Pipe A can fill an empty cistern in 6 hours while along with Pipe B it can fill it up in 4 hours. Only Pipe B is on for an hour after which Pipe A is also turned on. How much time will it take in all to fill up the cistern?
Understanding the Pipe and Cistern Problem
This question is a classic example of a 'Time and Work' problem, specifically applied to pipes filling a cistern. We are given the time it takes for Pipe A to fill the cistern alone and the time it takes for Pipe A and Pipe B to fill it together. We need to find the total time taken to fill the cistern under a specific sequence of operations: Pipe B runs alone for one hour, and then Pipe A joins it.
Calculating Individual and Combined Work Rates
In these types of problems, it's often helpful to think in terms of the fraction of the work done per unit of time. This is called the work rate.
- Pipe A fills the cistern in 6 hours. This means Pipe A fills \( \frac{1}{6} \) of the cistern in 1 hour.
- Pipe A and Pipe B together fill the cistern in 4 hours. This means Pipe A and B together fill \( \frac{1}{4} \) of the cistern in 1 hour.
Now, we can find the work rate of Pipe B alone. The combined rate of A and B is the sum of their individual rates. So, Rate of B = Rate of (A + B) - Rate of A.
Rate of Pipe B \( = \frac{1}{4} - \frac{1}{6} \)
To subtract these fractions, we find a common denominator, which is 12.
\( \frac{1}{4} = \frac{1 \times 3}{4 \times 3} = \frac{3}{12} \)
\( \frac{1}{6} = \frac{1 \times 2}{6 \times 2} = \frac{2}{12} \)
Rate of Pipe B \( = \frac{3}{12} - \frac{2}{12} = \frac{1}{12} \)
So, Pipe B fills \( \frac{1}{12} \) of the cistern in 1 hour.
| Pipe | Time to fill alone (hours) | Work Rate (fraction of cistern per hour) |
|---|---|---|
| A | 6 | \( \frac{1}{6} \) |
| A + B | 4 | \( \frac{1}{4} \) |
| B (calculated) | 12 | \( \frac{1}{12} \) |
Work Done by Pipe B Alone
The problem states that only Pipe B is on for the first hour. We know Pipe B fills \( \frac{1}{12} \) of the cistern per hour.
Work done by Pipe B in 1 hour \( = 1 \text{ hour} \times \frac{1}{12} \text{ (per hour)} = \frac{1}{12} \)
After the first hour, \( \frac{1}{12} \) of the cistern is filled.
Calculating the Remaining Work
The total work required is to fill the entire cistern, which we represent as 1 unit of work. After the first hour, \( \frac{1}{12} \) of the work is done.
Remaining work \( = \text{Total work} - \text{Work done by B alone} \)
Remaining work \( = 1 - \frac{1}{12} = \frac{12}{12} - \frac{1}{12} = \frac{11}{12} \)
So, \( \frac{11}{12} \) of the cistern still needs to be filled.
Time Taken by Pipes A and B Together for Remaining Work
After the first hour, Pipe A is also turned on. Now, both Pipe A and Pipe B are working together to fill the remaining \( \frac{11}{12} \) of the cistern.
The combined work rate of Pipe A and Pipe B is \( \frac{1}{4} \) of the cistern per hour.
Time taken \( = \frac{\text{Remaining Work}}{\text{Combined Work Rate}} \)
Time taken \( = \frac{11/12}{1/4} \)
To divide by a fraction, we multiply by its reciprocal:
Time taken \( = \frac{11}{12} \times \frac{4}{1} = \frac{44}{12} \)
Simplify the fraction:
Time taken \( = \frac{11 \times 4}{3 \times 4} = \frac{11}{3} \) hours
Converting Time to Hours and Minutes
The time taken by A and B together is \( \frac{11}{3} \) hours. Let's convert this into hours and minutes.
\( \frac{11}{3} \) hours \( = 3 \frac{2}{3} \) hours
This is 3 full hours plus \( \frac{2}{3} \) of an hour.
To convert the fraction of an hour to minutes, multiply by 60:
\( \frac{2}{3} \) hours \( = \frac{2}{3} \times 60 \text{ minutes} = 2 \times 20 \text{ minutes} = 40 \text{ minutes} \)
So, Pipes A and B together take 3 hours and 40 minutes to fill the remaining part of the cistern.
Calculating Total Time to Fill the Cistern
The total time is the sum of the time Pipe B worked alone and the time Pipes A and B worked together.
- Time Pipe B worked alone = 1 hour
- Time Pipes A and B worked together = 3 hours 40 minutes
Total time \( = 1 \text{ hour} + 3 \text{ hours } 40 \text{ minutes} = 4 \text{ hours } 40 \text{ minutes} \)
Thus, it will take 4 hours and 40 minutes in all to fill up the cistern.
Revision Table: Pipe and Cistern Calculation Steps
| Step | Description | Calculation | Result |
|---|---|---|---|
| 1 | Calculate work rate of A | \( \frac{1}{6} \) | \( \frac{1}{6} \) / hour |
| 2 | Calculate combined work rate of A & B | \( \frac{1}{4} \) | \( \frac{1}{4} \) / hour |
| 3 | Calculate work rate of B | \( \frac{1}{4} - \frac{1}{6} \) | \( \frac{1}{12} \) / hour |
| 4 | Work done by B in 1 hour | \( 1 \times \frac{1}{12} \) | \( \frac{1}{12} \) |
| 5 | Remaining work | \( 1 - \frac{1}{12} \) | \( \frac{11}{12} \) |
| 6 | Time for A & B to do remaining work | \( \frac{11/12}{1/4} \) | \( \frac{11}{3} \) hours |
| 7 | Convert remaining time to hours and minutes | \( \frac{11}{3} \) hrs \( = 3 \frac{2}{3} \) hrs \( = 3 \) hrs \( 40 \) mins | 3 hours 40 minutes |
| 8 | Total time | 1 hour (B alone) + 3 hours 40 minutes (A+B) | 4 hours 40 minutes |
Additional Information on Time and Work Problems
Time and work problems often involve calculating the efficiency or rate at which a task is completed. Here are some key concepts:
- Work Rate: If a person or pipe can complete a work in \( n \) units of time, their work rate is \( \frac{1}{n} \) of the work per unit of time.
- Total Work: The total work is usually considered as 1 unit or can be taken as the Least Common Multiple (LCM) of the individual times given. Using LCM can sometimes simplify calculations by avoiding fractions initially.
- Combined Work Rate: If multiple entities work together, their individual work rates are added to find the combined work rate (assuming they help each other and don't hinder).
- Time Taken Together: If entities with rates \( R_1, R_2, \dots \) work together, their combined rate is \( R_{total} = R_1 + R_2 + \dots \). The time taken to complete the whole work (1 unit) is \( \frac{1}{R_{total}} \).
- Work Done = Rate \( \times \) Time: This fundamental formula is used to calculate the amount of work done in a given time or the time taken to do a certain amount of work.
Understanding these basic principles allows you to solve various time and work problems, including those involving pipes filling or emptying cisterns.
Similar Questions
Two pipes A and B can fill an empty cistern in 32 and 48 hours, respectively. Pipe C can drain the entire cistern in 64 hours when no other pipe is in operation. Initially, when the cistern was empty Pipe A and Pipe C were turned on. After a few hours, Pipe A was turned off and Pipe B was turned on instantly. In all it took 112 hours to fill the cistern. For how many hours was Pipe B turned on?
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?
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?
Two pipes, when working one at a time can fill a cistern in 2 hours and 3 hours, respectively while a third pipe can drain the cistern empty in 6 hours. All the three pipes were opened together when the cistern was 1/6 full. How long will it take for the cistern to be completely full?
A tanker can fill a cistern in 12 hours. After half the cistern is filled, 2 more similar tankers are opened. What is the time taken to fill the remaining half of the tank?
A sump is filled by three tankers with uniform flow. The first two tankers operating simultaneously fill the sump in the same time during which the sump is filled by the third tanker alone. The second tanker fills the sump 5 hours faster than the first tanker and 4 hours slower than the third tanker. The time required by the first tanker is:
- 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?
Pipe A can fill a tank in 12 hours. Pipe B can empty it in X hours. If both the pipes are opened together, then the tank will be filled in 30 hours. Find X.
15 pipes, each having the same diameter, can fill a cistern in 12 minutes. How long will it take for 6 pipes, each with double the original diameter, to fill the same cistern at the same speed of the flow?
Two pipes namely A and B can fill a tank in 25 minutes and half an hour respectively and a pipe C can empty 3 gallons per minute. All three pipes working together can fill the tank in 15 minutes. The capacity of the tank is:
Important Questions from Pipe and Cistern
Two pipes A and B can independently fill a tank completely in 20 and 30 minutes respectively. If both the pipes are opened simultaneously, how much time will they take to fill the tank completely?
The compound interest on Rs. 64,000 for 3 years, compounded annually at 7.5% p.a. is
A water tank can be emptied in 40 minutes by a pipe of d 'diameter, so how long will it take for a 2d diameter pipe to be emptied?
Two pipes X and Y can fill an empty tank in 16 hours and 20 hours respectively. Pipe Z alone can empty the completely filled tank in 25 hours. Firstly both pipes X and Y are opened and after 6 hours pipe Z is also opened. What will be the total time (in hours) taken to completely fill the tank?
Pipe J can fill a tank in 48 hours and pipe K can fill the same tank in 72 hours. If both the pipes are opened alternately for one hour each and pipe K is opened for the first hour, then in how much time the tank will be full?
