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

$-\dfrac{2}{3}$

$-\dfrac{1}{4}$

$0$

$\dfrac{1}{4}$

$\dfrac{2}{3}$

Solución:

Podemos calcularlo de la siguiente manera: $\begin{gather*} [(1\otimes 2)\otimes 3]-[1\otimes (2\otimes 3)] \\ = [\dfrac{1^2}{2} \otimes 3] - [1 \otimes \dfrac{2^2}{3}] \\ = [\dfrac{1}{2} \otimes 3] - [1 \otimes \dfrac{4}{3}] \\ = \dfrac{(1 / 2)^2}{3} - \dfrac{1^2}{(4 / 3)} \\ = \dfrac{1}{4} \cdot \dfrac{1}{3} - \dfrac{3}{4} \\ = \dfrac{1}{12} - \dfrac{3}{4} \\= -\dfrac{2}{3}. \end{gather*}$

Así, A es la respuesta correcta.

We can calculate it as follows. $\begin{gather*} [(1\otimes 2)\otimes 3]-[1\otimes (2\otimes 3)] \\ = [\dfrac{1^2}{2} \otimes 3] - [1 \otimes \dfrac{2^2}{3}] \\ = [\dfrac{1}{2} \otimes 3] - [1 \otimes \dfrac{4}{3}] \\ = \dfrac{(1 / 2)^2}{3} - \dfrac{1^2}{(4 / 3)} \\ = \dfrac{1}{4} \cdot \dfrac{1}{3} - \dfrac{3}{4} \\ = \dfrac{1}{12} - \dfrac{3}{4} \\= -\dfrac{2}{3}. \end{gather*}$

Thus, A is the correct answer.

2000 AMC 8 Problema 17

Solución:

El Problema 17 en otros años