The operation ⊗ is defined for all nonzero numbers by a ⊗ b = a 2 / b . Determine [(1 ⊗ 2) ⊗ 3] − [1 ⊗ (2 ⊗ 3)] .

AMC 8 2000 Problem 17: Algebra Solution

Problem 17 · 2000 AMC 8 Hard

Algebra & Patterns custom-operation

The operation ⊗ is defined for all nonzero numbers by a ⊗ b = a² / b. Determine [(1 ⊗ 2) ⊗ 3] − [1 ⊗ (2 ⊗ 3)].

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Answer: A — −2/3.

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Hint 1 of 2

A made-up symbol is just a recipe — obey the brackets exactly, working the innermost operation first. The whole point of the problem is that the brackets are placed *differently* on the two sides.

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Hint 2 of 2

This operation is NOT associative: (a⊗b)⊗c and a⊗(b⊗c) genuinely differ. So you can't shuffle the parentheses — compute each side honestly and subtract.

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Approach: obey the brackets, innermost first, on each side

Left side: 1 ⊗ 2 = 1²/2 = ½, then (½) ⊗ 3 = (½)²/3 = (¼)/3 = 1/12.
Right side: 2 ⊗ 3 = 2²/3 = 4/3, then 1 ⊗ (4/3) = 1²/(4/3) = 3/4.
Difference = 1/12 − 3/4 = 1/12 − 9/12 = −2/3.
You'll see it again: for an unfamiliar operation, treat the definition as a literal substitution recipe and never assume the usual algebra rules (associativity, commutativity) carry over — that the two bracketings disagree is exactly the trap being tested.

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