Problem 11 · 2012 Math Kangaroo
Medium
Arithmetic & Operations
order-of-operations
In four of the following calculations you can swap the number 8 with another positive number without changing the answer to the sum. For which calculation does it not work?
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Answer: D — \(8 - (8 \div 8) + 8\)
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Hint 1 of 2
Try replacing every 8 by the same other number and see if the value changes.
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Hint 2 of 2
Four expressions simplify so the number cancels out; one does not.
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Approach: replace 8 by a variable and see which value depends on it
- Replace each 8 by the same number x and simplify each expression.
- (8+8−8)÷8 becomes x÷x = 1; 8+(8÷8)−8 becomes 1; 8÷(8+8+8) becomes 1/3; 8×(8÷8)÷8 becomes 1 — all independent of x.
- But 8−(8÷8)+8 becomes 2x−1, which changes when x changes.
- So swapping the 8 fails for D.
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