Averages, mixtures and alligation for SSC CGL

Averages look like the easiest topic in quant: add up, divide by the count. But SSC rarely asks you to add up. It asks what happens to an average when someone joins, leaves or is replaced, or what ratio two ingredients must be mixed in to get a target price. These are best solved by thinking about deviations from the average, not by adding long lists of numbers.

The basics

Average = sum of observations ÷ number of observations, so sum = average × number.

This second form is the key to most questions.

The deviation method

To find the average of 46, 48, 51, 53 and 57, assume an average of 50. The deviations are −4, −2, +1, +3 and +7, which total +5. Divided by 5 that is +1, so the average = 51.

Adding a member

Worked example: The average age of 24 students is 15 years. When the teacher's age is included, the average rises by 1 year. Find the teacher's age.
New average = 16 for 25 people. Teacher's age = 16 + 24 × 1 = 40 years.

(The teacher must be the new average, 16, plus enough extra years to raise each of the 24 students' share by 1.)

Replacing a member

Worked example: The average weight of 8 people increases by 2.5 kg when a person weighing 65 kg is replaced by a new person. Find the new person's weight.
Increase in total = 8 × 2.5 = 20 kg. New person = 65 + 20 = 85 kg.

Averages of consecutive numbers

• The average of the first n natural numbers = (n + 1)/2.
• The average of consecutive numbers = (first + last)/2.
• The average of the first n odd numbers = n; of the first n even numbers = n + 1.

Weighted averages

When groups of different sizes are combined:

Combined average = (n₁a₁ + n₂a₂)/(n₁ + n₂)

Worked example: Section A has 30 students with an average of 60 marks; Section B has 20 students with an average of 70. Combined average = (1,800 + 1,400)/50 = 64.

Alligation

Alligation is a quick way to find the ratio in which two ingredients must be mixed to get a mean value.

Write the cheaper value (c) and the dearer value (d), with the mean value (m) between them. Then:

Quantity of cheaper : quantity of dearer = (d − m) : (m − c)

Worked example: In what ratio must rice at ₹40/kg be mixed with rice at ₹55/kg to get a mixture worth ₹45/kg?
Cheaper : dearer = (55 − 45) : (45 − 40) = 10 : 5 = 2 : 1.

Alligation also works for the weighted average above: for averages of 60 and 70 combining to 64, the ratio of students = (70 − 64) : (64 − 60) = 6 : 4 = 3 : 2 — which matches 30 : 20.

Milk and water

Worked example: A milkman mixes water with milk costing ₹60/litre and sells the mixture at ₹60/litre, making a 20% profit. Find the ratio of milk to water.
The cost of the mixture = 60 ÷ 1.2 = ₹50/litre. Water costs 0.
Milk : water = (50 − 0) : (60 − 50) = 5 : 1.

Repeated replacement

If a vessel contains x litres of liquid and y litres are taken out and replaced with water n times, the liquid left is:

x × (1 − y/x)^n

Worked example: A 40-litre container is full of milk. 4 litres are removed and replaced with water. This is done twice more (three times in all). How much milk remains?
40 × (1 − 4/40)³ = 40 × (0.9)³ = 40 × 0.729 = 29.16 litres.

Common traps

Practice

1. The average of five numbers is 28. If one number is excluded, the average becomes 25. Find the excluded number.
2. The average age of a team of 11 is 26. When the captain (aged 36) is replaced by a new player, the average falls by 1. Find the new player's age.
3. Find the average of the first 50 natural numbers.
4. In what ratio must tea at ₹180/kg be mixed with tea at ₹240/kg to get a mixture worth ₹200/kg?
5. A 60-litre mixture has milk and water in the ratio 2 : 1. How much water must be added to make the ratio 1 : 1?
6. Two groups of 40 and 60 students have average scores of 72 and 82. Find the combined average.

Answers: 1. 40. 2. 25. 3. 25.5. 4. 2 : 1. 5. 20 litres. 6. 78.