Question

two cranes can unload a ship together in 6 hours. The faster crane in 5 hours less than the slower crane, if each unload the same ship alone. How long does it take for each crane to unload this ship?

Answer 1

Solution:

Let x be the time it takes for the slower crane to unload a ship.
So 1/x is the rate at which the slower crane works.
x-5 is the time it takes the faster crane to unloas a ship.
So 1/(x-5) is the rate at which the faster crane works.
The total rate of the two cranes when they work together is 1/x + 1/(x-5)
We also know that when they work together they unload a ship in 6 hours, so the rate they work together is 1/6.
The equation is then 1/x + 1/(x-5)=1/6
Multiply both sides by 6x(x-5):
6x-30 + 6x = x2-5x
Add 30-12x to both sides:
0=x2-17x+30
0=x2-15x-2x+30
0=x(x-15)-2(x-15)
0=(x-2)(x-15)

So x=2 or x=15, since x-5 is the time it takes for the faster crane to work alone, it can't be negative, so x=2 is ruled out leaving us with x=15 and x-5=10