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

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==Problem==
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==Problem 13==
 
Jamal has a drawer containing <math>6</math> green socks, <math>18</math> purple socks, and <math>12</math> orange socks. After adding more purple socks, Jamal noticed that there is now a <math>60\%</math> chance that a sock randomly selected from the drawer is purple. How many purple socks did Jamal add?
 
Jamal has a drawer containing <math>6</math> green socks, <math>18</math> purple socks, and <math>12</math> orange socks. After adding more purple socks, Jamal noticed that there is now a <math>60\%</math> chance that a sock randomly selected from the drawer is purple. How many purple socks did Jamal add?
  
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==Solution 2 (variant of Solution 1)==
 
==Solution 2 (variant of Solution 1)==
 
As in Solution 1, we have the equation <math>\frac{18+x}{36+x}=\frac{3}{5}</math>. Cross-multiplying yields <math>90+5x=108+3x \Rightarrow 2x=18 \Rightarrow x=9</math>. Thus, Jamal added <math>\boxed{\textbf{(B) }9}</math> purple socks.
 
As in Solution 1, we have the equation <math>\frac{18+x}{36+x}=\frac{3}{5}</math>. Cross-multiplying yields <math>90+5x=108+3x \Rightarrow 2x=18 \Rightarrow x=9</math>. Thus, Jamal added <math>\boxed{\textbf{(B) }9}</math> purple socks.
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==Solution 3 (non-algebraic)==
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<math>6</math> green socks and <math>12</math> orange socks together should be <math>100\%-60\% = 40\%</math> of the new total number of socks, so that new total must be <math>\frac{6+12}{0.4}= 45</math>. It follows that <math>45-6-18-12=\boxed{\textbf{(B) }9}</math> purple socks were added.
  
 
==Video Solution==
 
==Video Solution==
https://youtu.be/x9Di0yxUqeU
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https://youtu.be/xjwDsaRE_Wo
  
 
==See also==
 
==See also==
 
{{AMC8 box|year=2020|num-b=12|num-a=14}}
 
{{AMC8 box|year=2020|num-b=12|num-a=14}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 00:22, 10 January 2021

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 Jamal adds $x$ purple socks, he has $(18+x)$ purple socks and $6+18+12+x=(36+x)$ total socks. This means the probability of drawing a purple sock is $\frac{18+x}{36+x}$, so we obtain \[\frac{18+x}{36+x}=\frac{3}{5}\] Since $\frac{18+9}{36+9}=\frac{27}{45}=\frac{3}{5}$, the answer is $\boxed{\textbf{(B) }9}$.

Solution 2 (variant of Solution 1)

As in Solution 1, we have the equation $\frac{18+x}{36+x}=\frac{3}{5}$. Cross-multiplying yields $90+5x=108+3x \Rightarrow 2x=18 \Rightarrow x=9$. Thus, Jamal added $\boxed{\textbf{(B) }9}$ purple socks.

Solution 3 (non-algebraic)

$6$ green socks and $12$ orange socks together should be $100\%-60\% = 40\%$ of the new total number of socks, so that new total must be $\frac{6+12}{0.4}= 45$. It follows that $45-6-18-12=\boxed{\textbf{(B) }9}$ purple socks were added.

Video Solution

https://youtu.be/xjwDsaRE_Wo

See also

2020 AMC 8 (ProblemsAnswer KeyResources)
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

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