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.)

$2$

$3$

$4$

$5$

$6$

Solution:

From the dot plot, there are $7$ scores at least $85$ : two $85$ 's, three $90$ 's, one $95$ , and one $100$ . For the median to be $85$ , the $10$ th and $11$ th scores in increasing order must both be at least $85$ , so at least $11$ scores must be at least $85$ .

Each regraded student can gain only $5$ points, so the only scores below $85$ that can become at least $85$ are the $80$ 's. There are enough $80$ 's, and we need $11-7=4$ more scores at least $85$ . Regrading four students who originally scored $80$ achieves this.

Thus, the correct answer is C.

2022 AMC 8 Problem 19

Solution:

Problem 19 in Other Years