Core formula
Average = Sum of observations / Number of observations, so Sum = Average x Number. Almost every average question is solved by converting averages into sums. Target: 30 seconds.
Key results
| Situation | Result |
|---|---|
| First n natural numbers | (n + 1)/2 |
| First n odd numbers | n |
| First n even numbers | n + 1 |
| Consecutive numbers (odd count) | average = middle term |
| A person joins and average rises by d | new person = old average + (new count x d) |
| A person replaced and average rises by d | new person = old person + (count x d) |
| Combined average of groups | (n1 a1 + n2 a2)/(n1 + n2) |
Deviation method (fastest)
Pick an assumed average. Each value's excess or shortfall from it must balance out. Example: 7 technicians earn Rs. 12,000 and the rest Rs. 6,000; overall average Rs. 8,000. Technicians are +4,000 each (total +28,000); others are -2,000 each, so there must be 14 others. Total = 21 workers.
Overlap formula
If the average of the first k and the last k of (2k - 1) numbers are given, the middle number = (k x first avg + k x last avg) - (2k - 1) x overall avg.
Batsman / innings
If a batsman scores x in the (n + 1)th innings and the average rises by d: new average = x - n x d.
Worked example
The average weight of 8 men increases by 2.5 kg when one of them weighing 65 kg is replaced by a new man. Weight of the new man?
Increase in total = 8 x 2.5 = 20 kg. New man = 65 + 20 = 85 kg.
Worked example 2
The average of 5 numbers is 40. If one number is excluded, the average becomes 38. The excluded number?
200 - 4 x 38 = 200 - 152 = 48.
Average speed (link with SDT)
- Equal distances at speeds x and y: 2xy/(x + y).
- Equal times at speeds x and y: (x + y)/2.
Common traps
- Adding averages of groups of different sizes - always use sums.
- Counting the overlapping middle term only once.
- When the teacher is included, the count becomes n + 1.
- Ages: after t years every member's age rises by t, so the average also rises by t.
Worked example 3 (overlap)
The average of 11 numbers is 36. The first six average 32 and the last six average 40. Sixth number = 6 x 32 + 6 x 40 - 11 x 36 = 192 + 240 - 396 = 36.
Worked example 4 (combined average)
Section A has 30 students averaging 60 marks; section B has 20 students averaging 70. Combined average = (1,800 + 1,400)/50 = 64. Notice it is closer to 60 because section A is larger.