Averages and mixtures for CDS

Averages and mixtures are two views of one idea. An average is a total shared equally. A mixture's price or concentration is a weighted average of its ingredients. Alligation, the shortcut everyone learns for mixtures, is nothing more than a weighted average run backwards.

CDS questions here are usually short: a wrongly copied number, a new member joining a group, a score needed to lift an average, or two kinds of rice mixed to a target price. They are quick marks if you work with totals and slow, error-prone ones if you try to juggle averages in your head.

Averages: think in totals

Average = sum ÷ number of items, so sum = average × number. Almost every average question becomes easy once you convert the averages to totals, do the addition or subtraction, and convert back only at the end.

Useful facts:

• The average of consecutive numbers, or of any evenly spaced list, is the middle term, or the mean of the first and last terms.
• If every value rises by k, the average rises by k. If every value is multiplied by k, the average is multiplied by k.
• When one member of a group of n is replaced, the change in average = (new value − old value) ÷ n.

Weighted average

If group A has n₁ members with average a and group B has n₂ members with average b, the combined average is (n₁a + n₂b) ÷ (n₁ + n₂). It always lies between a and b, closer to the average of the larger group.

Alligation: a weighted average run backwards

Mix a cheaper item at price c with a dearer item at price d to get a mixture at mean price m. Then

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

Why: the cheaper item gains (m − c) per unit in the mixture and the dearer one loses (d − m) per unit. These gains and losses must cancel, so the quantities are in the inverse ratio of the gaps.

A simple way to lay it out:

Mixtures: track the quantity that does not change

When water is added to milk, the milk stays the same. When a mixture is boiled, the solute stays the same. Fix the unchanged quantity first, then set up the new ratio.

Repeated replacement

A vessel holds V litres of a pure liquid. Each time, x litres are taken out and replaced with water. After n such operations,

• pure liquid left = V × (1 − x/V)ⁿ

Why: each operation keeps the fraction (1 − x/V) of whatever pure liquid was there, so the fractions multiply, exactly like compound depreciation.

Worked questions

Question 1: The average of 10 numbers is 40. Later it is found that 58 was copied as 85. Find the correct average.

• Total used = 400. It was 85 − 58 = 27 too high.
• Correct total = 373, so the correct average = 37.3.

Question 2: The average weight of 8 people rises by 2.5 kg when one of them, who weighs 65 kg, is replaced by a new person. Find the new person's weight.

• The total rises by 8 × 2.5 = 20 kg.
• New person = 65 + 20 = 85 kg.

Question 3: A player averages 32 runs over 10 innings. How many runs are needed in the 11th innings to raise the average to 34?

• Runs so far = 320. Runs needed after 11 innings = 11 × 34 = 374.
• Required score = 374 − 320 = 54.

Question 4: How many kilograms of salt at ₹42 a kg must be mixed with 25 kg of salt at ₹24 a kg so that selling the mixture at ₹40 a kg gives a 25% profit?

• Cost price of the mixture = 40 ÷ 1.25 = ₹32 a kg.
• Alligation: ₹24 salt : ₹42 salt = (42 − 32) : (32 − 24) = 10 : 8.
• 25 kg corresponds to 10 parts, so the ₹42 salt = 8 parts = 20 kg.
• Check: (20 × 42 + 25 × 24) ÷ 45 = 1,440 ÷ 45 = 32.

Question 5: 40 litres of milk and water are in the ratio 3 : 1. How much water must be added to make the ratio 3 : 2?

• Milk = 30 litres, water = 10 litres. Milk stays at 30.
• For 3 : 2, water must be 20 litres, so add 10 litres.

Question 6: A container has 40 litres of milk. 4 litres are taken out and replaced with water. This is done three times in all. How much milk is left?

• Milk left = 40 × (1 − 4/40)³ = 40 × 0.9³ = 40 × 0.729 = 29.16 litres.

Which method, when

Practice set

1. Find the average of the first 20 even natural numbers.
2. The average age of 30 students is 15 years. When the teacher's age is included, the average becomes 16. Find the teacher's age.
3. In what ratio should tea at ₹20 a kg be mixed with tea at ₹32 a kg to get a mixture worth ₹28 a kg?
4. Section A has 40 students with an average of 60 marks, and section B has 60 students with an average of 70. Find the average of all 100 students.
5. The average of five consecutive odd numbers is 27. Find the largest.
6. The average of 11 results is 50. The average of the first six is 49 and of the last six is 52. Find the sixth result.
7. How much water must be added to 80 litres of a 25% acid solution to make it a 20% solution?
8. A vessel has 60 litres of milk. 6 litres are removed and replaced with water, and this is done twice in all. How much milk is left?

Answers

1. 21. The numbers run 2 to 40, so the average is (2 + 40) ÷ 2.
2. 46 years. 31 × 16 − 30 × 15 = 496 − 450.
3. 1 : 2. (32 − 28) : (28 − 20) = 4 : 8.
4. 66. (40 × 60 + 60 × 70) ÷ 100 = 6,600 ÷ 100.
5. 31. The middle number is 27, so the numbers are 23 to 31.
6. 56. 6 × 49 + 6 × 52 − 11 × 50 = 294 + 312 − 550. The sixth result is counted twice.
7. 20 litres. The acid (20 litres) stays fixed. At 20%, the total must be 100 litres.
8. 48.6 litres. 60 × 0.9² = 60 × 0.81.

What to do next

• Rewrite every average in your next ten questions as a total before solving
• Practise alligation with the cost price as the mean, including profit questions
• Revise ratio and proportion, since every mixture answer is a ratio
• Solve the average and mixture questions from recent CDS papers at a minute each