How long to fill the pool with just the hose?

Question

It takes 12 hours to fill a pool using a pipe. It takes 8 4/7 hours to fill it using a pipe and a hose. How long will it take to fill the pool using just the hose?

Answer 1

Combined work formula:
1/t₁ + 1/t₂ + 1/t₃ ... = 1/T

Where:
(1) t₁ is the amount of time it takes machine 1 (or person 1, or whatever) to do the job
(2) t₂ is the amount of time for machine 2 (or person 2, or whatever) to do the job
(3) t₃ is the amount of time for machine 3 (or person 3, or whatever) to do the job
(4) T is the amount of time it takes for the machines/people to complete the job while working simultaneously

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This formula can include as many people/machines/etc as needed, but this question only requires the formula as such:

1/t₁ + 1/t₂ = 1/T

 

Where: 

(1) t₁ is the amount of time it takes the pipe to fill the pool
(2) t₂ (our unknown) is the amount of time for the hose to fill the pool
(3) T is the amount of time it takes for both the hose & the pipe to fill the pool (working simultaneously)

 

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Plugging in the known information, and allowing using "x" to represent t₂, we get:
1/12 + 1/x = 1 / (8 4/7)

 

We can convert 8 4/7 to an improper fraction to simplify things. 

8 4/7 = 60/7

 

This makes our formula:

1/12 + 1/x = 1/(60/7)

1/ (60/7) is a complex fraction (a fraction within a fraction) and is equivalent to 1 x 7/60. [Remember that when dividing fractions, you flip the second fraction and change the division sign to multiplication.]

 

 

This makes our formula:
1/12 + 1/x = 7/60

 

Now we have a rational equation to solve.  The best way to solve these is by by multiplying both sides of the equation by the LCM of all of the denominators in the equation (12, x, and 60 in our case).  Our LCM is 60x.  Multiplying both sides of the equation by 60x will eliminate all of our denominators.

 

[60x] (1/12 + 1/x) = (7/60) [60x]

 

Don't forget to distribute on the left side of the equation.

 

(60x)/12 + (60x)/x = [(7)(60)(x)]/[60]

 

Many things will cancel out, leaving us with this:

5x + 60 = 7x

 

To solve for x, subtract 5x from both sides of the equation:

60 = 2x

 

Then divide both sides of the equation by 2 to completely isolate x.

30 = x


Remember that I used "x" to represent t₂, which was the time that it would take the hose alone to fill the pool.  Therefore, it takes 30 hours to fill the pool if you use the hose alone.

 

A similar problem on combined work rates can be found here...but it is a bit easier as the problem requires finding T (combined rate), rather than one of the individual rates (like your question asked you to do).

https://www.wyzant.com/resources/answers/94455/question_in_description

 

Answer 2

Let x = time (in hours) it takes for the hose to fill the pool alone

 

1/12 = amount of pool filled per hour by the pipe

1/x = amount of pool filled per hour by the hose

1/(8 4/7) = amount of pool filled per hour by the pipe and hose

 

Then, 1/12 + 1/x = 1/(8 4/7)

 

        1/12 + 1/x = 7/60

 

Multiply both sides by 60x:  5x + 60 = 7x

 

                                               2x = 60

 

                                                 x = 30 hours