Averages for AFCAT: the total method, replacements and weighted averages

Averages is one of the friendliest topics in AFCAT Numerical Ability. The questions are short, the numbers are usually clean, and almost every one of them falls to a single habit: turn the average into a total, work with totals, then turn back. With about 72 seconds per question on average (100 questions in 2 hours) and a mark lost for every wrong answer, this is a topic where you want a sure, fast method rather than an equation you build from scratch each time.

This guide gives you that method, the shortcuts that sit on top of it, and why each shortcut works. The same idea returns in mixtures and alligation and in average speed, so the time you spend here pays off twice.

The one idea: work with totals

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

An average hides the individual values but it never hides the total. When a question adds a person, removes a number, replaces a player or corrects a mistake, the total changes in a way you can see. The average does not. So:

1. Convert every average in the question into a total.
2. Do the adding or subtracting with totals.
3. Divide at the end if the question asks for an average.

Example: The average of five numbers is 20. One is removed and the average of the rest is 18. The removed number is 5 × 20 − 4 × 18 = 100 − 72 = 28.

Consecutive and evenly spaced numbers

For numbers that go up by the same step (5, 8, 11, 14…), the average is the middle value, or the average of the first and last. The values pair off around the centre: the first and last are as far below and above the middle as the second and second-last, so every pair averages to the same number.

So the average of 1 to 20 is 21/2 = 10.5, and the average of the first 7 odd numbers is 7.

A common twist: "the average of five consecutive odd numbers is 27; find the largest." The middle one is 27, so the numbers are 23, 25, 27, 29, 31 and the largest is 31.

The deviation shortcut for adding and replacing

When one item joins or is replaced and the average moves, you can skip the totals:

• Someone joins and the average of n people rises by d to become the new average: the newcomer = old average + (n + 1) × d.
• Someone is replaced and the average of n people rises by d: the new person = the person who left + n × d. (If the average falls, subtract.)

Why it works: in a replacement, the total must rise by n × d, and the only change in the group is one person going and one coming. So the newcomer must bring exactly n × d more than the leaver. When someone joins, the newcomer first has to "match" the old average and then pay for the rise of d in all n + 1 places.

Example: A class of 30 has an average age of 14. When the teacher is included, the average becomes 15. By totals, 31 × 15 − 30 × 14 = 465 − 420 = 45. By the shortcut, 14 + 31 × 1 = 45. Same answer, fewer steps.

Weighted (combined) averages

If group 1 has n₁ items with average a₁ and group 2 has n₂ items with average a₂, the combined average is

(n₁a₁ + n₂a₂) ÷ (n₁ + n₂)

This is just the total method again. The simple average of the two averages, (a₁ + a₂)/2, is correct only when the groups are the same size. The combined average always lies between a₁ and a₂, nearer the bigger group.

Worked examples

Example 1 (combined average). Section A has 30 students with an average of 60 marks. Section B has 20 students with an average of 70. Find the average of all 50.

• Totals: 30 × 60 = 1,800 and 20 × 70 = 1,400.
• Combined: 3,200 ÷ 50 = 64.
• Check: 64 lies between 60 and 70, nearer 60 because Section A is bigger.

Example 2 (a batter's average). A batter scores 87 runs in the 17th innings and raises the average by 3. What is the average after the 17th innings?

• Let the average after 16 innings be x. Then 16x + 87 = 17(x + 3).
• 16x + 87 = 17x + 51, so x = 36.
• New average = 36 + 3 = 39.
• Shortcut: the 87 runs covered the old average once and the rise of 3 seventeen times, so old average = 87 − 17 × 3 = 36.

Example 3 (replacement). The average weight of 8 people rises by 2.5 kg when a new person replaces one who weighs 65 kg. Find the new person's weight.

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

Example 4 (wrong entry). The average of 50 numbers was found to be 36. Later it was noticed that 48 had been copied as 23. Find the correct average.

• The total was short by 48 − 23 = 25.
• The average rises by 25 ÷ 50 = 0.5, so the correct average is 36.5.

Example 5 (overlapping groups). The average of 11 results is 50. The average of the first six is 49 and the average of the last six is 52. Find the sixth result.

• The sixth result is counted in both groups of six.
• 6 × 49 + 6 × 52 = 294 + 312 = 606. The total of all 11 is 550.
• The sixth result = 606 − 550 = 56.

Example 6 (ages over time). Three years ago, the average age of a family of four was 24. A child has been born since, and the average age of the five members today is 22. How old is the child?

• Today, the four original members total 4 × (24 + 3) = 108 years.
• The five members total 5 × 22 = 110 years.
• The child is 110 − 108 = 2 years old.

Common mistakes

• Averaging averages. Two groups of unequal size cannot be combined by (a₁ + a₂)/2. Weight each by its size.
• Using the wrong count. When one person joins a group of 30, the new total is divided by 31.
• Dropping the direction. If an average falls after a replacement, the new person is lighter, so subtract.
• Uniform changes. If every value rises by 5, the average rises by 5. If every value is multiplied by 3, the average is multiplied by 3. There is no need to recalculate.

Practice set

1. Find the average of 12, 15, 18, 21 and 24.
2. The average of 8 numbers is 25. A ninth number raises it to 26. Find the ninth number.
3. The average of 6 numbers is 30. If each number is multiplied by 3, what is the new average?
4. Find the average of the first 20 natural numbers.
5. The average of three numbers is 20, and the average of the first two is 18. Find the third.
6. The average of four consecutive even numbers is 27. Find the smallest.
7. The average marks of 40 students are 72. Ten students with an average of 60 leave. Find the average of the rest.
8. A group of 10 has an average age of 25. One member aged 34 leaves and a new member joins, and the average falls by 1. How old is the new member?

Answers:

1. 18. Evenly spaced, so the middle value.
2. 34. 9 × 26 − 8 × 25 = 234 − 200. Or 25 + 9 × 1.
3. 90. Multiplying every value by 3 multiplies the average by 3.
4. 10.5. (1 + 20)/2.
5. 24. 3 × 20 − 2 × 18 = 60 − 36.
6. 24. The numbers are 24, 26, 28 and 30, which average 27.
7. 76. (40 × 72 − 10 × 60) ÷ 30 = (2,880 − 600) ÷ 30 = 2,280 ÷ 30.
8. 24. The total falls by 10 × 1 = 10, so the new member = 34 − 10.

What to do next

• Redo the six worked examples without looking, using the total method first and the shortcut second.
• Learn the consecutive-number table until you can answer any row instantly.
• Solve 20 mixed average questions under a timer, aiming for under a minute each.
• Move on to mixtures and alligation, which is weighted averages run backwards, and revise ratio and proportion alongside.