John and Matt are going to fill a pool with 2 different sized hoses. John can fill the pool in 5 hours, while Matt can complete it in 10 hours. Explain each step in solving this equation. (10 points)

We have to find how much time will be consumed if both John and Matt fill the pool together and show the steps to reach at the answer.

Solution:

John fill the pool in 5 hours. This means in 1 hour John fills [tex] \frac{1}{5} [/tex] part of the pool. Matt fills the pool in 10 hours. This means in 1 hour Matt fill [tex] \frac{1}{10} [/tex] part of the pool.

If they both work together:
Portion of the pool filled in 1 hour = [tex] \frac{1}{5}+ \frac{1}{10}= \frac{3}{10} [/tex]

Next we can use proportion to solve this problem.

3/10 portion of the pool is filled in 1 hour.

3/10 of the pool is filled in time ⇒ 1 hour
Multiplying both sides by 10/3 we get

3/10 x 10/3 of the pool in time = 1 x 10/3

1 pool is filled in time = 10/3 hours

This means, if both John and Matt fill the pool together they will take 10/3 hours which is 3 hours and 20 minutes to fill the pool.

A shortcut to solve such questions is:

Time taken to fill the pool = (Product of individual times)/(Sum of individual times)

Time taken to fill the pool = [tex] \frac{5(10)}{5+10}= \frac{50}{15}= \frac{10}{3} [/tex]

This results in the same answer as we achieved above.