Pipe A can fill 50% of the tank in 6 hours and pipe B can completely fill the same tank in 18 hours. If both the pipes are opened at the same time, in how much time (in minutes) will the empty tank be completely filled?
432
Solving Pipe and Tank Filling Problems
This question asks us to find the total time it takes for two pipes, A and B, to fill an empty tank when working together. We are given the individual filling rates of both pipes.
Understanding Individual Pipe Filling Rates
First, let's determine the rate at which each pipe fills the tank individually.
- Pipe A: We are told Pipe A can fill 50% (or \(\frac{1}{2}\)) of the tank in 6 hours. If it fills half the tank in 6 hours, it will take twice as long to fill the entire tank.
Time taken by Pipe A to fill 100% of the tank = \(6 \text{ hours} \times 2 = 12 \text{ hours}\).
So, the rate of Pipe A is \(\frac{1}{12}\) of the tank filled per hour.
- Pipe B: We are told Pipe B can completely fill (100%) the same tank in 18 hours.
Time taken by Pipe B to fill 100% of the tank = \(18 \text{ hours}\).
So, the rate of Pipe B is \(\frac{1}{18}\) of the tank filled per hour.
Calculating the Combined Filling Rate
When both pipes are opened at the same time, their rates of filling the tank add up. The combined rate is the sum of their individual rates.
Combined rate of (A + B) = Rate of A + Rate of B
Combined rate = \(\frac{1}{12} + \frac{1}{18}\) (tank per hour)
To add these fractions, we need a common denominator. The least common multiple (LCM) of 12 and 18 is 36.
Combined rate = \(\frac{1 \times 3}{12 \times 3} + \frac{1 \times 2}{18 \times 2} = \frac{3}{36} + \frac{2}{36}\)
Combined rate = \(\frac{3 + 2}{36} = \frac{5}{36}\) of the tank filled per hour.
Determining the Time to Fill the Tank
The time taken to fill the entire tank is the reciprocal of the combined filling rate (since rate is the amount filled per unit time, time is the total amount / rate).
Time taken by (A + B) to fill the tank = \(\frac{1}{\text{Combined rate}}\)
Time taken = \(\frac{1}{\frac{5}{36}} = \frac{36}{5}\) hours.
Converting Time to Minutes
The question asks for the time in minutes. To convert hours to minutes, we multiply the number of hours by 60.
Time in minutes = Time in hours \(\times 60\) minutes/hour
Time in minutes = \(\frac{36}{5} \times 60\)
Time in minutes = \(36 \times \frac{60}{5} = 36 \times 12\)
Now, let's calculate \(36 \times 12\):
| Calculation | Result |
|---|---|
| \(36 \times 10\) | 360 |
| \(36 \times 2\) | 72 |
| \(360 + 72\) | 432 |
Time taken = 432 minutes.
Therefore, if both pipes are opened at the same time, the empty tank will be completely filled in 432 minutes.
Revision Table: Pipe Filling Summary
| Pipe | Time to fill 100% | Rate (Tank per hour) |
|---|---|---|
| A | 12 hours (from 50% in 6 hrs) | \(\frac{1}{12}\) |
| B | 18 hours | \(\frac{1}{18}\) |
| A + B (Combined) | \(\frac{36}{5}\) hours | \(\frac{5}{36}\) |
Additional Information on Work and Time Problems
Problems involving pipes filling or emptying tanks are similar to work and time problems. The main concept is that the rate of work (filling/emptying) is the reciprocal of the time taken to complete the entire work (fill/empty the entire tank).
- If a pipe can complete a task in 't' hours, its rate of work is \(\frac{1}{t}\) of the task per hour.
- If multiple pipes work together, their rates are added to find the combined rate.
- If some pipes fill and others empty, the emptying rate is subtracted from the filling rate to find the net combined rate.
- The total time taken to complete the work is 1 / (combined rate).
Understanding these basic principles helps solve a wide variety of problems involving pipes and cisterns or multiple workers doing a job together.
Similar Questions
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