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?
2 h
Understanding the Tanker Filling Problem
This question is about calculating the time taken to fill a cistern (tank) using one or more tankers, which is a classic time and work problem.
Initial Setup and Work Rate of One Tanker
We are given that a single tanker can fill the entire cistern in 12 hours.
This means the work rate of one tanker is:
- Rate of one tanker = \(\frac{1 \text{ cistern}}{12 \text{ hours}}\)
- Or, one tanker fills \(\frac{1}{12}\) of the cistern per hour.
Time Taken for the First Half of the Cistern
The problem states that the first half of the cistern is filled by the initial tanker. Since the tanker fills the whole cistern in 12 hours, it will take half the time to fill half the cistern.
- Time taken to fill the first half = Time for full cistern / 2
- Time taken to fill the first half = 12 hours / 2 = 6 hours
So, after 6 hours, half the cistern is filled.
Situation for the Remaining Half
After the first half is filled, 2 more similar tankers are opened. This means for the remaining half of the cistern, we have:
- Initial tanker + 2 more tankers = 1 + 2 = 3 tankers
All 3 tankers are working together to fill the remaining half of the cistern.
Combined Work Rate of Three Tankers
Since all tankers are similar, each has the same work rate of \(\frac{1}{12}\) of the cistern per hour. When multiple workers (tankers) work together, their rates add up.
- Combined rate of 3 tankers = Rate of tanker 1 + Rate of tanker 2 + Rate of tanker 3
- Combined rate of 3 tankers = \(\frac{1}{12} + \frac{1}{12} + \frac{1}{12}\) per hour
- Combined rate of 3 tankers = \(\frac{1+1+1}{12} = \frac{3}{12}\) per hour
- Combined rate of 3 tankers = \(\frac{1}{4}\) of the cistern per hour
So, 3 tankers working together can fill \(\frac{1}{4}\) of the cistern in one hour.
Calculating Time to Fill the Remaining Half
The remaining part of the cistern is the second half, which is \(\frac{1}{2}\) of the total cistern. The 3 tankers are filling this remaining \(\frac{1}{2}\) at a combined rate of \(\frac{1}{4}\) per hour.
The formula for time is: Time = Work Done / Work Rate
- Work to be done = Remaining half = \(\frac{1}{2}\) of the cistern
- Work rate = Combined rate of 3 tankers = \(\frac{1}{4}\) of the cistern per hour
Time taken to fill the remaining half = \(\frac{\text{Work to be done}}{\text{Work rate}}\)
Time = \(\frac{\frac{1}{2}}{\frac{1}{4}}\) hours
To divide fractions, we multiply the first fraction by the reciprocal of the second fraction:
Time = \(\frac{1}{2} \times \frac{4}{1}\) hours
Time = \(\frac{1 \times 4}{2 \times 1}\) hours
Time = \(\frac{4}{2}\) hours
Time = 2 hours
So, it takes 2 hours to fill the remaining half of the cistern with 3 tankers working together.
Final Answer Summary
The total time to fill the cistern would be the time for the first half plus the time for the remaining half:
- Time for first half (1 tanker) = 6 hours
- Time for remaining half (3 tankers) = 2 hours
Total time = 6 hours + 2 hours = 8 hours. However, the question only asks for the time taken to fill the remaining half.
The time taken to fill the remaining half of the tank is 2 hours.
| Task | Tankers | Portion Filled | Rate per Hour | Time Taken |
|---|---|---|---|---|
| Fill First Half | 1 | \(\frac{1}{2}\) | \(\frac{1}{12}\) | \(\frac{1/2}{1/12} = \frac{1}{2} \times 12 = 6\) hours |
| Fill Remaining Half | 3 | \(\frac{1}{2}\) | \(\frac{3}{12} = \frac{1}{4}\) | \(\frac{1/2}{1/4} = \frac{1}{2} \times 4 = 2\) hours |
Revision Table: Key Concepts in Time and Work
| Concept | Explanation | Formula (if applicable) |
|---|---|---|
| Work Rate | The amount of work done per unit of time. | Rate = \(\frac{\text{Work Done}}{\text{Time Taken}}\) |
| Time Taken | The total time required to complete a specific amount of work. | Time = \(\frac{\text{Work Done}}{\text{Rate}}\) |
| Total Work | Often represented as 1 unit or the total quantity to be filled/completed. | - |
| Combined Rate | When multiple workers work together, their individual rates are added. | Combined Rate = Rate1 + Rate2 + ... |
Additional Information: Understanding Inverse Proportion
In time and work problems involving multiple identical workers (like similar tankers), the relationship between the number of workers and the time taken to complete a certain amount of work is inversely proportional.
- If the number of workers increases, the time taken to complete the same work decreases.
- If the number of workers decreases, the time taken increases.
For example, filling the full tank with 1 tanker takes 12 hours. If we had 2 tankers, they would take roughly 12/2 = 6 hours (assuming linear relationship and no other factors). If we had 3 tankers, they would take approximately 12/3 = 4 hours.
In our specific problem for the *remaining half*:
- One tanker would take 6 hours to fill half the tank.
- With 3 tankers, the time taken is \(\frac{1}{3}\) of the time taken by 1 tanker for the same amount of work (filling half).
- Time for 3 tankers = Time for 1 tanker (to fill half) / Number of tankers
- Time for 3 tankers = 6 hours / 3 = 2 hours.
This confirms our calculation using work rates and helps illustrate the inverse relationship principle in time and work problems.
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?
Pipes A, B and C are attached to an empty cistern. While the first two can fill the cistern in 4 and 10 hours, respectively, the third can drain the cistern, when filled, in 6 hours. If all the three pipes are opened simultaneously when the cistern is three-fifth full, how many hours will be needed to fill the cistern?
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?
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?
P, Q and R are channels which discharge solutions A, B, C respectively in a tank. When the tank is empty and all the three channels are opened what is the proportion of the solution C in the tank after 3 minutes, if the channels P, Q and R can fill the tank from empty to full in 30 minutes and 20 minutes and 10 minutes respectively when they are opened one at a time.
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:
Pipe A can fill a tank in 6 hours. Pipe B can empty it in 15 hours. If both the pipes are opened together, then the tank will be filled in how many hours?
- 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?
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?
A pipe can fill a tank in 4 hours, while a leak which is at one-fourth of the height of the tank from bottom can empty upto that part in 2 hours. If both are operated simultaneously and initially the tank is full, then when it will be one-fourth full?
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?
