A softball team played ten games, scoring $1,2,3,4,5,6,7,8,9$, and $10$ runs. They lost by one run in exactly five games. In each of the other games, they scored twice as many runs as their opponent. How many total runs did their opponents score? 
$\textbf {(A) } 35 \qquad \textbf {(B) } 40 \qquad \textbf {(C) } 45 \qquad \textbf {(D) } 50 \qquad \textbf {(E) } 55$
To score twice as many runs as their opponent, the softball team must have scored an even number.
Therefore we can deduce that when they scored an odd number of runs, they lost by one, and when they scored an even number of runs, they won by twice as much.
Therefore, the total runs by the opponent is $(2+4+6+8+10)+(1+2+3+4+5) = 45$, which is $C$