SOLUTION: on a certain project the only grades awarded were 75 and 100. If 85 students completed the project and the average of their grades was 85, how many earned 100?

Algebra ->  Human-and-algebraic-language -> SOLUTION: on a certain project the only grades awarded were 75 and 100. If 85 students completed the project and the average of their grades was 85, how many earned 100?      Log On


   

Question 830910: on a certain project the only grades awarded were 75 and 100. If 85 students completed the project and the average of their grades was 85, how many earned 100?
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
Maybe there is a typo, but I will solve it as posted.

x= number of students who earned 100.
85-x= number of students who earned 70.
100x= sum of the grades of students who earned 100.
70%2A%2885-x%29= sum of the grades of students who earned 70.

The average can be calculated as
%28100x%2B70%2885-x%29%29%2F85
So our equation is
%28100x%2B70%2885-x%29%29%2F85=85

Solving:
%28100x%2B70%2885-x%29%29%2F85=85
100x%2B70%2885-x%29=85%2A85
100x%2B70%2A85-70x=7225
100x%2B5950-70x=7225
100x-70x=7225-5950
30x=1275
x=1275%2F30
x=42.5
42.5=42%261%2F2
What does it mean to find that the number of students who earned 100 is 42%261%2F2 ?
It could be that there is a typo and one of the numbers was wrong.
It could also be that the average was not exactly 85.00, but it was 85 after rounding. So maybe there were 42 students who earned a 100 in the project. Or maybe 43 students earned a 100 in the project.
The average can be calculated as

For x=42 that average is
%285950%2B30%2A42%29%2F85=%285950%2B1260%29%2F85=7210%2F85=84.82%22...%22
For x=43 that average is
%285950%2B30%2A43%29%2F85=%285950%2B1290%29%2F85=7240%2F85=85.17%22...%22
Both of those averages round to 85, so the number of students who earned a 100 in the project could have been highlight%2842%29 or highlight%2843%29 .
On the other hand, for x=41 we get an average that rounds down to 84,
and for x=44 we get an average that rounds up to 86.