Problem 6 · 1996 AJHSME
Medium
Algebra & Patterns
minimize
What is the smallest result that can be obtained from the following process? Choose three different numbers from the set {3, 5, 7, 11, 13, 17}, add two of them, then multiply their sum by the third number.
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Answer: C — 36.
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Hint 1 of 2
The multiplier matters most — it scales the whole sum. To keep the product small, which of the three jobs should the smallest number do: be added, or be the multiplier?
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Hint 2 of 2
Give the smallest number (3) the multiplying job, since multiplying blows things up the fastest. Then make the two added numbers as small as possible too.
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Approach: make the smallest number the multiplier
- The multiplier scales the entire sum, so it does the most 'damage' to keeping things small — give that role to the smallest number, 3. Then pick the next two smallest, 5 and 7, to add: (5 + 7) × 3 = 36.
- Sanity-check against the tempting alternative: swapping so 5 multiplies, (3 + 7) × 5 = 50 — bigger. Any other split is larger still, so the smallest is 36.
- Why this transfers: in (sum) × (factor), the factor controls the scale. To minimize, give the smallest value the most powerful role; to maximize, give it the weakest. Spot which slot amplifies.
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