2011 AMC 10B Problem 2

Averages are easiest to work with when you turn them into totals, so find the totals.

1. Her five scores add up to $90 + 80 + 70 + 60 + 85 = 385$.
2. Her average now is $385 \div 5 = 77$.
3. She wants to raise it by $3$, to $80$.
4. An average of $80$ over six tests means the six scores add up to $6 \times 80 = 480$.
5. She already has $385$ points, so the sixth test must give at least
$$480 - 385 = 95.$$
6. Check: $(385 + 95) \div 6 = 480 \div 6 = 80$. That is exactly $3$ above $77$.

The answer is $\boxed{\textbf{(E)}\ 95}$.

"The average of $6$ tests is $80$" means the same thing as "the $6$ tests add up to $480$." Totals can be added and subtracted; averages cannot. Another way to see the answer: the new score has to be $77$ plus enough extra to raise each of the six scores by $3$, which is $77 + 6 \times 3 = 95$.