2000 AMC 8 Problema 17

Question

La operación

\otimes

se define para todos los números distintos de cero por

a\otimes b =\dfrac{a^{2}}{b}.

Determina

[(1\otimes 2)\otimes 3]-[1\otimes (2\otimes 3)].

The operation

\otimes

is defined for all nonzero numbers by

a\otimes b =\dfrac{a^{2}}{b}.

Determine

[(1\otimes 2)\otimes 3]-[1\otimes (2\otimes 3)].

-\dfrac{2}{3}

-\dfrac{1}{4}

0

\dfrac{1}{4}

\dfrac{2}{3}

Evalúa cada expresión

\otimes

de adentro hacia afuera.

Evaluate each

\otimes

expression from the inside out

Conserva los paréntesis; la operación no es asociativa.

Keep the parentheses; the operation is not associative

Podemos calcularlo de la siguiente manera:

\begin{gather*} [(1\otimes 2)\otimes 3]-[1\otimes (2\otimes 3)] \\ = [\dfrac{1^2}{2} \otimes 3] - [1 \otimes \dfrac{2^2}{3}] \\ = [\dfrac{1}{2} \otimes 3] - [1 \otimes \dfrac{4}{3}] \\ = \dfrac{(\frac{1}{2})^2}{3} - \dfrac{1^2}{(\frac{4}{3})} \\ = \dfrac{1}{4} \cdot \dfrac{1}{3} - \dfrac{3}{4} \\ = \dfrac{1}{12} - \dfrac{3}{4} \\= -\dfrac{2}{3}. \end{gather*}

Así, A es la respuesta correcta.

We can calculate it as follows.

\begin{gather*} [(1\otimes 2)\otimes 3]-[1\otimes (2\otimes 3)] \\ = [\dfrac{1^2}{2} \otimes 3] - [1 \otimes \dfrac{2^2}{3}] \\ = [\dfrac{1}{2} \otimes 3] - [1 \otimes \dfrac{4}{3}] \\ = \dfrac{(\frac{1}{2})^2}{3} - \dfrac{1^2}{(\frac{4}{3})} \\ = \dfrac{1}{4} \cdot \dfrac{1}{3} - \dfrac{3}{4} \\ = \dfrac{1}{12} - \dfrac{3}{4} \\= -\dfrac{2}{3}. \end{gather*}

Thus, A is the correct answer.

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