2019 AMC 10B Problems/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?
of seniors do not play a musical instrument. If we denote as the number of seniors, then
Thus there are non-seniors. Since 70% of the non-seniors play a musical instrument, .
Let be the number of seniors, and be the number of non-seniors. Then
Multiplying both sides by gives us
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 .
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 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: By elimination, we get to be equal to . This means that . The answer is percent of . This is equal to Therefore, the answer is .
~Education, the Study of Everything
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