Pipe P can fill $\frac{2}{3}$ part of a tank in 16 hours, and pipe Q can fill $\frac{5}{6}$ part of the same tank in 20 hours. Both P and Q were kept open for 2 hours, then both were closed. Pipe R alone was then opened and it emptied the water in the tank in 8 hours. Pipes P, Q and R together can fill the empty tank in:
The correct answer is 16 hours.
Step-by-Step Explanation
1. Find the time taken by P and Q to fill the tank individually:
Pipe P: $\frac{2}{3}$ of the tank is filled in 16 hours.
Full tank time = $16 \times \frac{3}{2} = \mathbf{24 \text{ hours}}$.
Rate of P = $\frac{1}{24}$ tank/hour.
Pipe Q: $\frac{5}{6}$ of the tank is filled in 20 hours.
Full tank time = $20 \times \frac{6}{5} = \mathbf{24 \text{ hours}}$.
Rate of Q = $\frac{1}{24}$ tank/hour.
2. Calculate the amount filled in the first 2 hours:
Combined rate of P and Q = $\frac{1}{24} + \frac{1}{24} = \frac{2}{24} = \frac{1}{12}$ tank/hour.
Amount filled in 2 hours = $\frac{1}{12} \times 2 = \mathbf{\frac{1}{6} \text{ of the tank}}$.
3. Find the emptying rate of Pipe R:
Pipe R empties that $\frac{1}{6}$ part in 8 hours.
Rate of R = $\frac{1/6}{8} = \mathbf{\frac{1}{48} \text{ tank/hour}}$ (emptying).
4. Find the time when P, Q, and R work together:
Combined rate = Rate(P) + Rate(Q) $-$ Rate(R)
Combined rate = $\frac{1}{24} + \frac{1}{24} - \frac{1}{48}$
Combined rate = $\frac{2}{24} - \frac{1}{48} = \frac{1}{12} - \frac{1}{48}$
Combined rate = $\frac{4 - 1}{48} = \frac{3}{48} = \mathbf{\frac{1}{16} \text{ tank/hour}}$.
5. Final Result:
Time taken to fill the tank = $\frac{1}{1/16} = \mathbf{16 \text{ hours}}$.
Important Questions from Pipe and Cistern
- Pipe A can fill a tank in 18 hours, pipe B can fill the same tank in 30 hours and pipe C can fill the same tank in 18 hours. The time taken by them to fill the same tank if they operate together is::
Pipe P can fill $\frac{2}{3}$ part of a tank in 24 hours, and pipe Q can fill $\frac{2}{3}$ part of the same tank in 16 hours. Both P and Q were kept open for 4 hours, then both were closed. Pipe R alone was then opened and it emptied the water in the tank in 5 hours. Pipes P, Q and R together can fill the empty tank in:
- Pipe P can fill $\frac{5}{7}$ part of a tank in 30 hours, and pipe Q can fill $\frac{2}{5}$ part of the same tank in 28 hours. Both P and Q were kept open for 7 hours, then both were closed. Pipe R alone was then opened and it emptied the water in the tank in 8 hours. Pipes P, Q and R together can fill the empty tank in:
Pipe P can fill $\frac{3}{8}$ part of a tank in 21 hours and pipe Q can fill $\frac{2}{7}$ part of the same tank in 16 hours. Both P and Q were kept open for 2 hours, then both were closed. Pipe R alone was then opened and it emptied the water in the tank in 3 hours. Pipes P, Q and R together can fill the empty tank in:
- Two pipes can fill a cistern, individually, in 40 minutes and 14 minutes, respectively. There is a pipe located at the bottom of the cistern to empty it. If all the three pipes are opened simultaneously, then the empty cistern gets filled in 20 minutes. How long will the pipe at the bottom of the tank take to empty the completely filled cistern if no other pipe is then open?
