied by k, the average is also multiplied by k.
• The average always lies between the smallest and the largest value.

Typical Problems on Average

1. A person replaced. The average age of 8 persons increases by 2.5 years when a person aged 65 is replaced by a new person. The total increase is 8 × 2.5 = 20, so the new person is 65 + 20 = 85 years old.

2. A new member added. The average age of a class of 30 students is 14 years. When the teacher's age is included, the average becomes 15. The teacher's age = 31 × 15 − 30 × 14 = 465 − 420 = 45 years.

3. Batsman's average. A batsman scores 48 runs in his 12th innings, and his average increases by 2. Let the old average be x for 11 innings. Then (11x + 48)/12 = x + 2, which gives 11x + 48 = 12x + 24, so x = 24 and the new average is 26.

4. Wrong value corrected. The average of 10 numbers is 15, but one number 25 was read as 52. The correct total = 150 − 52 + 25 = 123, so the correct average is 12.3.

5. Weighted average. One group of 40 students has an average of 50 marks, and another group of 60 students has an average of 60. The combined average = (40 × 50 + 60 × 60)/100 = (2,000 + 3,600)/100 = 56. It is closer to the average of the larger group.

6. Average speed. If a person goes at 40 km/h and returns at 60 km/h over the same distance, the average speed is 2 × 40 × 60/(40 + 60) = 48 km/h, not the simple average.

Mixture: Combining Quantities

A mixture problem involves mixing two or more items (milk and water, two types of rice, two solutions of different strengths) in a certain ratio. The central idea is that the quantity of the pure substance does not change when only water is added or removed. Let us look at some basics.

Adding water. A 40 litre mixture has milk and water in the ratio 3:1. So milk is 30 L and water is 10 L. How much water must be added to make the ratio 3:2? Milk stays 30 L; for the ratio 3:2, water must be 20 L. So 10 L must be added.

Two solutions. If 4 L of a 30 percent alcohol solution is mixed with 6 L of a 20 percent solution, the total alcohol = 1.2 + 1.2 = 2.4 L, and the total volume is 10 L, so the strength is 24 percent.

Mixing different prices. If two types of rice at 50 and 60 rupees per kg are mixed in the ratio 2:3, the price of the mixture is (2 × 50 + 3 × 60)/5 = 280/5 = 56 rupees per kg.

Alligation: The Shortcut

Alligation is a fast method to find the ratio in which two items must be mixed to get a mixture of a required mean value. If the cheaper item has a value c, the costlier item has a value d and the required mean is m, then:

The easy way is the "cross" method: write c and d on top, the mean m in the middle, then subtract diagonally. The mean must lie between c and d.

Example 1. In what ratio should sugar costing 20 rupees per kg be mixed with sugar costing 30 rupees per kg to get a mixture worth 24 rupees per kg? The ratio is (30 − 24):(24 − 20) = 6:4 = 3:2.

Example 2. In what ratio should a 10 percent solution be mixed with a 30 percent solution to get a 22 percent solution? The ratio is (30 − 22):(22 − 10) = 8:12 = 2:3.

Example 3. A trader mixes two types of tea costing 40 and 50 rupees per kg in the ratio 3:2. At what price should he sell the mixture to get a 20 percent profit? The cost price = (3 × 40 + 2 × 50)/5 = 220/5 = 44 rupees per kg. The selling price = 44 × 1.2 = 52.8 rupees per kg.

Replacement Formula

When a part of a mixture is repeatedly taken out and replaced with something else (usually water), use this formula. If a container has x units of a pure liquid and y units are taken out and replaced with water, and the process is repeated n times, then:

Pure liquid left = x × (1 − y/x)^n.

Example. A container has 50 litres of pure milk. 10 litres are taken out and replaced with water, and the process is repeated once more. The milk left = 50 × (1 − 10/50)² = 50 × (0.8)² = 50 × 0.64 = 32 litres. The final ratio of milk to water is 32:18 = 16:9.

Profit from Dishonest Mixing

A common question is that a seller mixes water in milk and sells the mixture at the cost price of milk. The gain percent equals (water/milk) × 100. If he adds 1 litre of water to 4 litres of milk, he sells 5 litres for the price of 4 litres, so the gain = 1/4 × 100 = 25 percent. If he sells at a profit above the cost, calculate step by step.

Average of Groups and Alligation Together

Alligation is also used for averages. If the average age of boys in a class is 12 and that of girls is 14, and the overall average is 13.2, the ratio of boys to girls is (14 − 13.2):(13.2 − 12) = 0.8:1.2 = 2:3. The same rule works for the average speed, marks and prices. This is a major shortcut; practise it with several examples.

Tips and Common Mistakes

Always convert the average into a sum before solving. In mixture questions, track the pure quantity, which does not change. For alligation, make sure the mean lies between the two values, and subtract in the cross manner without mixing up which quantity gets which difference. When the question is about the "ratio of quantities", the alligation answer is directly the required ratio; but if the question asks for the ratio of the "cost" or the "value", adjust accordingly. Common errors include taking a simple average instead of a weighted one, forgetting to include the replaced item in the count and subtracting in the wrong order. Check the answer by putting the values back into the original question.

Quick Revision

• Sum = average × number; weighted average depends on group sizes.
• Alligation: cheaper : costlier = (d − m) : (m − c).
• Replacement: pure liquid left = x(1 − y/x)^n.
• Pure quantity stays constant when only water is added; dishonest sale gain = water/milk × 100.

With the help of these three chapters, you can solve a large set of arithmetic questions rapidly. Learn the formulas, solve each type at least five times and revise your mistakes, and the chapter will become a source of quick marks.

Related Notes to Read Next

Continue your preparation with these related notes from the same subject:

• Mensuration Notes for Competitive Exams
• Time, Speed and Distance Notes for Competitive Exams
• Simplification and Approximation Notes
• All Quantitative Aptitude notes

Frequently Asked Questions (FAQ)

What is the formula for average?

Average equals the sum of all items divided by the number of items, so the sum equals average times the number of items.

What is the average of the first n natural numbers?

The average of the first n natural numbers is (n + 1)/2.

What is weighted average?

Weighted average takes into account the size or importance of each group, for example the number of students in each class, when finding the overall average.

How do I find the average speed for equal distances?

The average speed is 2xy/(x + y), where x and y are the two speeds.

What is alligation?

Alligation is a quick method of finding the ratio in which two items of different values should be mixed to get a mixture of a given mean value.

What is the alligation rule?

The ratio of the cheaper to the costlier quantity is (costlier value − mean) : (mean − cheaper value).

What is the replacement formula in mixtures?

If x units of pure liquid have y units replaced by water n times, the pure liquid left is x × (1 − y/x) raised to the power n.

What stays constant when water is added to a mixture?

The quantity of the pure substance (such as milk or alcohol) stays the same when only water is added.

How do I calculate the gain of a milkman who adds water and sells at cost price?

The gain percent is the quantity of water divided by the quantity of milk, multiplied by 100.

What happens to the average if all numbers increase by 5?

The average also increases by 5.

How do I correct an average when one number was wrongly read?

Find the old total, subtract the wrong number, add the correct number and divide by the number of items.

Where is alligation used besides mixtures?

Alligation can also be used to find the ratio of groups when their averages and the overall average are known.