Problem 18 · 1988 AJHSME
Hard
Ratios, Rates & Proportions
weighted-average
The average weight of 6 boys is 150 pounds and the average weight of 4 girls is 120 pounds. The average weight of the 10 children is
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Answer: C — 138 pounds.
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Hint 1 of 2
You can't just average 150 and 120 — there are more boys than girls, so the boys pull the combined average closer to their side. The only safe move is to go back to the raw totals.
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Hint 2 of 2
An average is always (total of everything) ÷ (how many things). Find the total weight of all 10 kids, then divide by 10.
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Approach: rebuild from total weight ÷ total count
- Recover the totals each average came from: boys weigh 6 × 150 = 900, girls weigh 4 × 120 = 480. All 10 children together weigh 900 + 480 = 1380.
- Average = 1380 ⁄ 10 = 138 pounds.
- Trap to avoid: (150 + 120) ⁄ 2 = 135 (choice A) treats the groups as equal in size. Because the 6 boys outnumber the 4 girls, the real average sits *above* the midpoint 135 — and 138 leans toward the heavier, larger group, exactly as it should.
- Why this transfers: to combine two averages, never average the averages. Undo each average into a sum, add the sums, and divide by the combined count — the group sizes do the weighting for you automatically.
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