Question

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 $\text{rem}\left(\dfrac{3}{8},-\dfrac{2}{5}\right)?$

$-\dfrac{3}{8}$

$-\dfrac{1}{40}$

$0$

$\dfrac{3}{8}$

$\dfrac{31}{40}$

Answer: B

Concepts:custom operation floor and ceiling functions

Difficulty rating: 1200

Solution:

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 Problem 3

Answer: B

Difficulty rating: 1200

Solution:

Problem 3 in Other Years