Understanding Pipe Filling Rates
First, let's determine the rate at which each pipe fills the tank. The rate is the fraction of the tank filled per hour.
- Rate of Pipe A = $ \frac{1}{8} $ tank per hour
- Rate of Pipe B = $ \frac{1}{12} $ tank per hour
- Rate of Pipe C = $ \frac{1}{24} $ tank per hour
Calculating Work Done in Stages
The filling process occurs in three distinct stages.
Stage 1: Pipe A Alone (1 hour)
Pipe A works alone for the first hour.
- Work done by A in 1 hour = Rate of A $ \times $ Time = $ \frac{1}{8} \times 1 = \frac{1}{8} $ tank.
- Tank capacity remaining = $ 1 - \frac{1}{8} = \frac{7}{8} $ tank.
Stage 2: Pipes A and B Together (2 hours)
Next, pipes A and B work together for 2 hours.
- Combined rate of A and B = $ \frac{1}{8} + \frac{1}{12} $.
- To add these fractions, find a common denominator (24): $ \frac{3}{24} + \frac{2}{24} = \frac{5}{24} $ tank per hour.
- Work done by A and B in 2 hours = Combined rate $ \times $ Time = $ \frac{5}{24} \times 2 = \frac{10}{24} = \frac{5}{12} $ tank.
- Total work done after 2 hours = Work from Stage 1 + Work from Stage 2 = $ \frac{1}{8} + \frac{5}{12} $.
- Common denominator (24): $ \frac{3}{24} + \frac{10}{24} = \frac{13}{24} $ tank.
- Tank capacity remaining = $ 1 - \frac{13}{24} = \frac{11}{24} $ tank.
Stage 3: Pipes A, B, and C Together
Finally, all three pipes work together until the tank is full.
- Combined rate of A, B, and C = $ \frac{1}{8} + \frac{1}{12} + \frac{1}{24} $.
- Common denominator (24): $ \frac{3}{24} + \frac{2}{24} + \frac{1}{24} = \frac{6}{24} = \frac{1}{4} $ tank per hour.
Determining Final Filling Time
Now, calculate the time needed for Stage 3 and the total time.
- Time required for Stage 3 = Remaining capacity / Combined rate of A, B, C
- Time for Stage 3 = $ \frac{11/24}{1/4} = \frac{11}{24} \times 4 = \frac{44}{24} = \frac{11}{6} $ hours.
- Total time to fill the tank = Time (Stage 1) + Time (Stage 2) + Time (Stage 3)
- Total time = $ 1 \text{ hour} + 2 \text{ hours} + \frac{11}{6} \text{ hours} = 3 + \frac{11}{6} $ hours.
- Total time = $ \frac{18}{6} + \frac{11}{6} = \frac{29}{6} $ hours.
Convert the total time into hours and minutes:
- $ \frac{29}{6} $ hours = 4 whole hours and $ \frac{5}{6} $ of an hour.
- $ \frac{5}{6} $ hour = $ \frac{5}{6} \times 60 $ minutes = 50 minutes.
- Therefore, the total time is 4 hours and 50 minutes.
Important Questions from Pipe and Cistern
A pipe can fill a cistern in 20 minutes where as the cistern when full can be emptied by a leak in 28 minutes. When both are opened, The time taken to fill the cistern is:
‘A’ pipe can empty a tank in 20 minutes. The second pipe ‘B’ has a diameter twice as that of ‘A’. If both A & B pipe are attached to the tank how much time will be required to empty the tank?
Pipes A and B can empty a full tank in 16 hours and 24 hours, respectively. Pipe C alone can fill the empty tank in 4 hours. If A, B and C are opened together, the tank will be 35% full after :
Pipes A and B can fill a tank in 36 minutes and 45 minutes, respectively. Both these pipes were opened simultaneously. After 20 minutes, a leak at the bottom of the tank was spotted which was immediately sealed. The tank was full in another 15 minutes. The leak alone can empty the full tank in:
A cistern has a leak which would empty it in 6 hours. A tap is turned on which admits 10 litres of water per minute into the cistern. When it is full it is now emptied in 10 hours. What is the capacity (in litres) of the cistern?
