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<div>Hey! Wassup. Friend me.</div>Vbugattihttps://artofproblemsolving.com/wiki/index.php?title=2019_AMC_10B_Problems/Problem_3&diff=1419262019 AMC 10B Problems/Problem 32021-01-12T00:58:48Z<p>Vbugatti: /* Solution 1 */</p>
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<div>==Problem==<br />
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In a high school with <math>500</math> students, <math>40\%</math> of the seniors play a musical instrument, while <math>30\%</math> of the non-seniors do not play a musical instrument. In all, <math>46.8\%</math> of the students do not play a musical instrument. How many non-seniors play a musical instrument?<br />
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<math>\textbf{(A) } 66 \qquad\textbf{(B) } 154 \qquad\textbf{(C) } 186 \qquad\textbf{(D) } 220 \qquad\textbf{(E) } 266</math><br />
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==Solution 1==<br />
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<math>60\%</math> of seniors do not play a musical instrument. If we denote <math>x</math> as the number of seniors, then <cmath>\frac{3}{5}x + \frac{3}{10}\cdot(500-x) = \frac{468}{1000}\cdot500</cmath><br />
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<cmath>\frac{3}{5}x + 150 - \frac{3}{10}x = 234</cmath><br />
<cmath>\frac{3}{10}x = 84</cmath><br />
<cmath>x = 84\cdot\frac{10}{3} = 280</cmath><br />
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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{\textbf{(B) } 154}</math>.<br />
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~IronicNinja<br />
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==Solution 2==<br />
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Let <math>x</math> be the number of seniors, and <math>y</math> be the number of non-seniors. Then <cmath>\frac{3}{5}x + \frac{3}{10}y = \frac{468}{1000}\cdot500 = 234</cmath><br />
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Multiplying both sides by <math>10</math> gives us<br />
<cmath>6x + 3y = 2340</cmath><br />
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Also, <math>x + y = 500</math> because there are 500 students in total.<br />
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Solving these system of equations give us <math>x = 280</math>, <math>y = 220</math>.<br />
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Since <math>70\%</math> of the non-seniors play a musical instrument, the answer is simply <math>70\%</math> of <math>220</math>, which gives us <math>\boxed{\textbf{(B) } 154}</math>.<br />
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== Solution 3 (using the answer choices) == <br />
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We can clearly deduce that <math>70\%</math> of the non-seniors do play an instrument, but, since the total percentage of instrument players is <math>46.8\%</math>, the non-senior population is quite low. By intuition, we can therefore see that the answer is around <math>\text{B}</math> or <math>\text{C}</math>. Testing both of these gives us the answer <math>\boxed{\textbf{(B) } 154}</math>.<br />
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==Video Solution==<br />
https://youtu.be/J8UdaSHyWJI<br />
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~savannahsolver<br />
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==See Also==<br />
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{{AMC10 box|year=2019|ab=B|num-b=2|num-a=4}}<br />
{{MAA Notice}}</div>Vbugatti