Problem 1 · 1999 AMC 8
Easy
Arithmetic & Operations
work-backward
(6 ? 3) + 4 − (2 − 1) = 5. To make this statement true, the question mark between the 6 and the 3 should be replaced by
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Answer: A — ÷ (division).
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Hint 1 of 2
Don't test all four operations from the front — clean up the part that has no mystery in it first, and see what the 6 ? 3 must equal.
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Hint 2 of 2
This is work backward: when one slot is unknown, simplify everything else so the unknown stands alone on one side.
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Approach: isolate the unknown operation by simplifying the rest
- Start with the certain part. 4 − (2 − 1) = 4 − 1 = 3 — no operation choice touches it, so settle it first.
- Now the equation reads (6 ? 3) + 3 = 5, so 6 ? 3 must be 2. Only division does that: 6 ÷ 3 = 2, the sign is ÷.
- Why this transfers: in any "fill the blank" equation, evaluate every known piece first so the unknown is left alone — then you solve one tiny question instead of testing every option.
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