2022 AMC 8 Problems/Problem 19

Problem

Mr. Ramos gave a test to his class of $20$ students. The dot plot below shows the distribution of test scores. [asy] //diagram by pog . give me 1,000,000,000 dollars for this diagram size(5cm); defaultpen(0.7); dot((0.5,1)); dot((0.5,1.5)); dot((1.5,1)); dot((1.5,1.5)); dot((2.5,1)); dot((2.5,1.5)); dot((2.5,2)); dot((2.5,2.5)); dot((3.5,1)); dot((3.5,1.5)); dot((3.5,2)); dot((3.5,2.5)); dot((3.5,3)); dot((4.5,1)); dot((4.5,1.5)); dot((5.5,1)); dot((5.5,1.5)); dot((5.5,2)); dot((6.5,1)); dot((7.5,1)); draw((0,0.5)--(8,0.5),linewidth(0.7)); defaultpen(fontsize(10.5pt)); label("$65$", (0.5,-0.1)); label("$70$", (1.5,-0.1)); label("$75$", (2.5,-0.1)); label("$80$", (3.5,-0.1)); label("$85$", (4.5,-0.1)); label("$90$", (5.5,-0.1)); label("$95$", (6.5,-0.1)); label("$100$", (7.5,-0.1)); [/asy]

Later Mr. Ramos discovered that there was a scoring error on one of the questions. He regraded the tests, awarding some of the students $5$ extra points, which increased the median test score to $85$. What is the minimum number of students who received extra points?

(Note that the median test score equals the average of the $2$ scores in the middle if the $20$ test scores are arranged in increasing order.)

$\textbf{(A)} ~2\qquad\textbf{(B)} ~3\qquad\textbf{(C)} ~4\qquad\textbf{(D)} ~5\qquad\textbf{(E)} ~6\qquad$

Solution

We set up our cases as solution 1 showed, realizing that only the second case is possible.

We notice that $13$ students have scores under $85$ currently and only $5$ have scores over $85$. We find the median of these two numbers, getting:

\[13-5=8\] \[\frac{8}{2}=4\] \[13-4=9\]

Thus, we realize that $4$ students must have their score increased by $5$.

So, the correct answer is $\boxed{\textbf{(C)}4}$.

Video Solution 1

https://youtu.be/oUEa7AjMF2A?si=Mnek8vOzVX77YLIS&t=3389

~Math-X

Video Solution 2

https://youtu.be/eSueEOf15c8

~Education, the Study of Everything

Video Solution 3

https://youtu.be/Ij9pAy6tQSg?t=1741

~Interstigation

https://www.youtube.com/watch?v=VuiX0JcXR7Q

~David

Video Solution 4

https://youtu.be/hs6y4PWnoWg?t=294

~STEMbreezy

Video Solution 5

https://youtu.be/jXx0fc-DgQM

~savannahsolver

See Also

2022 AMC 8 (ProblemsAnswer KeyResources)
Preceded by
Problem 18
Followed by
Problem 20
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All AJHSME/AMC 8 Problems and Solutions