2019 AMC 10B Problems/Problem 3
Problem 3
In a high school with 
students, 
of the seniors play a musical instrument, while 
of the non-seniors do not play a musical instrument. In all, 
of the students do not play a musical instrument. How many non-seniors play a musical instrument?
Solution
60% of seniors do not play a musical instrument. If we denote x as the number of seniors, then ![\[\frac{3}{5}x + \frac{3}{10}\cdot(500-x) = \frac{468}{1000}\cdot500\]](http://latex.artofproblemsolving.com/3/9/b/39beb27287fd9606cb72594254ed23d8f84d808a.png)
![\[\frac{3}{5}x + 150 - \frac{3}{10}x = 234\]](http://latex.artofproblemsolving.com/3/b/b/3bbf60a030fd69addddedd21963193e62f2f4e54.png)
![\[\frac{3}{10}x = 84\]](http://latex.artofproblemsolving.com/a/e/0/ae0b9942f3104e50463964898645e3071ad35d75.png)
![\[x = 84\cdot\frac{10}{3} \Rightarrow 280\]](http://latex.artofproblemsolving.com/3/5/4/354f89942e46b6a7d82da881d8f728d151b59b28.png)
Thus there are 
non-seniors. Since 70% of the non-seniors play a musical instrument, 
iron
