Tank Filling Time Calculation

This problem involves calculating the time taken by a specific tap (Tap B) to fill a tank, given the combined filling time of multiple taps and the relationships between their speeds.

Understanding Tap Speeds and Relationships

Let the rate at which each tap fills the tank be represented by $R_A$, $R_B$, and $R_C$ for taps A, B, and C, respectively. The problem states the following relationships:

• Tap C is thrice as fast as Tap B: $R_C = 3 \times R_B$
• Tap B is twice as fast as Tap A: $R_B = 2 \times R_A$

From the second relationship, we can express the rate of Tap A in terms of Tap B's rate: $R_A = \frac{R_B}{2}$.

Calculating Combined Rate

All three taps (A, B, and C) together fill the tank in 8 hours. The combined rate of filling is the sum of their individual rates: $R_{Combined} = R_A + R_B + R_C$.

Now, let's express all rates in terms of $R_B$:

• $R_A = \frac{R_B}{2}$
• $R_B = R_B$
• $R_C = 3 \times R_B$

Substituting these into the combined rate equation:

$R_{Combined} = \frac{R_B}{2} + R_B + 3 \times R_B$

To add these, we find a common denominator (which is 2):

$R_{Combined} = \frac{R_B}{2} + \frac{2 R_B}{2} + \frac{6 R_B}{2}$

$R_{Combined} = \frac{R_B + 2 R_B + 6 R_B}{2}$

$R_{Combined} = \frac{9 R_B}{2}$

Determining Total Work Done

Let the total work (filling one tank) be represented by $W$. The relationship between work, rate, and time is $W = \text{Rate} \times \text{Time}$.

Since the combined rate of taps A, B, and C fills the tank in 8 hours, we have:

$W = R_{Combined} \times 8$

Substitute the combined rate we found:

$W = \left( \frac{9 R_B}{2} \right) \times 8$

$W = 9 \times R_B \times 4$

$W = 36 \times R_B$

This equation tells us that the total work required to fill the tank is equivalent to 36 times the rate of Tap B.

Calculating Time Taken by Tap B Alone

We want to find the time taken by Tap B alone to complete the same work $W$. Let this time be $T_B$. The equation is:

$W = R_B \times T_B$

We already found that $W = 36 \times R_B$. Now we can set the two expressions for $W$ equal to each other:

$R_B \times T_B = 36 \times R_B$

To find $T_B$, we can divide both sides by $R_B$ (assuming $R_B$ is not zero, which is true since it's filling the tank):

$T_B = \frac{36 \times R_B}{R_B}$

$T_B = 36$ hours

Conclusion

Therefore, Tap B alone will take 36 hours to fill the tank.