Problem 2 · 1997 AJHSME
Easy
Algebra & Patterns
optimization
Ahn chooses a two-digit integer, subtracts it from 200, and doubles the result. What is the largest number Ahn can get?
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Answer: D — 380.
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Hint 1 of 2
Don't try numbers at random — ask which way the dial turns. You subtract Ahn's number, so a SMALLER number subtracted means a BIGGER result.
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Hint 2 of 2
To maximize an answer, drive each piece to its best extreme: pick the smallest legal two-digit number.
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Approach: push the variable to its best extreme
- The result is 2 × (200 − n). Subtracting n then doubling, so the only way to make it big is to make n as small as possible — the value drops as n grows.
- The smallest two-digit number is 10 (not 00 or 1, which aren't two digits), giving 2 × (200 − 10) = 2 × 190 = 380.
- Trap check: the choices include 398, which you'd get from n = 1 — but 1 isn't a two-digit number. Re-reading the rules catches it.
- You'll see it again: for 'largest/smallest possible value' problems, find which direction helps and slam each free choice to its boundary.
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