Next-door neighbors Bob and Jim use hoses from both houses to fill Bob's swimming pool. They know that it takes 12 h using both hoses. They also know that Bob's hose, used alone, takes 20% less time than Jim's hose alone. How much time is required to fill the pool by each hose alone?

We have been given that Bob's hose, used alone, takes 20% less time than Jim's hose alone. This means that Bob's hose take will take 80% time of Jim's hose that is [tex]0.8t[/tex].

Part of pool filled by Bob's hose in 1 hour would be [tex]\frac{1}{0.8t}[/tex].

We are also told that it takes 12 hours using both hoses. Part of pool filled by both hoses in 1 hour would be [tex]\frac{1}{12}[/tex].

Now, we will add rates of both hoses and equate with [tex]\frac{1}{12}[/tex] as:

[tex]\frac{1}{t}+\frac{1}{0.8t}=\frac{1}{12}[/tex]

[tex]\frac{0.8}{0.8t}+\frac{1}{0.8t}=\frac{1}{12}[/tex]

[tex]\frac{0.8+1}{0.8t}=\frac{1}{12}[/tex]

[tex]\frac{1.8}{0.8t}=\frac{1}{12}[/tex]

Cross multiply:

[tex]0.8t*1=1.8(12)[/tex]

[tex]0.8t=21.6[/tex]

[tex]\frac{0.8t}{0.8}=\frac{21.6}{0.8}[/tex]

[tex]t=27[/tex]

Therefore, it will take 27 hours for Jim's hose to fill the pool alone.

Time take by Jim's hose alone: [tex]0.8t\Rightarrow 0.8(27)=21.6[/tex].

Therefore, it will take 21.6 hours for Bob's hose to fill the pool alone.