Answer:
84 <= x <= 102
Step-by-step explanation:
Fiona scored 76 on her 1st midterm. Lets call x the score she will get on her 2nd exam.
We know she needs between 80 and 89 inclusive so get a b. So the question is, how much does she need to get on the 2nd test? Or, what values x can take for her to get a b?
The average she gets between both test is the arithmetic mean, this is, the sum of both tests divided by 2:
(76+x)/2
So, she need this value to be between 80 and 89. Lets use inequalities:
80 <= (76+x)/2 <= 89
We can solve it in a similar way we do for equalities. Lets start by multiplying terms in every side by 2, to eliminate the 2 dividing the central term:
2*80 <= 76+x <= 89*2
160 <= 76+x <= 178
Now, lets subtract 76 in every side so we get x alone in the center:
160 - 76 <= x <= 178 - 76
84 <= x <= 102
This is, for getting a b she need her second score to be equal or greater than 84 and less or equal to 102.
Lets verify for some values. For example, if she gets 88:
(76+84)/2 = 160/2 =80---> verifies!
And any value greater than 84 will give her a better score. Lets see what happens with 102"
(76+102)/2 = 178/2 = 89 ---> verifies
Any score greater than 102 will giver here a total sore greater than 89 so it won't be a b.
So, if she scores 102 she will get 89 (the statement does not say the range of scores, it is not usually common that the pass 100, but as the statement does not restrict it we consider here. We the maximum score is 100 we will use 84 <= x <= 100).
. Filling in accordingly, 
. This simplifies to 
and 36=6x. Therefore, x = 6 so the bases are 6 and 12.