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== | ||
− | After | + | 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. | |
− | |||
− | ==Solution 3== | + | ==Solution 3 (non-algebraic)== |
+ | <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. | ||
− | + | ==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== | ==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
Contents
- 1 Problem 13
- 2 Solution 1
- 3 Solution 2 (variant of Solution 1)
- 4 Solution 3 (non-algebraic)
- 5 Video Solution by NiuniuMaths (Easy to understand!)
- 6 Video Solution by Math-X (First understand the problem!!!)
- 7 Video Solution (🚀Very Quick🚀)
- 8 Video Solution by North America Math Contest Go Go Go
- 9 Video Solution by WhyMath
- 10 Video Solution
- 11 Video Solution by Interstigation
- 12 See also
Problem 13
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.
Solution 3 (non-algebraic)
green socks and orange socks together should be of the new total number of socks, so that new total must be . Therefore, 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🚀)
~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
~savannahsolver
Video Solution
Video Solution by Interstigation
https://youtu.be/YnwkBZTv5Fw?t=546
~Interstigation
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.