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 (ProblemsAnswer KeyResources)
Preceded by
Problem 16
Followed by
Problem 18
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All AJHSME/AMC 8 Problems and Solutions

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