KA-0066Means, and working backwards from a mean
4 points, difficulty 2 of 3Level 5–6about 75sThe mean of four numbers is 9. Three of them are 5, 8 and 12. What is the fourth?
Hints
Take them one at a time. The first gives nothing away.
1A nudge
Turn the mean into a total first.
2The strategy
Four numbers with a mean of 9 have a fixed total. Work that total out, add up the three you know, and the difference is the one you do not.
3The full solution
The total is 4 x 9 = 36. The three known numbers add to 5 + 8 + 12 = 25, so the fourth is 36 - 25 = 11.
Solution
The reliable way
A mean of 9 across four numbers means the total is 4 x 9 = 36. The three given numbers add to 25, so the missing one is 36 - 25 = 11. Check: 5 + 8 + 12 + 11 = 36, and 36 divided by 4 is 9. The transferable idea: a mean is only useful once you have turned it into a total.
The elegant way
Measure each number against the mean: 5 is 4 below, 8 is 1 below, 12 is 3 above, a net 2 below. The last number must be 2 above the mean, so 11.
Why this is on the test: Almost every contest mean question runs backwards from the mean, and going through the total is the only route that does not guess.