AMC 8 2007 Problem 20

Before the district play, the Unicorns had won $45$% of their basketball games. During district play, they won six more games and lost two, to finish the season having won half their games. How many games did the Unicorns play in all? $\textbf{(A)}\ 48 \textbf{(B)}\ 50 \textbf{(C)}\ 52 \textbf{(D)}\ 54 \textbf{(E)}\ 60$