The one formula
Average = sum ÷ count, so sum = average × count. Almost every average question becomes easy once you convert averages back into totals.
Example. The average of 5 numbers is 27, so their sum is 135. If one number is removed and the average of the rest is 25, their sum is 100, so the removed number is 35.
Adding a member
Example. 30 students average 15 years. With the teacher, 31 people average 16 years. The teacher's age is 31 × 16 − 30 × 15 = 496 − 450 = 46.
Replacement
When one member is replaced, the total changes by (new − old), and the average changes by (new − old) ÷ count.
Example. The average weight of 8 people rises by 2.5 kg when someone weighing 65 kg is replaced. The total rises by 8 × 2.5 = 20 kg, so the newcomer weighs 85 kg.
Weighted average
When groups of different sizes combine, weight each average by its group size.
Example. 30 students averaging 60 and 20 students averaging 70: (1800 + 1400) ÷ 50 = 64. The combined average is always between the two, nearer the larger group.
Consecutive numbers
For equally spaced numbers, the average is the middle value (or the mean of the first and last).
The average of 1 to n is (n + 1)/2. For 1 to 20 it is 10.5.
Three consecutive even numbers averaging 24 are 22, 24, 26.
Average speed
For equal distances at speeds a and b, the average speed is 2ab / (a + b), not (a + b)/2.
Example. 40 km/h one way and 60 km/h back: 2 × 40 × 60 / 100 = 48 km/h.
Overlapping groups
Example. 11 results average 50. The first six average 49 and the last six average 52. The sixth result is counted in both groups, so it equals 294 + 312 − 550 = 56.
Shortcuts and tricks
The standard way is total = average × count, then work with totals. The deviation method below works with the change in the average instead, which keeps the numbers small.
Deviation from the average
Example: the average of 10 numbers is 50. An 11th number, 72, is added. Find the new average.
Standard way: the new total is 500 + 72 = 572, and 572 ÷ 11 = 52.
Shortcut: 72 is 22 above the old average. Shared among 11 numbers, that raises the average by 2, to 52.
Replacement
Example: in a group of 6, someone weighing 60 kg is replaced and the average rises by 1.5 kg.
Standard way: the total rises by 6 × 1.5 = 9 kg, so the newcomer weighs 60 + 9 = 69 kg.
Shortcut: newcomer = old weight + count × change = 60 + 6 × 1.5 = 69 kg.
Weighted average by the ratio of the groups
Example: 20 students average 50 and 30 students average 60. Find the overall average.
Standard way: (20 × 50 + 30 × 60) ÷ 50 = 2,800 ÷ 50 = 56.
Shortcut: the groups are in the ratio 2 : 3, so the average lies 3/5 of the way from 50 to 60: 50 + 6 = 56.
Equal distances at two speeds
Example: 30 km/h out and 20 km/h back.
Standard way: take 60 km each way: 2 h out and 3 h back, so 120 km in 5 h = 24 km/h.
Shortcut: average speed = 2xy/(x + y) = 2 × 30 × 20/50 = 24 km/h.
Exam tips
A new value above the average raises it; below the average lowers it.
In cricket questions, the runs needed to raise the average are easiest with totals.
Don't average averages unless the groups are the same size.