Calculating Combined Tank Filling Time
This problem involves calculating the time taken to fill a tank when one tap fills it and another simultaneously empties it. We need to find the net filling rate.
Step 1: Determine Individual Rates
- Tap A fills the tank in 40 minutes. Its filling rate is $ \frac{1}{40} $ of the tank per minute.
- Tap B empties the tank in 60 minutes. Its emptying rate is $ \frac{1}{60} $ of the tank per minute.
Step 2: Calculate the Net Filling Rate
When both taps are open, the net filling rate is the difference between the filling rate and the emptying rate:
Net Rate = (Rate of Tap A) - (Rate of Tap B)
Net Rate = $ \frac{1}{40} \, \text{tank/min} - \frac{1}{60} \, \text{tank/min} $
To subtract these fractions, find a common denominator, which is 120:
Net Rate = $ \left( \frac{1 \times 3}{40 \times 3} \right) - \left( \frac{1 \times 2}{60 \times 2} \right) \, \text{tank/min} $
Net Rate = $ \frac{3}{120} - \frac{2}{120} \, \text{tank/min} $
Net Rate = $ \frac{1}{120} \, \text{tank/min} $
Step 3: Calculate the Total Time Taken
The time taken to fill the tank is the reciprocal of the net filling rate:
Time = $ \frac{1}{\text{Net Rate}} $
Time = $ \frac{1}{\frac{1}{120} \, \text{tank/min}} $
Time = $ 120 $ minutes
Conclusion
Therefore, the time taken to fill the tank when both taps are opened simultaneously is 120 minutes.
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