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Difference between revisions of "2020 AMC 8 Problems/Problem 13"
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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 1==
 
==Solution 1==
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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
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After Jamal adds <math>x</math> purple socks, he has <math>(18+x)</math> purple socks and <math>6+18+12+x=(36+x)</math> total socks. This means the probability of drawing a purple sock is <math>\frac{18+x}{36+x}</math>, so we obtain <cmath>\frac{18+x}{36+x}=\frac{3}{5}</cmath> Since <math>\frac{18+9}{36+9}=\frac{27}{45}=\frac{3}{5}</math>, the answer is <math>\boxed{\textbf{(B) }9}</math>.
  
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==Solution 2==
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==Solution 2 (variant of Solution 1)==
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The total number of socks that Jamal has is <math>\, 6+18+12=36</math> socks. We are trying to determine how many purple socks he added to his drawer. Let's say he adds <math>x</math> purple socks. This means that the total number of purple socks in his drawer will be <math>(18+x)</math> and the new total number of socks in his drawer will be <math>(36+x)</math>. The ratio of purple socks to total socks in his drawer is now <math>\frac{60}{100}=\frac{3}{5}</math>. This leads to the equation <math>\frac{18+x}{36+x}=\frac{3}{5}</math>. Cross multiplying this equation gives us <math>90+5x=108+3x \implies 2x=18 \implies x=9</math>. Thus, Jamal added 9 purple socks <math>\implies\boxed{\textbf{(B) }9}</math>.<br>
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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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~[http://artofproblemsolving.com/community/user/jmansuri junaidmansuri]
 
  
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==Solution 3==
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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>. Therefore, <math>45-6-18-12=\boxed{\textbf{(B) }9}</math> purple socks were added.
  
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Let Jamal add <math>x</math> more purple socks. Then we are told that <math>\frac{18+x}{6+18+12+x}=\frac35</math>.  Cross multiplying and simplifying tells us that <math>x=\textbf{(B)}9</math>.
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==Video Solution by NiuniuMaths (Easy to understand!)==
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https://www.youtube.com/watch?v=bHNrBwwUCMI
  
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-franzliszt
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~NiuniuMaths
  
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==Video Solution==
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==Video Solution by Math-X (First understand the problem!!!)==
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https://youtu.be/UnVo6jZ3Wnk?si=6Ux3fNTMn5tSUnP3&t=1833
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~Math-X
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==Video Solution (🚀Very Quick🚀)==
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https://youtu.be/0nnyYRYi-40
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~Education, the Study of Everything
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==Video Solution by North America Math Contest Go Go Go==
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https://www.youtube.com/watch?v=u81EWYcC0Wg
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~North America Math Contest Go Go Go
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==Video Solution by WhyMath==
 
https://youtu.be/x9Di0yxUqeU
 
https://youtu.be/x9Di0yxUqeU
  
 
~savannahsolver
 
~savannahsolver
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==Video Solution==
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https://youtu.be/xjwDsaRE_Wo
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==Video Solution by Interstigation==
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https://youtu.be/YnwkBZTv5Fw?t=546
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~Interstigation
  
 
==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}}

Latest revision as of 15:34, 26 January 2024

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 $\frac{6+12}{0.4}= 45$. Therefore, $45-6-18-12=\boxed{\textbf{(B) }9}$ purple socks were added.

Video Solution by NiuniuMaths (Easy to understand!)

https://www.youtube.com/watch?v=bHNrBwwUCMI

~NiuniuMaths

Video Solution by Math-X (First understand the problem!!!)

https://youtu.be/UnVo6jZ3Wnk?si=6Ux3fNTMn5tSUnP3&t=1833

~Math-X

Video Solution (🚀Very Quick🚀)

https://youtu.be/0nnyYRYi-40

~Education, the Study of Everything

Video Solution by North America Math Contest Go Go Go

https://www.youtube.com/watch?v=u81EWYcC0Wg

~North America Math Contest Go Go Go

Video Solution by WhyMath

https://youtu.be/x9Di0yxUqeU

~savannahsolver

Video Solution

https://youtu.be/xjwDsaRE_Wo

Video Solution by Interstigation

https://youtu.be/YnwkBZTv5Fw?t=546

~Interstigation

See also

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