2019 AMC 10 B

Complete problem set with solutions and individual problem pages

Problem 3 Easy

In a high school with 500 students, 40\% of the seniors play a musical instrument, while 30\% of the non-seniors do not play a musical instrument. In all, 46.8\% of the students do not play a musical instrument. How many non-seniors play a musical instrument? (2019 AMC 10B Problem, Question#3)

  • A.

    66

  • B.

    154

  • C.

    186

  • D.

    220

  • E.

    266

Answer:B

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} \cdot 500,

\frac{3}{5}x+150- \frac{3}{10}x=234 \frac{3}{10}x=84x=84 \cdot \frac{10}{3}=280,

Thus there are 500-x= 220 non-seniors. Since 70\% of the non-seniors play a musical instrument, 220 \cdot \frac{7}{10}=\left( \text {B}\right)154.

Let x be the number of seniors, and y be the number of non-seniors.Then\frac{3}{5}x+ \frac{3}{10}y= \frac{468}{1000} \cdot 500=234,

Multiplying both sides by 10 gives us 6x+3y=2340,

Also，x+y=500 because there are 500 students in total.

Solving these system of equations give us x=280，y=220.

Since 70\% of the non-seniors play a musical instrument, the answer is simply 70\% of 220, which gives us \left( \text {B}\right)154.

We can clearly deduce that 70\% of the non-seniors do play an instrument, but, since the total percentage of instrument players is 46.8\%, the non-senior population is quite low. By intuition, we can therefore see that the answer is around B or C. Testing both of these gives us the answer \left( \text {B}\right)154.

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