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?
10
Understanding the Tank Filling and Emptying Problem
This question involves two pipes, one filling a tank and the other emptying it. To solve this, we need to understand the rate at which each pipe works individually and then calculate their combined effect when working together.
Calculating Individual Pipe Rates
- Pipe A is a filling pipe. It fills the tank in 6 hours. This means in 1 hour, Pipe A fills \(\frac{1}{6}\) of the tank. The filling rate of Pipe A is \(\frac{1}{6}\) tank per hour.
- Pipe B is an emptying pipe. It empties the tank in 15 hours. This means in 1 hour, Pipe B empties \(\frac{1}{15}\) of the tank. The emptying rate of Pipe B is \(\frac{1}{15}\) tank per hour. Note that emptying is the opposite of filling, so its contribution is negative in the combined calculation.
Finding the Combined Rate of Filling
When both pipes are opened together, their rates combine. Since Pipe A is filling and Pipe B is emptying, the net rate at which the tank is being filled is the difference between the filling rate and the emptying rate.
Combined rate = (Rate of Pipe A) - (Rate of Pipe B)
Combined rate = \(\frac{1}{6} - \frac{1}{15}\) tank per hour.
Calculating the Combined Rate
To subtract these fractions, we need a common denominator. The least common multiple (LCM) of 6 and 15 is 30.
- Convert \(\frac{1}{6}\) to a fraction with denominator 30: \(\frac{1}{6} \times \frac{5}{5} = \frac{5}{30}\)
- Convert \(\frac{1}{15}\) to a fraction with denominator 30: \(\frac{1}{15} \times \frac{2}{2} = \frac{2}{30}\)
Now, subtract the fractions:
Combined rate = \(\frac{5}{30} - \frac{2}{30} = \frac{5 - 2}{30} = \frac{3}{30}\)
Simplify the combined rate:
Combined rate = \(\frac{3}{30} = \frac{1}{10}\) tank per hour.
This means that when both pipes are open, \(\frac{1}{10}\) of the tank is filled every hour.
Determining the Time to Fill the Tank
If \(\frac{1}{10}\) of the tank is filled in 1 hour, then the total time taken to fill the entire tank (which is 1 whole tank) is the reciprocal of the combined rate.
Time taken = \(\frac{1}{\text{Combined rate}}\)
Time taken = \(\frac{1}{\frac{1}{10}}\) hours
Time taken = \(1 \times \frac{10}{1}\) hours
Time taken = 10 hours.
Therefore, if both pipes are opened together, the tank will be filled in 10 hours.
Revision Table: Key Concepts
| Concept | Description | Formula |
|---|---|---|
| Individual Rate (Filling) | Fraction of work done by a filling pipe in one unit of time (e.g., 1 hour). | \(\frac{1}{\text{Time taken to fill}}\) |
| Individual Rate (Emptying) | Fraction of work done by an emptying pipe in one unit of time. Treated as negative work. | \(\frac{-1}{\text{Time taken to empty}}\) |
| Combined Rate | The net fraction of work done by all pipes together in one unit of time. Sum of individual rates. | Sum of individual rates (filling rates are positive, emptying rates are negative) |
| Time Taken (Combined) | The total time required to complete the work (fill the tank) at the combined rate. | \(\frac{1}{\text{Combined rate}}\) (if combined rate is positive, meaning filling is faster than emptying) |
Additional Information on Tank Filling Problems
Problems involving pipes and tanks are often solved using the concept of 'work rate'. Here's a bit more detail:
- Work Rate: If a job (like filling a tank) can be completed in 'T' units of time, the rate of work is \(1/T\) of the job per unit of time.
- Multiple Workers (Pipes): When multiple pipes work together, their individual work rates are added to find the combined work rate.
- Filling vs. Emptying: Filling pipes contribute positively to the work (filling the tank), while emptying pipes contribute negatively (emptying the tank). So, when both types are involved, you subtract the emptying rate from the filling rate.
- Total Work: The total work is typically considered as '1 unit' (representing the full tank).
- Time = Total Work / Combined Rate: This fundamental relationship is used to find the time taken when the combined rate is known.
It's important to ensure the combined rate is positive for the tank to actually fill up. If the combined rate were negative (meaning the emptying rate is higher than the filling rate), the tank would be emptied, not filled, and the question might ask for the time taken to empty a full tank instead.
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