Difference between revisions of "2016 AMC 10A Problems/Problem 4"
Math101010 (talk | contribs) (Created page with "The remainder can be defined for all real numbers <math>x</math> and <math>y</math> with <math>y \neq 0</math> by <cmath>\text{rem} (x ,y)=x-y\left \lfloor \frac{x}{y} \right ...") |
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| + | ==Problem== | ||
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The remainder can be defined for all real numbers <math>x</math> and <math>y</math> with <math>y \neq 0</math> by <cmath>\text{rem} (x ,y)=x-y\left \lfloor \frac{x}{y} \right \rfloor</cmath>where <math>\left \lfloor \tfrac{x}{y} \right \rfloor</math> denotes the greatest integer less than or equal to <math>\tfrac{x}{y}</math>. What is the value of <math>\text{rem} (\tfrac{3}{8}, -\tfrac{2}{5} )</math>? | The remainder can be defined for all real numbers <math>x</math> and <math>y</math> with <math>y \neq 0</math> by <cmath>\text{rem} (x ,y)=x-y\left \lfloor \frac{x}{y} \right \rfloor</cmath>where <math>\left \lfloor \tfrac{x}{y} \right \rfloor</math> denotes the greatest integer less than or equal to <math>\tfrac{x}{y}</math>. What is the value of <math>\text{rem} (\tfrac{3}{8}, -\tfrac{2}{5} )</math>? | ||
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| + | <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> | ||
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| + | ==Solution== | ||
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| + | The value, by definition, is <cmath>\frac{3}{8}-\left(-\frac{2}{5}\right)\lfloor{\frac{3}{8}*\frac{-5}{2}\rfloor=-\frac{15}{16}}=\frac{3}{8}-\frac{2}{5}=\boxed{\textbf{(B) } -\frac{1}{40}.}</cmath> | ||
Revision as of 18:29, 3 February 2016
Problem
The remainder can be defined for all real numbers 
and 
with 
by ![\[\text{rem} (x ,y)=x-y\left \lfloor \frac{x}{y} \right \rfloor\]](http://latex.artofproblemsolving.com/9/5/c/95c21b1ce04c5f3c86f558991d39aa35a13bca21.png)
where 
denotes the greatest integer less than or equal to 
. What is the value of 
?
Solution
The value, by definition, is ![\[\frac{3}{8}-\left(-\frac{2}{5}\right)\lfloor{\frac{3}{8}*\frac{-5}{2}\rfloor=-\frac{15}{16}}=\frac{3}{8}-\frac{2}{5}=\boxed{\textbf{(B) } -\frac{1}{40}.}\]](http://latex.artofproblemsolving.com/d/2/c/d2ca06774f3be48b85c7660adbc5828a37071a9a.png)
