# 2000 AMC 8 Problems/Problem 17

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## Problem

The operation $\otimes$ is defined for all nonzero numbers by $a\otimes b =\frac{a^{2}}{b}$. Determine $[(1\otimes 2)\otimes 3]-[1\otimes (2\otimes 3)]$. $\text{(A)}\ -\frac{2}{3}\qquad\text{(B)}\ -\frac{1}{4}\qquad\text{(C)}\ 0\qquad\text{(D)}\ \frac{1}{4}\qquad\text{(E)}\ \frac{2}{3}$

## Solution

Follow PE(MD)(AS), doing the innermost parentheses first. $[(1\otimes 2)\otimes 3]-[1\otimes (2\otimes 3)]$ $[\frac{1^2}{2}\otimes 3]-[1\otimes \frac{2^2}{3}]$ $[\frac{1}{2}\otimes 3]-[1\otimes \frac{4}{3}]$ $[\frac{(\frac{1}{2})^{2}}{3}]-[\frac{1^2}{(\frac{4}{3})}]$ $[\frac{1}{4} \cdot \frac{1}{3}]-[\frac{3}{4}]$ $\frac{1}{12} - \frac{3}{4}$ $\frac{1}{12} - \frac{9}{12}$ $\frac{-8}{12}$ $-\frac{2}{3}$, which is answer $\boxed{A}$

## See Also

 2000 AMC 8 (Problems • Answer Key • Resources) Preceded byProblem 16 Followed byProblem 18 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 All AJHSME/AMC 8 Problems and Solutions

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