Difference between revisions of "2020 AMC 8 Problems/Problem 13"

Revision as of 19:04, 19 November 2020

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 Jamal adds $x$ purple socks, he has $18+x$ purple socks and $6+18+12+x=x+36$ total socks, for a probability of drawing a purple sock of $\dfrac{18+x}{36+x}=\dfrac{3}{5}.$ Since $\dfrac{18+9}{36+9}=\dfrac{27}{45}=\dfrac35$ , the answer is $\boxed{\textbf{(B) }9}$ . ~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

Solution 3

Let Jamal add $x$ more purple socks. Then we are told that $\frac{18+x}{6+18+12+x}=\frac35$ . Cross multiplying and simplifying tells us that $x=\textbf{(B)}9$ .

-franzliszt

Video Solution

https://youtu.be/x9Di0yxUqeU

~savannahsolver

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

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

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 ==Solution 1==
-After he adds <math>x</math> purple socks, the probability becomes <cmath>\frac{18+x}{6+18+12+x}\implies\frac{18+x}{36+x}=\frac{3}{5}.</cmath> Then, the answer is <math>\textbf{(B) }9</math> because <math>\frac{18+9}{36+9}=\frac{27}{45}=\frac{3}{5}</math>. ~icematrix
+After Jamal adds <math>x</math> purple socks, he has <math>18+x</math> purple socks and <math>6+18+12+x=x+36</math> total socks, for a probability of drawing a purple sock of <cmath>\dfrac{18+x}{36+x}=\dfrac{3}{5}.</cmath> Since <math>\dfrac{18+9}{36+9}=\dfrac{27}{45}=\dfrac35</math>, the answer is <math>\boxed{\textbf{(B) }9}</math>. ~icematrix
 ==Solution 2==

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