SOLUTION: Two taps can each fill a tank in 10 and 15 minutes respectively. Both were opened simultaneously and at the time the tank was supposed to be full it was found tha the drain pip

Question 24157:
Two taps can each fill a tank in 10 and 15 minutes respectively. Both were opened simultaneously and at the time the tank was supposed to be full it was found tha the drain pipe was also open. It was then closed and the tank became full in another 2 minutes. In how manutes will the drain pipe alone empty the tank?

Answer by AnlytcPhil(1810) (Show Source): You can put this solution on YOUR website!

Two taps can each fill a tank in 10 and 15 minutes respectively. Both were
opened simultaneously and at the time the tank was supposed to be full it was
found tha the drain pipe was also open. It was then closed and the tank became
full in another 2 minutes. In how manutes will the drain pipe alone empty the
full tank? 

Let t = how many minutes the drain pipe alone can empty
the full tank.

Therefore the drain rate of the pipe is 1 tank per t minutes
or the fraction 1/t tanks per minute.

The fill rate of the 1st tap is 1 tank per 10 minutes, or
(1 tank)/(10 minutes) or 1/10 tank/minute

The fill rate of the 2nd tap is 1 tank per 15 minutes, or
(1 tank)/(15 minutes) or 1/15 tank/minute

The combined rate of two taps = (1/10 + 1/15) tank/minute
= (3/30 + 2/30) tank/minute = 5/30 tank/minute = 1/6 tank/minute.

Therefore if the drain had been closed the two taps would 
have filled 1/6 of the tank in 1 minute or the whole tank 
in 6 minutes.

>>...Both were opened simultaneously and at the time the tank was supposed to
be full it was found that the drain pipe was also open...<<

We now know it was 6 minutes later that the drain was 
discovered to be open. 

>>...(the drain) was then closed and the tank became full in another 2 minutes...<<

So it took 6+2 or 8 minutes total to fill the tank, starting
at the very beginning.

During each of those 8 minutes the first tap filled
1/10 of the tank; therefore in all 8 minutes it filled
8/10 or 4/5 of the tank.

During each of those 8 minutes the second tap filled 
1/15 of the tank; therefore in all 8 minutes it filled 
8/15 of the tank.

During each of the first 6 of those 8 minutes, the drain 
took out the fraction 1/t of the tank; therefore in all
6 of those minutes it was open, the drain pipe took out
the fraction 6/t of the tank.  

The equation is gotten from this reasoning:

Fraction of      Fraction of     Fraction of 
tank 1st tap     tank 2nd tap    tank the drain  
put in        +  put in       -  took out        = 1 full tank
during those     during those    during the
8 mins.          8 mins.         first 6 of 
                                 those 8 mins.

 4/5         +     8/15      -      6/t          = 1

                    4/5 + 8/15 - 6/t = 1  

We solve that and get 18 minutes.

Edwin
AnlytcPhil@aol.com

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