Solving the Tank Filling Problem with Drainpipes
This problem involves calculating the number of drainpipes when the total number of pipes, their individual filling/draining rates, and the overall tank filling time are known.
1. Define Variables and Rates
- Let $N$ be the total number of pipes, $N=28$.
- Let $x$ be the number of filling pipes.
- Let $y$ be the number of drainpipes.
- We know $x + y = 28$.
- Rate of one filling pipe = $\frac{1}{14}$ tank per hour.
- Rate of one drainpipe = $\frac{1}{35}$ tank per hour.
2. Calculate Combined Work Rate
The total rate at which the tank fills is the combined rate of all filling pipes minus the combined rate of all drainpipes.
- Rate of $x$ filling pipes = $x \times \frac{1}{14} = \frac{x}{14}$ tank/hour.
- Rate of $y$ drainpipes = $y \times \frac{1}{35} = \frac{y}{35}$ tank/hour.
- Combined rate = $\frac{x}{14} - \frac{y}{35}$ tank/hour.
The tank is filled in $2.5$ hours, which means the net filling rate is:
Net Rate = $\frac{1 \text{ tank}}{2.5 \text{ hours}} = \frac{1}{5/2} = \frac{2}{5}$ tank/hour.
3. Formulate and Solve the Equation
Equating the combined rate to the net rate:
$ \frac{x}{14} - \frac{y}{35} = \frac{2}{5} $
Substitute $x = 28 - y$ into the equation:
$ \frac{28 - y}{14} - \frac{y}{35} = \frac{2}{5} $
Simplify the equation:
$ \left( \frac{28}{14} - \frac{y}{14} \right) - \frac{y}{35} = \frac{2}{5} $
$ 2 - \frac{y}{14} - \frac{y}{35} = \frac{2}{5} $
Rearrange the terms to solve for $y$:
$ 2 - \frac{2}{5} = \frac{y}{14} + \frac{y}{35} $
$ \frac{10 - 2}{5} = y \left( \frac{1}{14} + \frac{1}{35} \right) $
$ \frac{8}{5} = y \left( \frac{5}{70} + \frac{2}{70} \right) $
$ \frac{8}{5} = y \left( \frac{7}{70} \right) $
$ \frac{8}{5} = y \left( \frac{1}{10} \right) $
Solve for $y$:
$ y = \frac{8}{5} \times 10 = 8 \times 2 = 16 $
4. Conclusion
There are 16 drainpipes.
Similar Questions
- Pipe A can fill a tank in 12 hours, pipe B can fill the same tank in 22 hours and pipe C can fill the same tank in 8 hours. The time taken by them to fill the same tank if they operate together is:
- Pipe A can fill a tank in 18 hours, pipe B can fill the same tank in 28 hours and pipe C can fill the same tank in 11 hours. The time taken by them to fill the same tank if they operate together is:
- Two pipes can fill a cistern, individually, in 36 min and 45 min, 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 70 min. 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?
- Pipe A can fill a tank in 19 hours, pipe B can fill the same tank in 21 hours and pipe C can fill the same tank in 8 hours. The time taken by them to fill the same tank if they operate together is:
- Tap A can fill a tank in 6 h, whereas Tap B can fill it in 8 h. Tap C can empty the same tank in 4 h. If all the taps are opened together, how long will it take to fill the tank?
- An inlet pipe can fill a tank in 4 h and an outlet pipe can empty the tank in 6 h. By mistake, both the pipes are kept open. Find the number of hours in which the tank will be half-full.
- 6 pipes, all of same type, are required to fill a tank in 1 h 20 min. How long will it take if only 5 pipes of the same type are used?
- Pipe 1 can empty a tank in 6 h while pipe 2 can do so in 18 h. If both are working together, in how much time will they empty the full tank?
- Tap A can fill a tank in 40 min and Tap B can empty the tank in 60 min. If the taps are opened at the same time, then the time taken to fill the tank will be:
- Pipes A and B are fitted to a tank. A is the filling pipe and B can be used for filling or emptying at the same rate. When B is used for filling, it takes time 't' along with A to fill the tank. If it is used for emptying when A is filling the tank, the time taken for the tank to fill up would be '5t'. Find the ratio of the rates of A and B.
Important Questions from Pipe and Cistern
Pipes A, B and C can fill a tank in 20, 30 and 60 hours, respectively. Pipes A, B and C are opened at 7 a.m., 8 a.m., and 9 a.m., respectively, on the same day. When will the tank be full?
There are two water taps in a tank which can fill the empty tank in 12 hours and 18 hours respectively. It is seen that there is a leakage point at the bottom of the tank which can empty the completely filled tank in 36 hours. If both the water taps are opened at the same time to fill the empty tank and leakage point was repaired after 1 hour, then in how much time the empty tank will be completely filled?
Two pipes A and B can fill a tank in 12 minutes and 24 minutes, respectively, while a third pipe C can empty the full tank in 32 minutes. All the three pipes are opened simultaneously. However, pipe C is closed 2 minutes before the tank is filled. In how much time (in minutes) will the tank be full?
Pipes A and B can fill a tank in 12 hours and 16 hours respectively and pipe C can empty the full tank in 24 hours. All three pipes are opened together, but after 4 hours pipe B is closed. In how many hours, the empty tank will be completely filled?
Pipes A and B can fill a tank in 43.2 minutes and 108 minutes, respectively. Pipe C can empty it at 3 litres/minute. When all the three pipes are opened together, they fill the tank in 54 minutes. The capacity (in litres) of the tank is:
