Tank Fill Time Calculation with Multiple Taps
This problem requires calculating the total time to fill a tank when multiple taps, some filling and some emptying, operate simultaneously.
Tap Rates Explained
We first determine the rate at which each tap works, expressed as the fraction of the tank filled or emptied per hour:
- Tap A fills the tank in 6 hours. Its filling rate is $1/6$ tank per hour.
- Tap B fills the tank in 8 hours. Its filling rate is $1/8$ tank per hour.
- Tap C empties the tank in 4 hours. Its emptying rate is $1/4$ tank per hour. This contributes negatively to filling.
Combined Rate Calculation
When all taps are open together, the net rate of filling is the sum of the filling rates minus the emptying rate.
Combined Rate = (Rate of Tap A) + (Rate of Tap B) - (Rate of Tap C)
Combined Rate = $1/6 + 1/8 - 1/4$ tank per hour.
To combine these fractions, we find the Least Common Multiple (LCM) of the denominators (6, 8, 4), which is 24.
Combined Rate = $4/24 + 3/24 - 6/24$
Combined Rate = $(4 + 3 - 6) / 24$
Combined Rate = $1/24$ tank per hour.
Time Calculation
The time needed to fill the entire tank (1 whole tank) is the reciprocal of the combined filling rate.
Time = $1 / (\text{Combined Rate})$
Time = $1 / (1/24)$ hours
Time = $24$ hours.
Similar Questions
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