Problem 13 · 1999 AMC 8
Medium
Arithmetic & Operations
averagetotals
The average age of the 40 members of a computer science camp is 17 years. There are 20 girls, 15 boys, and 5 adults. If the average age of the girls is 15 and the average age of the boys is 16, what is the average age of the adults?
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Answer: C — 28 years.
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Hint 1 of 2
You can't average the averages — averages of different-sized groups don't combine directly. Convert each average back into a TOTAL (average × count), because totals do add.
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Hint 2 of 2
Find the whole camp's total age, subtract the girls' and boys' totals, and what's left is the adults' total. Then divide by 5.
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Approach: averages → totals (totals are what's allowed to add)
- Turn averages into sums: whole camp 40 × 17 = 680, girls 20 × 15 = 300, boys 15 × 16 = 240.
- Adults' total age = 680 − 300 − 240 = 140, so their average is 140 ÷ 5 = 28.
- Why this transfers: the golden rule for averages is "un-average into totals first." Sums combine and split cleanly; averages don't, because they hide how many people are behind each number.
Another way — balance the deviations from 17:
- Measure each group against the overall average of 17. Girls are 2 below (×20 = −40 total), boys are 1 below (×15 = −15), so the camp is 55 'age-years' short of 17-each.
- The 5 adults must carry that whole +55 surplus above 17: 55 ÷ 5 = 11 above, so 17 + 11 = 28.
- This 'see-saw around the mean' view is fast and shows the adults pulling the average up to balance the younger campers.
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