Averages

An average is the level everything would sit at if you evened it out. Work with deviations from that level instead of adding raw numbers.

Start with this

Five friends have Rs 10, 20, 30, 40 and 50. If they pool it and share equally, each gets Rs 30. The average is the 'equal share'.

Averages appear on their own and inside DI, mixtures and ages. Most questions are one identity: total = average × count.

Average = total / count, so total = average × count

Worked example

The average of 6 numbers is 24. If one number is removed, the average of the rest is 22. What number was removed?

1. Total of 6 = 6 × 24 = 144.
   Why: Go to totals.
2. Total of 5 = 5 × 22 = 110.
3. Removed = 144 - 110 = 34.

Answer: 34

Someone joins or leaves

When a person joins and the average changes, compare totals. Or think of the newcomer as 'giving' or 'taking' from everyone.

A new person raises the average of 10 people by 2: the newcomer brings the old average plus 2 for each of the 11 people now in the group.

Worked example

The average age of 30 students is 15. When the teacher's age is included, the average becomes 16. Teacher's age?

1. Old total = 30 × 15 = 450.
2. New total = 31 × 16 = 496.
   Why: 31 people now.
3. Teacher = 496 - 450 = 46.

Answer: 46

Combined averages and consecutive numbers

Two groups together: combined average = (n1 a1 + n2 a2) / (n1 + n2). It lies between the two averages, closer to the bigger group.

Consecutive numbers (or any evenly spaced list): the average is the middle term, which is also (first + last)/2.

Worked example

Class A has 30 students averaging 60 marks; class B has 20 averaging 70. Combined average?

1. Totals: 1800 and 1400.
   Why: Average × count.
2. 3200 / 50 = 64.

Answer: 64

Worked example

The average of 7 consecutive odd numbers is 27. Find the largest.

1. The middle (4th) number is 27.
   Why: Evenly spaced: the average is the middle.
2. Largest = 27 + 3 × 2 = 33.

Answer: 33

Shortcut: Deviation method

Use it when: Five 2- or 3-digit numbers clustered together.

1. Assume 100: 92, 97, 105, 108, 88 → -8, -3, +5, +8, -12 = -10.
2. -10/5 = -2 → 98.

Deviation method: try it

Average of 92, 97, 105, 108, 88

1. 98.

Answer: 98

Shortcut: New member shifts the average

Use it when: Teacher joins a class, batsman plays one more innings.

1. Everyone, old and new, gets +d. The newcomer paid for all of it.

New member shifts the average: try it

30 students average 45. With the teacher the average is 46. Teacher's age?

1. 45 + 31 × 1.

Answer: 76

Common mistakes

Your turn medium

Average of 11 numbers is 60. First six average 58, last six average 63. The 6th number?

1. 348 + 378 - 660.

Answer: 66

Your turn medium

A batsman scores 85 in his 17th innings and raises his average by 3. New average?

1. 85 - 16 × 3.

Answer: 37

Your turn hard

The average of 4 numbers is 42. The first is twice the second, the second is twice the third, and the fourth is 24 more than the third. Find the fourth.

1. Third = x: 4x + 2x + x + (x + 24) = 168.
2. 8x = 144 → x = 18.
3. Fourth = 42.

Answer: 42

Your turn hard

The average of 30 marks was 60, but an 80 was read as 50. Correct average?

1. Total short by 30 → +1 on average.

Answer: 61

Recap

• always move to totals
• find the age or mark of someone who joins or leaves
• combine two groups' averages

Now practise

11 questions with full solutions.

Open practice Back to the lesson