Difference between revisions of "2022 AMC 8 Problems/Problem 19"
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Case #<math>1</math>: The middle two scores are <math>80</math> and <math>90</math>. To do this, we firstly suppose that the two students who got <math>85</math> are awarded the extra <math>5</math> points. We then realize that this case will have a lot of students who receive the extra points, therefore we reject this case. | Case #<math>1</math>: The middle two scores are <math>80</math> and <math>90</math>. To do this, we firstly suppose that the two students who got <math>85</math> are awarded the extra <math>5</math> points. We then realize that this case will have a lot of students who receive the extra points, therefore we reject this case. | ||
− | Case #<math>2</math>: The middle two scores are both <math>85</math>. To do this, we simply need to suppose that some of the students who got <math>80</math> are awarded the extra <math>5</math> points. Note that there are 8 students who got <math>75</math> or less. Therefore there must be only <math>1</math> student who got <math>80</math> so that the middle two scores are both <math>85</math>. Therefore our answer is <math>\boxed{\textbf{(C) }4}</math>. | + | Case #<math>2</math>: The middle two scores are both <math>85</math>. To do this, we simply need to suppose that some of the students who got <math>80</math> are awarded the extra <math>5</math> points. Note that there are <math>8</math> students who got <math>75</math> or less. Therefore there must be only <math>1</math> student who got <math>80</math> so that the middle two scores are both <math>85</math>. Therefore our answer is <math>\boxed{\textbf{(C) }4}</math>. |
Revision as of 18:54, 28 January 2022
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
test scores are arranged in increasing order.)
Solution
The median is the average of the and
scores. In order to minimize the number of changed scores, the
score must be
. It is easiest to move the people whose scores are
since it only takes
move.
In progress please do not edit
Solution 2
Before Mr. Ramos added scores, the median was . There are two cases now:
Case #: The middle two scores are
and
. To do this, we firstly suppose that the two students who got
are awarded the extra
points. We then realize that this case will have a lot of students who receive the extra points, therefore we reject this case.
Case #: The middle two scores are both
. To do this, we simply need to suppose that some of the students who got
are awarded the extra
points. Note that there are
students who got
or less. Therefore there must be only
student who got
so that the middle two scores are both
. Therefore our answer is
.