AMC 12 2005a Problem 6

Josh and Mike live $13$ miles apart. Yesterday Josh started to ride his bicycle toward Mike's house. A little later Mike started to ride his bicycle toward Josh's house. When they met, Josh had ridden for twice the length of time as Mike and at four-fifths of Mike's rate. How many miles had Mike ridden when they met?

$\textbf{(A) } 4 \textbf{(B) } 5 \textbf{(C) } 6 \textbf{(D) } 7 \textbf{(E) } 8$