Problem 9 · 2004 AMC 8
Easy
Arithmetic & Operations
average-from-totals
The average of the five numbers in a list is 54. The average of the first two numbers is 48. What is the average of the last three numbers?
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Answer: D — 58.
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Hint 1 of 2
An average hides a total inside it (average × count = sum). Convert each average back into its sum — sums you can add and subtract, averages you cannot.
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Hint 2 of 2
The key move is the average ↔ sum conversion: never combine averages directly. Turn them into totals, do the arithmetic, then convert back at the very end.
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Approach: work in totals, convert back at the end
- Recover the totals: all five sum to 5 × 54 = 270; the first two sum to 2 × 48 = 96. This is the whole trick — averages can't be subtracted, but the sums behind them can.
- So the last three sum to 270 − 96 = 174, and their average is 174 ÷ 3 = 58.
- Sanity check: the first two pulled the average down (48 < 54), so the last three must pull it back up — above 54, and 58 is. Common slip: averaging 54 and 48 gives 51, which is wrong because the groups have different sizes.
Another way — balance around the mean:
- The two low numbers each sit 54 − 48 = 6 below the overall mean, a deficit of 2 × 6 = 12.
- The three remaining numbers must cover that deficit, so together they sit 12 above the mean, i.e. 12 / 3 = 4 above each. Their average is 54 + 4 = 58.
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