¶ 2022 AMC8 Problem 19
Problem:
Mr. Ramos gave a test to his class of 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 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 20 test scores are arranged in increasing order.)
Answer Choices:
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B.
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E.
Solution:
If the scores are arranged in increasing order, the median will equal the average of the numbers in the and positions. The dot plot shows that after the initial round of grading, the middle scores were
so the median was . After regrading, the median increased to , so the scores in the and positions must have increased by points, along with the scores of in the next positions:
Other scores might have increased as well but those changes would not have affected the median.Therefore a minimum of students received extra points.
After the first round of grading, there were scores of or greater. As a result of the regrade,the median increased to , so the middle numbers must have been and the scores to the right must have been the same or greater. Therefore a minimum of students received extra points.
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions