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Difference between revisions of "2019 AMC 8 Problems/Problem 10"

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==Problem==
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The diagram shows the number of students at soccer practice each weekday during last week. After computing the mean and median values, Coach discovers that there were actually <math>21</math> participants on Wednesday. Which of the following statements describes the change in the mean and median after the correction is made?
  
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<center><asy>
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=Problem 10=
 
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The diagram shows the number of students at soccer practice each weekday during last week. After computing the mean and median values, Coach discovers that there were actually <math>21</math> participants on Wednesday. Which of the following statements describes the change in the mean and median after the correction is made?
 
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<asy>
 
 
unitsize(2mm);
 
unitsize(2mm);
 
defaultpen(fontsize(8bp));
 
defaultpen(fontsize(8bp));
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i=i+4) { label(string(i),(i,1));
 
i=i+4) { label(string(i),(i,1));
 
}label("Number of students at soccer practice",(14,3.5));
 
}label("Number of students at soccer practice",(14,3.5));
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</asy>
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</asy></center>
  
 
<math>\textbf{(A) }</math>The mean increases by <math>1</math> and the median does not change.
 
<math>\textbf{(A) }</math>The mean increases by <math>1</math> and the median does not change.
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<math>\textbf{(E) }</math>The mean increases by <math>5</math> and the median increases by <math>5</math>.
 
<math>\textbf{(E) }</math>The mean increases by <math>5</math> and the median increases by <math>5</math>.
  
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==Solution==
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On Monday, <math>20</math> people come. On Tuesday, <math>26</math> people come. On Wednesday, <math>16</math> people come. On Thursday, <math>22</math> people come. Finally, on Friday, <math>16</math> people come. <math>20+26+16+22+16=100</math>, so the mean is <math>20</math>. The median is <math>(16, 16, 20, 22, 26)</math> <math>20</math>. The coach figures out that actually <math>21</math> people come on Wednesday. The new mean is <math>21</math>, while the new median is <math>(16, 20, 21, 22, 26)</math> <math>21</math>. Also, the median increases by <math>1</math> because now the median is <math>21</math> instead of <math>20</math>. The median and mean both change, so the answer is <math>\boxed{\textbf{(B)}}</math>.
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Another way to compute the change in mean is to notice that the sum increased by <math>5</math> with the correction. So, the average increased by <math>5/5 = 1</math>.
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==Video Solution by Math-X (First fully understand the problem!!!)==
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https://youtu.be/IgpayYB48C4?si=Al-ATwscPJRaKEtF&t=3094
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~Math-X
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The Learning Royal : https://youtu.be/8njQzoztDGc
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== Video Solution 2 ==
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Solution detailing how to solve the problem: https://www.youtube.com/watch?v=vkm1ZXuuQcc&list=PLbhMrFqoXXwmwbk2CWeYOYPRbGtmdPUhL&index=11
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==Video Solution 3==
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https://youtu.be/8d0VqXncuXY
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~savannahsolver
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== Video Solution by OmegaLearn ==
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https://youtu.be/TkZvMa30Juo?t=2171
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~ pi_is_3.14
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== Video Solution (CREATIVE THINKING!!!)==
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https://youtu.be/zunojG4lZxY
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~Education, the Study of Everything
  
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==Video Solution by The Power of Logic(1 to 25 Full Solution)==
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https://youtu.be/Xm4ZGND9WoY
  
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==Solution 1==
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~Hayabusa1
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On Monday, <math>20</math> people come. On Tuesday, <math>26</math> people come. On Wednesday, <math>16</math> people come. On Thursday, <math>22</math> people come. Finally, on Friday, <math>16</math> people come. <math>20</math>+<math>26</math>+<math>16</math>+<math>22</math>+<math>16</math>=<math>100</math>, so the mean is <math>20</math>. The median is (<math>16</math>,<math>16</math>,<math>20</math>,<math>22</math>,<math>26</math>) <math>20</math>. The coach figures out that actually <math>21</math> people come on Wednesday. The new mean is <math>21</math>, while the new median is (<math>16</math>,<math>20</math>,<math>21</math>,<math>22</math>,<math>26</math>)<math>21</math>. The median and mean both change, so the answer is <math>\boxed{\textbf{(B)}}</math>~heeeeeeheeeee
 
  
 
==See Also==
 
==See Also==

Latest revision as of 09:31, 9 November 2024

Problem

The diagram shows the number of students at soccer practice each weekday during last week. After computing the mean and median values, Coach discovers that there were actually $21$ participants on Wednesday. Which of the following statements describes the change in the mean and median after the correction is made?

[asy] unitsize(2mm); defaultpen(fontsize(8bp)); real d = 5; real t = 0.7; real r; int[] num = {20,26,16,22,16}; string[] days = {"Monday","Tuesday","Wednesday","Thursday","Friday"}; for (int i=0; i<30; i=i+2) { draw((i,0)--(i,-5*d),gray); }for (int i=0; i<5; ++i) { r = -1*(i+0.5)*d; fill((0,r-t)--(0,r+t)--(num[i],r+t)--(num[i],r-t)--cycle,gray); label(days[i],(-1,r),W); }for(int i=0; i<32; i=i+4) { label(string(i),(i,1)); }label("Number of students at soccer practice",(14,3.5)); [/asy]

$\textbf{(A) }$The mean increases by $1$ and the median does not change.

$\textbf{(B) }$The mean increases by $1$ and the median increases by $1$.

$\textbf{(C) }$The mean increases by $1$ and the median increases by $5$.

$\textbf{(D) }$The mean increases by $5$ and the median increases by $1$.

$\textbf{(E) }$The mean increases by $5$ and the median increases by $5$.

Solution

On Monday, $20$ people come. On Tuesday, $26$ people come. On Wednesday, $16$ people come. On Thursday, $22$ people come. Finally, on Friday, $16$ people come. $20+26+16+22+16=100$, so the mean is $20$. The median is $(16, 16, 20, 22, 26)$ $20$. The coach figures out that actually $21$ people come on Wednesday. The new mean is $21$, while the new median is $(16, 20, 21, 22, 26)$ $21$. Also, the median increases by $1$ because now the median is $21$ instead of $20$. The median and mean both change, so the answer is $\boxed{\textbf{(B)}}$.

Another way to compute the change in mean is to notice that the sum increased by $5$ with the correction. So, the average increased by $5/5 = 1$.

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

https://youtu.be/IgpayYB48C4?si=Al-ATwscPJRaKEtF&t=3094

~Math-X

The Learning Royal : https://youtu.be/8njQzoztDGc

Video Solution 2

Solution detailing how to solve the problem: https://www.youtube.com/watch?v=vkm1ZXuuQcc&list=PLbhMrFqoXXwmwbk2CWeYOYPRbGtmdPUhL&index=11

Video Solution 3

https://youtu.be/8d0VqXncuXY

~savannahsolver

Video Solution by OmegaLearn

https://youtu.be/TkZvMa30Juo?t=2171

~ pi_is_3.14

Video Solution (CREATIVE THINKING!!!)

https://youtu.be/zunojG4lZxY

~Education, the Study of Everything

Video Solution by The Power of Logic(1 to 25 Full Solution)

https://youtu.be/Xm4ZGND9WoY

~Hayabusa1

See Also

2019 AMC 8 (Problems • Answer Key • Resources)
Preceded by
Problem 9 		Followed by
Problem 11
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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