Question

La función resto puede definirse para todos los números reales $x$ e $y$ con $y\neq 0$ mediante $\text{rem}(x,y)=x-y\left\lfloor \dfrac{x}{y}\right\rfloor,$ donde $\left\lfloor \dfrac{x}{y}\right\rfloor$ denota el mayor entero menor o igual que $\dfrac{x}{y}.$ ¿Cuál es el valor de $\text{rem}\left(\dfrac{3}{8},-\dfrac{2}{5}\right)$ ?

The remainder function can be defined for all real numbers $x$ and $y$ with $y\neq 0$ by $\text{rem}(x,y)=x-y\left\lfloor \dfrac{x}{y}\right\rfloor,$ where $\left\lfloor \dfrac{x}{y}\right\rfloor$ denotes the greatest integer less than or equal to $\dfrac{x}{y}.$ What is the value of $\text{rem}\left(\dfrac{3}{8},-\dfrac{2}{5}\right)?$

$-\dfrac{3}{8}$

$-\dfrac{1}{40}$

$0$

$\dfrac{3}{8}$

$\dfrac{31}{40}$

Solución:

Primero, $\begin{gathered} \dfrac{x}{y}=\dfrac{3/8}{-2/5}\\ =\dfrac{3}{8}\cdot\left(-\dfrac{5}{2}\right)\\ =-\dfrac{15}{16}, \end{gathered}$ y $\left\lfloor -\dfrac{15}{16}\right\rfloor=-1.$

Por consiguiente $\begin{gathered} \text{rem}\left(\dfrac{3}{8},-\dfrac{2}{5}\right)\\ =\dfrac{3}{8}-\left(-\dfrac{2}{5}\right)(-1)\\ =\dfrac{3}{8}-\dfrac{2}{5}\\ =\dfrac{15-16}{40}\\ =-\dfrac{1}{40}. \end{gathered}$

Por lo tanto, la respuesta correcta es B.

First, $\begin{gathered} \dfrac{x}{y}=\dfrac{3/8}{-2/5}\\ =\dfrac{3}{8}\cdot\left(-\dfrac{5}{2}\right)\\ =-\dfrac{15}{16}, \end{gathered}$ and $\left\lfloor -\dfrac{15}{16}\right\rfloor=-1.$

Therefore $\begin{gathered} \text{rem}\left(\dfrac{3}{8},-\dfrac{2}{5}\right)\\ =\dfrac{3}{8}-\left(-\dfrac{2}{5}\right)(-1)\\ =\dfrac{3}{8}-\dfrac{2}{5}\\ =\dfrac{15-16}{40}\\ =-\dfrac{1}{40}. \end{gathered}$

Thus, the correct answer is B.

2016 AMC 12A Problema 3

Solución:

El Problema 3 en otros años