Problem 13 · 2008 AMC 8
Medium
Algebra & Patterns
system-of-equations
Mr. Harman needs to know the combined weight in pounds of three boxes he wants to mail. However, the only available scale is not accurate for weights less than 100 pounds or more than 150 pounds. So the boxes are weighed in pairs in every possible way. The results are 122, 125 and 127 pounds. What is the combined weight in pounds of the three boxes?
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Answer: C — 187 pounds.
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Hint 1 of 2
Don't solve for the three weights separately — you only need their total. Add all three pair-readings and see what you've actually counted.
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Hint 2 of 2
With three boxes, every box sits in exactly 2 of the 3 pairs, so adding the pair-sums counts each box twice.
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Approach: add the pair-sums, then halve
- Each box (a, b, c) appears in two of the three pairings, so 122 + 125 + 127 = (a+b) + (a+c) + (b+c) = 2(a+b+c). The grand total double-counts everyone.
- So 374 = 2·(total), giving total = 374 ÷ 2 = 187. No need to untangle the individual weights.
- Why this transfers: when items are summed in overlapping groups, add all the group-sums and divide by how many times each item was counted — a clean shortcut to the grand total.
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