# 2019 AMC 10B Problems/Problem 3

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## Problem

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?

$\textbf{(A) } 66 \qquad\textbf{(B) } 154 \qquad\textbf{(C) } 186 \qquad\textbf{(D) } 220 \qquad\textbf{(E) } 266$

## Solution 1

$60\%$ of seniors do not play a musical instrument. If we denote $x$ as the number of seniors, then $$\frac{3}{5}x + \frac{3}{10}\cdot(500-x) = \frac{468}{1000}\cdot500$$

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

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

~IronicNinja

## Solution 2

Let $x$ be the number of seniors, and $y$ be the number of non-seniors. Then $$\frac{3}{5}x + \frac{3}{10}y = \frac{468}{1000}\cdot500 = 234$$

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 $\boxed{\textbf{(B) } 154}$.

## Solution 3 (using the answer choices)

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 $\text{B}$ or $\text{C}$. Testing both of these gives us the answer $\boxed{\textbf{(B) } 154}$.

## Solution 4

We know that $40\%$ of the seniors play a musical instrument, and $30\%$ of the non-seniors do not. In addition, we know that the number of people who do not play a musical instrument is $$46.8\% \cdot 500 = 46.8 \cdot 5 = \frac{468}{2} = 234$$ We can also conclude that $60\%$ of the seniors do not play an instrument, $70\%$ of the non seniors do play an instrument, and $500-234 = 266$ people do play an instrument.

We can now set up the following equations, where $s$ is the number of seniors and $n$ is the number of non-seniors: $$0.3n + 0.6s = 234$$ $$0.7n + 0.4s = 266$$ By elimination, we get $1.5n$ to be equal to $330$. This means that $n = \frac{330}{1.5} = 220$. The answer is $70$ percent of $220$. This is equal to $$0.7*220 = 7*22 = 154$$ Therefore, the answer is $\boxed{\textbf{(B) } 154}$.

~TheGoldenRetriever

## Video Solution

~Education, the Study of Everything

~savannahsolver

## See Also

 2019 AMC 10B (Problems • Answer Key • Resources) Preceded byProblem 2 Followed byProblem 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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