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

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Thus there are <math>500-x = 220</math> non-seniors. Since 70% of the non-seniors play a musical instrument, <math>220 \cdot \frac{7}{10} = \boxed{B) 154}</math>
 
Thus there are <math>500-x = 220</math> non-seniors. Since 70% of the non-seniors play a musical instrument, <math>220 \cdot \frac{7}{10} = \boxed{B) 154}</math>
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==Solution 2==
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Let x be the number of seniors, and y be the number of non-seniors. Then <cmath>\frac{3}{5}x + \frac{3}{10}y = \frac{468}{1000}\cdot500 = 234</cmath>
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Multiplying 10 to every term gives us
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<cmath>6x + 3y = 2340</cmath>
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Also, <math>x + y = 500</math> because there are 500 students in total.
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Solving these system of equations give us <math>x = 280</math>,  <math>y = 220</math>
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Since 70% of the non-seniors play a musical instrument, we simply get 70% of 220, which gives us <math>\boxed{B) 154}</math>
  
 
==See Also==
 
==See Also==

Revision as of 21:18, 15 February 2019

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

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

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

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

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 10 to every term 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, we simply get 70% of 220, which gives us $\boxed{B) 154}$

See Also

2019 AMC 10B (ProblemsAnswer KeyResources)
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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