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 :
2 \(\frac{2}{5}\) hours
Understanding Pipe Filling and Emptying Rates
This problem involves calculating the time taken for multiple pipes, some filling and some emptying a tank, to reach a specific level (35% full). We need to determine the individual rates of each pipe and then their combined rate.
Calculating Individual Pipe Rates
The rate of a pipe is the fraction of the tank it can fill or empty in one unit of time (in this case, one hour).
- Pipe A empties the tank in 16 hours. Its rate is negative because it empties the tank. Rate of Pipe A = $-\frac{1}{16}$ tank per hour.
- Pipe B empties the tank in 24 hours. Its rate is also negative. Rate of Pipe B = $-\frac{1}{24}$ tank per hour.
- Pipe C fills the empty tank in 4 hours. Its rate is positive because it fills the tank. Rate of Pipe C = $+\frac{1}{4}$ tank per hour.
Calculating the Combined Rate
When all three pipes are opened together, their rates add up. The combined rate tells us how much of the tank is filled or emptied per hour when all pipes operate simultaneously.
Combined Rate = Rate of A + Rate of B + Rate of C
Combined Rate = $\left(-\frac{1}{16}\right) + \left(-\frac{1}{24}\right) + \left(+\frac{1}{4}\right)$
To add these fractions, we find a common denominator. The least common multiple (LCM) of 16, 24, and 4 is 48.
Combined Rate = $-\frac{1 \times 3}{16 \times 3} - \frac{1 \times 2}{24 \times 2} + \frac{1 \times 12}{4 \times 12}$
Combined Rate = $-\frac{3}{48} - \frac{2}{48} + \frac{12}{48}$
Combined Rate = $\frac{-3 - 2 + 12}{48}$
Combined Rate = $\frac{7}{48}$ tank per hour.
Since the combined rate is positive ($\frac{7}{48}$), the pipes working together will eventually fill the tank.
Calculating Time to Fill 35% of the Tank
We need to find the time it takes to fill 35% of the tank. First, let's express 35% as a fraction:
35% = $\frac{35}{100} = \frac{7}{20}$
The time taken to fill a certain fraction of the tank is calculated as: Time = $\frac{\text{Fraction of Tank}}{\text{Combined Rate}}$
Time = $\frac{\frac{7}{20} \text{ tank}}{\frac{7}{48} \text{ tank per hour}}$
Time = $\frac{7}{20} \times \frac{48}{7}$ hours
The 7s cancel out:
Time = $\frac{48}{20}$ hours
Now, simplify the fraction $\frac{48}{20}$ by dividing both numerator and denominator by their greatest common divisor, which is 4:
Time = $\frac{48 \div 4}{20 \div 4} = \frac{12}{5}$ hours
To express this as a mixed number:
Time = $2 \frac{2}{5}$ hours
Therefore, if pipes A, B, and C are opened together, the tank will be 35% full after $2 \frac{2}{5}$ hours.
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 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?
A tank has two inlets, A and B, which can fill it in 15 hours and 20 hours, respectively. An outlet C can empty the full tank in 12 hours. If A, B and C are opened together when the tank is empty, then in how much time will the tank be filled?
