Difference between revisions of "2022 AMC 8 Problems/Problem 19"
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== Solution == | == Solution == | ||
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| − | + | We notice that <math>13</math> students have scores under <math>85</math> currently and only <math>5</math> have scores over <math>85</math>. We find the median of these two numbers, getting: | |
| − | + | <cmath>13-5=8</cmath> | |
| + | <cmath>\frac{8}{2}=4</cmath> | ||
| + | <cmath>13-4=9</cmath> | ||
| − | + | Thus, we realize that <math>4</math> students must have their score increased by <math>5</math>. | |
| − | < | + | So, the correct answer is <math>\boxed{\textbf{(C)}4}</math>. |
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| − | </ | ||
| − | ==Video Solution== | + | ==Video Solution (A Clever Explanation You’ll Get Instantly)== |
| + | https://youtu.be/tYWp6fcUAik?si=V8hv_zOn_zYOi9E5&t=2364 | ||
| + | ~hsnacademy | ||
| + | |||
| + | ==Video Solution 1== | ||
| + | https://youtu.be/oUEa7AjMF2A?si=Mnek8vOzVX77YLIS&t=3389 | ||
| + | |||
| + | ~Math-X | ||
| + | |||
| + | ==Video Solution 2== | ||
| + | https://youtu.be/eSueEOf15c8 | ||
| + | |||
| + | <i>~Education, the Study of Everything</i> | ||
| + | |||
| + | ==Video Solution 3== | ||
https://youtu.be/Ij9pAy6tQSg?t=1741 | https://youtu.be/Ij9pAy6tQSg?t=1741 | ||
| + | |||
~Interstigation | ~Interstigation | ||
https://www.youtube.com/watch?v=VuiX0JcXR7Q | 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 | ||
| + | |||
| + | ==Video Solution by Dr. David== | ||
| + | https://youtu.be/OjDLxi66Uzg | ||
==See Also== | ==See Also== | ||
{{AMC8 box|year=2022|num-b=18|num-a=20}} | {{AMC8 box|year=2022|num-b=18|num-a=20}} | ||
| + | {{MAA Notice}} | ||
| + | |||
| + | [[Category:Introductory Algebra Problems]] | ||
Latest revision as of 17:27, 4 June 2025
Contents
Problem
Mr. Ramos gave a test to his class of 
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]](http://latex.artofproblemsolving.com/8/f/1/8f168ff6d1e7635e68e0139420bed9ecfc2e4993.png)
Later Mr. Ramos discovered that there was a scoring error on one of the questions. He regraded the tests, awarding some of the students 
extra points, which increased the median test score to 
. What is the minimum number of students who received extra points?
(Note that the median test score equals the average of the 
scores in the middle if the test scores are arranged in increasing order.)
Solution
We notice that 
students have scores under currently and only have scores over . We find the median of these two numbers, getting:
![\[13-5=8\]](http://latex.artofproblemsolving.com/a/e/a/aea3e8150f45440100b8d998970e1d8fd6d61859.png)
![\[\frac{8}{2}=4\]](http://latex.artofproblemsolving.com/d/b/0/db0178224a8588827d38ca0711bf889750cf0e66.png)
![\[13-4=9\]](http://latex.artofproblemsolving.com/8/d/1/8d127b922e0e675e4f34eebea490cbd3c3d38322.png)
Thus, we realize that 
students must have their score increased by .
So, the correct answer is 
.
Video Solution (A Clever Explanation You’ll Get Instantly)
https://youtu.be/tYWp6fcUAik?si=V8hv_zOn_zYOi9E5&t=2364 ~hsnacademy
Video Solution 1
https://youtu.be/oUEa7AjMF2A?si=Mnek8vOzVX77YLIS&t=3389
~Math-X
Video Solution 2
~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
~savannahsolver
Video Solution by Dr. David
See Also
| 2022 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 18 |
Followed by Problem 20 | |
| 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 | ||
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. 
