2020 AMC 8 Problems/Problem 13

Jamal has a drawer containing $6$ green socks, $18$ purple socks, and $12$ orange socks. After adding more purple socks, Jamal noticed that there is now a $60\%$ chance that a sock randomly selected from the drawer is purple. How many purple socks did Jamal add?

$\textbf{(A) }6 \qquad \textbf{(B) }9 \qquad \textbf{(C) }12 \qquad \textbf{(D) }18 \qquad \textbf{(E) }24$

Solution 1

After he adds $x$ blue socks, the probability becomes $\frac{18+x}{6+18+12+x}\implies\frac{18+x}{36+x}=\frac{3}{5}.$ Then, the answer is $\textbf{(B) }9$ because $\frac{18+9}{36+9}=\frac{27}{45}=\frac{3}{5}$ . ~icematrix

Solution 2

The total number of socks that Jamal has is $\, 6+18+12=36$ socks. We are trying to determine how many purple socks he added to his drawer. Let's say he adds $x$ purple socks. This means that the total number of purple socks in his drawer will be $(18+x)$ and the new total number of socks in his drawer will be $(36+x)$ . The ratio of purple socks to total socks in his drawer is now $\frac{60}{100}=\frac{3}{5}$ . This leads to the equation $\frac{18+x}{36+x}=\frac{3}{5}$ . Cross multiplying this equation gives us $90+5x=108+3x \implies 2x=18 \implies x=9$ . Thus, Jamal added 9 purple socks $\implies\boxed{\textbf{(B) }9}$ .
~ junaidmansuri

2020 AMC 8 (Problems • Answer Key • Resources)
Preceded by Problem 12	Followed by Problem 14
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2020 AMC 8 Problems/Problem 13

Solution 1

Solution 2

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