Two pipes fill a tank when working individually in 25 and 40 hours, respectively while a third pipe can drain the filled tank in 16 hours. If all the three pipes are turned on at the same time when the tank is empty, how long will it take to fill the tank completely?
Understanding Pipe and Tank Filling Problems
This problem involves calculating the time taken to fill a tank when multiple pipes are working simultaneously, some filling the tank and others draining it. The key concept here is understanding the work rate of each pipe. The work rate is the amount of work done per unit of time. In this case, the "work" is filling (or draining) the tank, and the unit of time is hours.
Calculating Individual Work Rates
If a pipe can fill a tank in $T$ hours, its work rate is $\frac{1}{T}$ of the tank filled per hour. If a pipe can drain a tank in $T$ hours, its work rate is $-\frac{1}{T}$ of the tank drained per hour (negative to indicate draining).
- Pipe 1 fills the tank in 25 hours. Its filling rate is $\frac{1}{25}$ tank per hour.
- Pipe 2 fills the tank in 40 hours. Its filling rate is $\frac{1}{40}$ tank per hour.
- Pipe 3 drains the tank in 16 hours. Its draining rate is $-\frac{1}{16}$ tank per hour.
Calculating Combined Work Rate
When all three pipes are turned on at the same time, their work rates are combined. Filling rates are added, and draining rates are subtracted.
Combined rate = (Rate of Pipe 1) + (Rate of Pipe 2) + (Rate of Pipe 3)
Combined rate = $\frac{1}{25} + \frac{1}{40} - \frac{1}{16}$ tank per hour.
To add and subtract these fractions, we find a common denominator for 25, 40, and 16.
- Prime factorization of 25 is $5^2$.
- Prime factorization of 40 is $2^3 \times 5$.
- Prime factorization of 16 is $2^4$.
The least common multiple (LCM) of 25, 40, and 16 is $2^4 \times 5^2 = 16 \times 25 = 400$.
Now, we express each fraction with the denominator 400:
- $\frac{1}{25} = \frac{1 \times 16}{25 \times 16} = \frac{16}{400}$
- $\frac{1}{40} = \frac{1 \times 10}{40 \times 10} = \frac{10}{400}$
- $-\frac{1}{16} = -\frac{1 \times 25}{16 \times 25} = -\frac{25}{400}$
Combined rate = $\frac{16}{400} + \frac{10}{400} - \frac{25}{400} = \frac{16 + 10 - 25}{400} = \frac{26 - 25}{400} = \frac{1}{400}$ tank per hour.
Calculating Total Time to Fill the Tank
The total time taken to fill the tank is the reciprocal of the combined work rate.
Time = $\frac{1}{\text{Combined rate}} = \frac{1}{\frac{1}{400}}$ hours.
Time = 400 hours.
Converting Hours to Days and Hours
We need to convert 400 hours into days and hours. There are 24 hours in a day.
Divide 400 by 24:
$400 \div 24$
$400 = 24 \times 16 + 16$
This means 400 hours is equal to 16 full days and 16 remaining hours.
So, it will take 16 days and 16 hours to fill the tank completely when all three pipes are working together.
Revision Table: Pipe and Tank Calculations
| Pipe | Time (hours) | Type | Rate (tank/hour) |
|---|---|---|---|
| Pipe 1 | 25 | Filling | $\frac{1}{25}$ |
| Pipe 2 | 40 | Filling | $\frac{1}{40}$ |
| Pipe 3 | 16 | Draining | $-\frac{1}{16}$ |
| Combined | 400 | Filling (Net) | $\frac{1}{400}$ |
Additional Information: Work and Time Concepts
Problems involving pipes and tanks are a common application of work and time concepts. The fundamental principle is that the total work done is equal to the rate of work multiplied by the time taken.
In these problems:
- The "work" is often considered as completing one task, like filling or draining a tank (representing 1 unit of work).
- The "rate" is how much of the tank is filled or drained per unit of time.
- If multiple agents (pipes) are working, their rates are combined (added for agents doing the same work, subtracted for agents doing opposite work).
If a pipe's filling time is $T$, its rate is $\frac{1}{T}$. The time taken for the combined work is $\frac{1}{\text{Combined Rate}}$. This relationship is inverse; a higher rate means less time to complete the work.
Similar Questions
Three pipes A, B and C can fill a tank in 12 hours, 18 hours and 24 hours, respectively. A leak at the bottom can empty the full tank in 36 hours. If all the pipes are opened together, in how many hours will the tank be filled? (Round off your answer to two decimal places.)
Pipes A and B can fill an entire tank in 8 hours and 12 hours, respectively. The water tank is one-fourth full. If both the pipes are opened together, then how long will it take to fill the remaining part of the tank?
A pump can fill a tank in 3 hours. Due to a leak in the tank, it takes 4.5 hours to fill the tank. In how much time can the leak empty the full tank if no other entry or exit routes are open?
Two pipes, when working one at a time, can fill a cistern in 3 hours and 4 hours, respectively while a third pipe can drain the cistern empty in 8 hours. All the three pipes were opened together when the cistern was 1/12 full. How long did it take for the cistern to be completely full?
Two valves A and B can fill a sump in \(37\frac{1}{2}\) minutes and 45 minutes respectively. Both valves are opened. The sump will be filled in just 30 minutes, if valve B is turned off after?
One pipe can fill an empty cistern in 4 hours while another can drain the cistern when full in 10 hours. Both the pipes were turned on when the cistern was half-empty. How long will it take the cistern to be full?
A pipe, working at full speed, can fill an empty cistern in 1 hour. However, during the first hour it worked at one-twelfth of its capacity, during the second hour at one-ninth of its capacity, during the third hour at one-sixth of its usual capacity, during the fourth hour at one- fourth of its usual capacity and during the fifth hour it was only one-third as efficient as it was supposed to be. A second pipe also displayed similar performance, but if it worked at full speed would have filled the empty cistern in 2 hours. Together with a drain pipe that drained water out of the tank at a constant rate, the empty cistern could be filled in 5 hours, all the three pipes working concurrently. How many hours will it take the drain pipe to empty the filled cistern if no other pipe was functioning during the time?
Pipes A and C can fill an empty cistern in 16 and 24 hours respectively while Pipe B can drain the filled cistern in 12 hours. If the three pipes are turned on together when the cistern is empty, how many hours will it take for the cistern to be full?
Pipes A and C can fill an empty cistern in 32 and 48 hours, respectively while pipe B can drain the filled cistern in 24 hours. If the three pipes are turned on together when the cistern is empty, how many hours will it take for the cistern to be 2/3 full?
One of the two inlet pipes works twice as efficiently as the other. The two, working alongside a drain pipe that can empty a cistern all by itself in 8 hours, can fill the empty cistern in 8 hours. How many hours will the less efficient inlet pipe take to fill the empty cistern by itself?
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?
