The mean of five numbers is 12. One of the numbers is removed and the mean of the remaining four is 11. What number was removed?
Hints
Take them one at a time. The first gives nothing away.
1A nudge
Turn each mean into a total.
2The strategy
A mean and a count give you a total. Find the total before the removal and the total after it; the difference between those totals is the number that left.
3The full solution
Five numbers with mean 12 total 60. Four numbers with mean 11 total 44. The removed number is 60 - 44 = 16.
Solution
The reliable way
Convert both means into totals. Five numbers averaging 12 have a total of 5 x 12 = 60. The remaining four averaging 11 have a total of 4 x 11 = 44. The number removed is 60 - 44 = 16. The transferable idea: a mean is only useful once you turn it into a total.
The elegant way
Removing the number dropped the mean of the other four by 1 each, so it carried 12 plus 4 extra, which is 16.
Why this is on the test: Contest questions almost always use the mean backwards, and going through totals is the only route that does not guess.