Difference between revisions of "2020 AMC 8 Problems/Problem 13"
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+ | ==Problem== | ||
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== | ||
− | After Jamal adds <math>x</math> purple socks, he has <math>18+x</math> purple socks and <math>6+18+12+x=x | + | 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>. |
− | ==Solution 2== | + | ==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. | |
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==Video Solution== | ==Video Solution== | ||
https://youtu.be/x9Di0yxUqeU | https://youtu.be/x9Di0yxUqeU | ||
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==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 08:57, 20 November 2020
Problem
Jamal has a drawer containing green socks,
purple socks, and
orange socks. After adding more purple socks, Jamal noticed that there is now a
chance that a sock randomly selected from the drawer is purple. How many purple socks did Jamal add?
Solution 1
After Jamal adds purple socks, he has
purple socks and
total socks. This means the probability of drawing a purple sock is
, so we obtain
Since
, the answer is
.
Solution 2 (variant of Solution 1)
As in Solution 1, we have the equation . Cross-multiplying yields
. Thus, Jamal added
purple socks.
Video Solution
See also
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.