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
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