Problem 22 · 2014 Math Kangaroo
Stretch
Logic & Word Problems
casework
A set of scales does not always show the correct mass. If something weighs less than 1000 g it shows the exact mass; when something weighs 1000 g or more it shows some mass over 1000 g. You have 5 balls with masses A g, B g, C g, D g and E g, each less than 1000 g. Weighing them in pairs, the scales show: \(B+D=1200\), \(C+E=2100\), \(B+E=800\), \(B+C=900\), \(A+E=700\). Which ball is the heaviest?
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Answer: D — D
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Hint 1 of 2
A reading is trustworthy only when the true pair-sum is under 1000g; otherwise it just signals 'over 1000'.
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Hint 2 of 2
Pick out the readings below 1000 as exact and see which ball that leaves as the heaviest.
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Approach: keep only the readings under 1000g as exact
- Readings under 1000g are exact; readings of 1200 and 2100 only mean 'the true sum is over 1000g'.
- The trustworthy exact sums are B+E = 800, B+C = 900 and A+E = 700, so B, C, E, A are all fairly light.
- Since B+D exceeds 1000g while B is small, D must be large; combined with the small reliable sums, D comes out heaviest.
- The heaviest ball is D.
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