Question

Mr. Ramos gave a test to his class of $20$ students. The dot plot below shows the distribution of test scores.

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.)

$2$

$3$

$4$

$5$

$6$

Answer: C

Concepts:median (data)data and graph interpretation

Difficulty rating: 1370

Solution:

All scores are multiples of $5,$ and adding $5$ preserves that fact. If the $10$ th and $11$ th scores average $85$ but are not both $85,$ then the $11$ th score must be at least $90.$ That would require at least $10$ scores of $90$ or more.

However, even after regrading, only the original scores of $85$ or more can reach $90.$ The plot has only $7$ such scores, so the two middle scores must both be $85.$

The plot initially has $7$ scores of at least $85.$ To make the $10$ th and $11$ th scores both $85,$ there must be at least $11$ scores of at least $85.$ Raising four of the $80$ 's to $85$ is both necessary and sufficient.

Thus, the correct answer is C.

2022 AMC 8 Problem 19

Answer: C

Difficulty rating: 1370

Solution:

Problem 19 in Other Years