Problem 17 · 2012 Math Kangaroo
Medium
Arithmetic & Operations
order-of-operations
In which of the following expressions can one exchange each number 8 with 8 different sets of equal positive numbers without changing the result?
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Answer: E — \((8 + 8 - 8) \div 8\)
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Hint 1 of 2
Replacing each 8 by equal parts means scaling all the 8s by the same factor.
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Hint 2 of 2
Which expression's value stays the same when every 8 is multiplied by the same number?
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Approach: scale-invariance
- Splitting each 8 into equal positive parts is the same as replacing 8 by a multiple of itself everywhere at once.
- The result is unchanged only if the expression is unaffected by scaling all the 8s — i.e. it is a ratio of equal degree.
- (8 + 8 − 8) ÷ 8 = 8 ÷ 8 = 1 stays 1 under any common scaling, so option E works.
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