The problem states: On a daily homework assignment the student has earned the maximum score of 10 on 15 of 40 assignments. The mode of the 40 scores is 7, and the median is 9. What is the least arithmetic mean could be? Do I start with a ratio of scores or an algabraic expression?

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mrcard wrote:On a daily homework assignment the student has earned the maximum score of 10 on 15 of 40 assignments. The mode of the 40 scores is 7, and the median is 9. What is the least arithmetic mean could be?

That's an interesting exercise....

Okay, you know the student got 10 points each on 15 of the 40 papers. On the other twenty-five papers, he got less. Since the mode is 7, then this was the most-common score, so he got at least eleven papers scored at 7 points each. This leaves, at most, sixteen other papers.

Since the median is 9, then the two middle values in the ordered (of which 9 is the average) are either 7 and 11 (but this isn't possible) or 8 and 10 (but this would mean that the top 11 scores were 10, then an 8, and then 7s, so the median would have been 7) or 9 and 9. Since the other options don't work, then the middle two scores must be 9 and 9.

So we have the top 20 scores being fifteen 10s and at least six 9s (so the twentieth and twenty-first values are 9).

We have to have at least eleven 7s, but we're trying to minimize the arithmetic mean. So try to find a combination of scores which includes the above and also has the lowest average you can manage....