Two cranes can unload a ship together in 15 hours. The faster crane takes 16 hours less than the slower crane, if each were to unload the same ship alone. How long does it take for each crane to unload a ship

Let's assume that the slower crane takes x hours to unload the ship on its own.

According to the given information, the faster crane takes 16 hours less than the slower crane to unload the ship alone. So, the faster crane would take (x - 16) hours to unload the ship on its own.

Now, let's calculate the rate at which each crane unloads the ship. The rate can be calculated by dividing the work (unloading the ship) by the time taken. Since both cranes working together can unload the ship in 15 hours, their combined rate is 1/15 of the ship per hour.

The rate of the slower crane would be 1/x, and the rate of the faster crane would be 1/(x - 16).

As both cranes are working together, their rates are added, so:

1/x + 1/(x - 16) = 1/15

To solve this equation, we need to find the common denominator for the fractions. In this case, the common denominator is 15x(x - 16). Multiplying every term in the equation by the common denominator, we get:

15(x - 16) + 15x = x(x - 16)

Now, let's simplify the equation:

15x - 240 + 15x = x^2 - 16x

Combine like terms:

30x - 240 = x^2 - 16x

Rearrange the equation:

x^2 - 46x + 240 = 0

Now, we can solve this quadratic equation to find the value of x.

Using factoring or the quadratic formula, we find that x = 40 hours or x = 6 hours. Since we are looking for positive values of x, we can discard the x = 6 hours solution.

Therefore, the slower crane takes 40 hours to unload the ship on its own, and the faster crane takes 40 - 16 = 24 hours to unload the ship on its own.