We study the paths of minimal cost for first-passage percolation in the Z2 lattice. We obtain an exponential bound on the tail probability of the ratio of the length of the longest and shortest of these. This inequality permits us to answer a question of Hammersley and Welsh (1965) of differentiability of the time constant and to answer a question of Smythe and Wierman (1978) concerning the asymptotic behavior of the length of minimum-time paths. This is joint work with J. Michael Steele.