2019 AMC 10B Problems/Problem 3
Contents
Problem
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 1
of seniors do not play a musical instrument. If we denote 
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)
![\[x = 84\cdot\frac{10}{3} = 280\]](http://latex.artofproblemsolving.com/e/9/1/e917d53d7f0e260365f5b83bbee0d622ee11793d.png)
Thus there are 
non-seniors. Since $70%$ (Error compiling LaTeX. Unknown error_msg) of the non-seniors play a musical instrument, 
.
~IronicNinja
Solution 2
Let be the number of seniors, and 
be the number of non-seniors. Then ![\[\frac{3}{5}x + \frac{3}{10}y = \frac{468}{1000}\cdot500 = 234\]](http://latex.artofproblemsolving.com/c/7/f/c7f0504b3bfa775132523a004e290ee0d48efc5e.png)
Multiplying both sides by 
gives us
![\[6x + 3y = 2340\]](http://latex.artofproblemsolving.com/e/e/d/eed3fa790604393aa68809319e61bca0eba6aee2.png)
Also, 
because there are 500 students in total.
Solving these system of equations give us 
, 
.
Since 
of the non-seniors play a musical instrument, the answer is simply of 
, which gives us 
.
Solution 3 (using the answer choices)
We can clearly deduce that of the non-seniors do play an instrument, but, since the total percentage of instrument players is , the non-senior population is quite low. By intuition, we can therefore see that the answer is around 
or 
. Testing both of these gives us the answer .
Solution 4
We know that of the seniors play a musical instrument, and of the non-seniors do not. In addition, we know that the number of people who do not play a musical instrument is
![\[46.8\% \cdot 500 = 46.8 \cdot 5 = \frac{468}{2} = 234\]](http://latex.artofproblemsolving.com/d/1/b/d1ba74ccb0796b66da894348f94b057c6aeaa0ab.png)
We can also conclude that of the seniors do not play an instrument, of the non seniors do play an instrument, and 
people do play an instrument.
We can now set up the following equations, where 
is the number of seniors and 
is the number of non-seniors:
![\[0.3n + 0.6s = 234\]](http://latex.artofproblemsolving.com/1/b/c/1bc3a4b46bf7b776685a58971584aa2e3edd22fe.png)
![\[0.7n + 0.4s = 266\]](http://latex.artofproblemsolving.com/b/5/7/b57a3f4d7adffd93a2b18f9a356db07edd498fb6.png)
By elimination, we get 
to be equal to 
. This means that 
.
The answer is 
percent of . This is equal to
![\[0.7*220 = 7*22 = 154\]](http://latex.artofproblemsolving.com/9/4/a/94af09c5f582c161f13d632343037e257b9ea43a.png)
Therefore, the answer is .
~TheGoldenRetriever
Video Solution
~Education, the Study of Everything
Video Solution
~savannahsolver
See Also
| 2019 AMC 10B (Problems • Answer Key • Resources) | ||
| Preceded by Problem 2 |
Followed by Problem 4 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
| All AMC 10 Problems and Solutions | ||
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