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

Revision as of 14:10, 1 January 2024

Problem 10

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 1

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 $5$ because now the median is $21$ instead of $16$ . The median and mean both change, so the answer is $\boxed{\textbf{(C)}}$ .

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

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(Problem 1 to 25 Full Solution)

https://youtu.be/Xm4ZGND9WoY

~Hayabusa1

@@ Line 34: / Line 34: @@
 ==Solution 1==
-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 5 because now the median is 21 instead of 16. The median and mean both change, so the answer is <math>\boxed{\textbf{(C)}}</math>.
+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>5</math> because now the median is <math>21</math> instead of <math>16</math>. The median and mean both change, so the answer is <math>\boxed{\textbf{(C)}}</math>.
 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>.

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