Question

A garden hose can fill a swimming pool in 7 days, and a larger hose can fill the pool in 4 days. How long will it take to fill the pool if both hoses are used?

Answer 1

Hose 1 takes 7 days.

Hose 2 takes 4 days.

That means in a single day, Hose 1 fills 1/7 of the pool and Hose 2 fills 1/4 of the pool. So in one day, the two hoses will fill 1/7 + 1/4 of the pool.

If t is the number of days it takes to fill the pool with both hoses being used, then

`t\cdot \left((1)/(7) + (1)/(4)\right) =1`

will tell us how many days it will take to fill 1 pool

Solving this:

`\begin{aligned} t\cdot \left((1)/(7)\cdot(4)/(4) + (1)/(4)\cdot (7)/(7)\right) &=1 \\[0.5em]t\cdot \left((4)/(28) + (7)/(28)\right) &=1 \\[0.5em]t\cdot \left((11)/(28) \right) &=1 \\[0.5em]t &=(28)/(11) \\[0.5em]\end{aligned}`

It'll take 28/11 days, or 2 and 6/11 days, or about 2.54 days.