Difference between revisions of "2016 AMC 12A Problems/Problem 3"
(→Solution) |
(→Solution) |
||
| Line 1: | Line 1: | ||
| + | |||
| + | ==Problem== | ||
| + | The remainder function can be defined for all real numbers <math>x</math> and <math>y</math> with <math>y\ne 0</math> by | ||
| + | <math>\text{rem}(x,y)=x-y\Big\lfloor\frac{x}{y}\Big\rfloor}</math>, | ||
| + | where <math>\Big\lfloor\frac{x}{y}\Big\rfloor</math> denotes the greatest integer less than or equal to <math>\frac{x}{y}</math>. What is the value of <math>\text{rem}\left(\frac{3}{8},-\frac{2}{5}\right)</math>? | ||
| + | |||
| + | <math>\textbf{(A)}\ -\frac{3}{8}\qquad\textbf{(B)}\ -\frac{1}{40}\qquad\textbf{(C)}\ 0\qquad\textbf{(D)}\ \frac{3}{8}\qquad\textbf{(E)}\ \frac{31}{40}</math> | ||
==Solution== | ==Solution== | ||
Revision as of 23:43, 3 February 2016
Problem
The remainder function can be defined for all real numbers 
and 
with 
by
$\text{rem}(x,y)=x-y\Big\lfloor\frac{x}{y}\Big\rfloor}$ (Error compiling LaTeX. Unknown error_msg),
where 
denotes the greatest integer less than or equal to 
. What is the value of 
?
Solution
![\[\text{rem}\left(\frac{3}{8},-\frac{2}{5}\right)\]](http://latex.artofproblemsolving.com/4/b/c/4bc95cf8469a4cfdb6d3b62fdf68c7f9a237ba3b.png)
![\[=\frac{3}{8}-\left(-\frac{2}{5}\right)\Big\lfloor\frac{\frac{3}{8}}{-\frac{2}{5}}\Big\rfloor\]](http://latex.artofproblemsolving.com/7/1/9/7199ee4077d312c29bb69bff4eb734c8d19afbee.png)
![\[=\frac{3}{8}+\left(\frac{2}{5}\right)\Big\lfloor -\frac{15}{16}\Big\rfloor\]](http://latex.artofproblemsolving.com/5/f/8/5f8b7c703033544a74733c002937ae58f2e4eeaf.png)
![\[=\frac{3}{8}+\left(\frac{2}{5}\right)\left(-1\right)\]](http://latex.artofproblemsolving.com/c/e/1/ce191ca7c81c2c3d36c3c92798070445a4a58ca3.png)
![\[=\frac{3}{8}-\frac{2}{5}\]](http://latex.artofproblemsolving.com/0/e/8/0e82baa90861d265d469d327e817acadd6168a85.png)
![\[=\boxed{\textbf{(B)}-\frac{1}{40}}\]](http://latex.artofproblemsolving.com/f/f/e/ffe2006d31961de958874647aa4dcb4b583657fd.png)
