Problem 7 · 2007 AMC 8
Easy
Arithmetic & Operations
average-update
The average age of 5 people in a room is 30 years. An 18-year-old person leaves the room. What is the average age of the four remaining people?
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Answer: D — 33.
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Hint 1 of 2
You can't average ages directly when someone leaves — rebuild the total first. An average is just a total in disguise: 5 people at 30 means 150 years of age in the room.
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Hint 2 of 2
Average problems run on the total: convert average→total, adjust the total, then divide by the new count.
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Approach: work with the total of ages, not the averages
- Convert to a total: 5 people averaging 30 means 5 · 30 = 150 years all together.
- The 18-year-old walks out, taking 18 with them: 150 − 18 = 132 years among the 4 left.
- New average = 132 ÷ 4 = 33.
- Intuition check: removing someone younger than average should pull the average up — and 33 > 30, as expected.
Another way — see why it rises by 3:
- The leaver was 18 — that's 12 below the old average of 30.
- Those 12 'missing low' years get redistributed over the 4 who remain: 12 ÷ 4 = 3 added each, so 30 + 3 = 33.
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