question in description

Question

a small pump can drain a pool in 8 hours. a large pump could drain the same pool in 7 hours. how long (to the nearest minute) will it take to drain the pool if both pumps are used simultaneously?

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

 

Plugging in the known information, we get:

1/8 + 1/7 = 1/T

 

The LCM, 56, will be the common denominator on the left side.  This will allow us to add the fractions together and simplify the left side. We get:

 

(7/7) 1/8 + (8/8) 1/7 = 1/T

7/56 + 8/56 = 1/T

15/56 = 1/T

 

Now we have a proportion and we can solve for T by using cross multiplication.  This gives us:

(15)(T) = (56)(1)

15T = 56

 

To isolate "T," we must divide both sides of the equation by 15.  This gives us:

 

T = 56/15 hours

T = 3.733333333333333 hours

 

We know that the answer is about 3 hours and 45 minutes because 0.75 hours = 45 minutes.  To get the answer exactly to the nearest minute, remem that there are 60 minutes in an hour. Multiply 

0.733333333333333 by 60 to figure out how many minutes that is.  

 

(60)(0.733333333333333) = 44 minutes 

 

So, the answer is 3 hours and 44 minutes.

 

If you want the answer in strictly minutes, add the 44 minutes to 180 minutes (3 hours).  You'll get 224 minutes.

 

So, it takes 3.73 hours aka 3 hours 44 minutes aka 224 minutes to drain the pool when both pumps work together.

 

 

 

Answer 2

Small pump can drain a pool in 8 hours  ==>  in 1 hour small pump drains 1/8 of the pool  

Large pump can drain a pool in 7 hours  ==>  in 1 hour large pump drains 1/7 of the pool  

 

So in 1 hour both pumps together will drain (1/8 + 1/7) of the pool 

     in T hours both pumps will drain T*(1/8 + 1/7) of the pool  

 

The pool will be completely drained when T*(1/8 + 1/7) = 1  

 

T*(7/56 + 8/56) = 1

T*(15/56) = 1

T = 56/15 = 3 11/15 hours

 

(11/15) hours * (60 minutes / 1 hour) = 4*11 minute = 44 minutes 

 

So T = 3 hours 44 minutes