2012 Cayley Problem 17

A string has been cut into 4 pieces, all of different lengths. The length of each piece is 2 times the length of the next smaller piece. What fraction of the original string is the longest piece?

\textbf{(A)}\ \frac{8}{15}\quad \textbf{(B)}\ \frac{2}{5}\quad \textbf{(C)}\ \frac{1}{2}\quad \textbf{(D)}\ \frac{6}{13}\quad \textbf{(E)}\ \frac{1}{4}

If there are no answer choices shown, enter a numerical answer.

Full credit to this problem is given to the CEMC, you may view all Cayley contests here.