2cranes can unload a ship together in 4hrs. The faster crane takes 6hrs less than the slowr crane, if each were to unload the same ship alone. How long does it take for each crane to unload this ship?

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Together, they can do 1 ship in 4 hours: 1/4Slow Crane can do 1 ship in x hours: 1/xFast Crane can do 1 ship in (x - 6) hours: 1/(x - 6)For our equation we have their separate rates adding together to equal their Together rate:1/x + 1/(x - 6) = 1/4Multiply both sides by the LCD of these fractions, which is 4x(x - 6).  Then we have:4(x - 6) + 4x = x(x - 6)Distribute.4x - 24 + 4x = x^2 - 6xCombine like terms.8x - 24 = x^2 - 6xGet all terms to one side and 0 on the other.0 = x^2 - 14x + 24Factor the right side.0 = (x - 12)(x - 2)Set both of these factors equal to 0 and solve for the two x values and you get 12 and 2.  We know that the slow one can't possibly load a ship in two hours because that would be less time than they can do it together.  So, the slow one takes 12 hours and the fast one takes: x - 6 = 12 - 6 = 6 hours.  Let's check these solutions by plugging them into the original equation:1/12 + 1/6 = 1/41/12 + 2/12 = 1/43/12 = 1/41/4 = 1/4Checks out.