# Difference between revisions of "2019 AMC 10B Problems/Problem 3"

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

~savannahsolver