The final counts as 2/3 the grade so the other 4 tests count 1/3. Thus:
where g is the overall grade and f is the score on the final.
Let me explain this beast a little. This is something called a "weighted" average. We are weighting the first 4 scores with a value of 1/3 and the last score with a value of 2/3. The numerator represents the total number of points actually acquired, and the denominator represents the total number of points possible. That last 100 I tacked on at the end merely converts the average to a percentage.
a) We wish to find a value of f for g = 80.
Let's get the denominator first, since this is just a number:
Multiply both sides by 200/100:
So she needs to get a 70 on the final.
b) g = 90. Do the same thing. I get f = 100.