2016 AMC 10A Problem 4

Problem 4 of 25EasierAlgebra

The remainder function can be defined for all real numbers

x

and

y

with

y \neq 0

\operatorname{rem}(x,y)=x-y\left\lfloor\frac{x}{y}\right\rfloor,

where

\left \lfloor \frac{x}{y} \right \rfloor

denotes the greatest integer less than or equal to

\frac{x}{y}.

What is the value of

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

Show solution

Solution

Using the formula, we get

\begin{aligned} \text{rem} \left(\dfrac{3}{8}, -\dfrac{2}{5}\right) &= \dfrac{3}{8} \\ &{}+ \dfrac{2}{5} \left \lfloor \dfrac{\frac{3}{8}}{-\frac{2}{5}}\right \rfloor \\ &= \dfrac{3}{8} \\ &{}+ \dfrac{2}{5} \left \lfloor -\dfrac{15}{16} \right \rfloor \\ &= \dfrac{3}{8} + \dfrac{2}{5} \cdot -1 \\ &= \dfrac{3}{8} - \dfrac{2}{5} \\ &= - \dfrac{1}{40} \end{aligned}

Thus, the correct answer is B .