Problem 1 · 2014 AMC 8
Easy
Arithmetic & Operations
order-of-operations
Harry and Terry are each told to calculate 8 − (2 + 5). Harry gets the correct answer. Terry ignores the parentheses and calculates 8 − 2 + 5. If Harry's answer is H and Terry's answer is T, what is the difference H − T?
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Answer: A — −10.
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Hint 1 of 2
A minus sign in front of a parenthesis flips the sign of everything inside. Dropping the parentheses doesn't just add the 5 differently — it changes 8 − 2 − 5 into 8 − 2 + 5.
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Hint 2 of 2
Compute both values; the gap between them is exactly twice the number that got its sign flipped (the 5).
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Approach: subtracting a group flips every sign inside it
- Harry (correct): H = 8 − (2 + 5) = 8 − 7 = 1. Subtracting the whole group means subtracting both the 2 and the 5.
- Terry (drops the parentheses): T = 8 − 2 + 5 = 6 + 5 = 11. Now the 5 is added instead of subtracted.
- H − T = 1 − 11 = −10.
- Why this transfers: only the 5 changed roles — it went from −5 to +5, a swing of 10. That's why the answers differ by exactly 10. Whenever you remove parentheses after a minus sign, every term inside must flip.
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