Problem 2 · 1996 AJHSME
Easy
Arithmetic & Operations
order-of-operations
Jose, Thuy, and Kareem each start with the number 10. Jose subtracts 1 from 10, doubles his answer, and then adds 2. Thuy doubles 10, subtracts 1 from her answer, and then adds 2. Kareem subtracts 1 from 10, adds 2 to his number, and then doubles the result. Who gets the largest final answer?
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Answer: C — Kareem.
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Hint 1 of 2
All three do the same three operations — double, subtract 1, add 2 — just in different orders. Notice WHO doubles last. Doubling magnifies whatever you've built up, so saving it for last should help.
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Hint 2 of 2
Multiplying last scales up the full amount you've accumulated; doing it early means the later +2 isn't doubled. So predict the person who multiplies last wins — then just confirm.
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Approach: spot who doubles last
- Doubling is the only operation that scales the running total, so whoever doubles last gets their biggest number doubled. Kareem is the one who multiplies at the very end — so he should win.
- Confirm with quick arithmetic: Jose (10 − 1)·2 + 2 = 20, Thuy 10·2 − 1 + 2 = 21, Kareem (10 − 1 + 2)·2 = 22. The largest is Kareem's 22.
- Why this transfers: when the same steps run in different orders, ask which step amplifies — a ×2 or +2 done last lands on a bigger base. The order, not the operations, is the whole game.
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