Question 830910
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*(85-x)}}}= sum of the grades of students who earned 70.
 
The average can be calculated as
{{{(100x+70(85-x))/85}}}
So our equation is
{{{(100x+70(85-x))/85=85}}}
 
Solving:
{{{(100x+70(85-x))/85=85}}}
{{{100x+70(85-x)=85*85}}}
{{{100x+70*85-70x=7225}}}
{{{100x+5950-70x=7225}}}
{{{100x-70x=7225-5950}}}
{{{30x=1275}}}
{{{x=1275/30}}}
{{{x=42.5}}}

{{{42.5=42&1/2}}} 
What does it mean to find that the number of students who earned 100 is {{{42&1/2}}} ?
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
{{{(100x+70(85-x))/85=(100x+70*85-70x)=(100x+5950-70x)/85=(5950+30x)/85}}}
For {{{x=42}}} that average is
{{{(5950+30*42)/85=(5950+1260)/85=7210/85=84.82}}}{{{"..."}}}
For {{{x=43}}} that average is
{{{(5950+30*43)/85=(5950+1290)/85=7240/85=85.17}}}{{{"..."}}}
Both of those averages round to {{{85}}}, so the number of students who earned a 100 in the project could have been {{{highlight(42)}}} or {{{highlight(43)}}} .
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.