Question

A swimming pool can be emptied in 6 hours using a 10-horsepower pump along with a 6-horsepower pump. The 6-horsepower pump requires 5 hours more than the 10-horsepower pump to empty the pool when working by itself. How long will it take to empty the pool using just the 10-horsepower pump?

Answer 1

Answer: 10 hours

Explanation:

The 10hp pump takes x hours to empty the pool which means it gets
`(1)/(x)` of the job done in one hour.

The 6hp pump takes x+5 hours to empty the pool which means it gets
`(1)/(x+5)` of the job done in one hour.

Together, they can get
`(1)/(x)+(1)/(x+5)` of the job done in one hour.

It is given that together they get the job done in 6 hours which means they get
`(1)/(6)` of the job done in one hour.

10 hp pump + 6 hp pump = Together

`(1)/(x)\quad +\quad (1)/(x+5)\quad =\quad (1)/(6)`

Multiply by 6x(x+5) to eliminate the denominator:

`(1)/(x)(6x)(x+5) +(1)/(x+5)(6x)(x+5) = (1)/(6)(6x)(x+5)`

Simplify and solve for x:

6(x + 5) + 6x = x(x + 5)

6x + 30 + 6x = x² + 5x

12x + 30 = x² + 5x

0 = x² - 7x - 30

0 = (x - 10)(x + 3)

0 = x - 10 0 = x + 3

10 = x -3 = x

Since the number of hours cannot be negative, disregard x = -3.

So, the only valid answer is x = 10.