The efficiency of pipe A is twice as good as pipe B and the efficiency of pipe B is thrice as good as C. If all the pipes are open together the tank will be fill in 10 hours. If pipe B open separately then in how many times the tank will be fill?

Answer

To solve the problem step by step, we will first determine the efficiencies of the pipes and then calculate how long it will take for pipe B to fill the tank alone.

### Step 1: Establish the Ratios of Efficiencies
Given:
- Efficiency of pipe A (E_A) is twice that of pipe B (E_B).
- Efficiency of pipe B (E_B) is thrice that of pipe C (E_C).

From the information provided, we can express the efficiencies as:
- E_A = 2E_B
- E_B = 3E_C

Let’s denote the efficiency of pipe C as E_C = 1 unit. Then:
- E_B = 3E_C = 3 * 1 = 3 units
- E_A = 2E_B = 2 * 3 = 6 units

Thus, the efficiencies can be summarized as:
- E_A : E_B : E_C = 6 : 3 : 1

### Step 2: Calculate Total Efficiency
Now, we can calculate the total efficiency when all pipes are open together:
- Total Efficiency = E_A + E_B + E_C = 6 + 3 + 1 = 10 units

### Step 3: Determine the Work Done
The problem states that the tank can be filled in 10 hours when all pipes are open together. The total work done to fill the tank can be calculated as:
- Total Work = Total Efficiency × Time = 10 units × 10 hours = 100 units of work

### Step 4: Calculate the Time Taken by Pipe B Alone
Now, we need to find out how long it will take for pipe B to fill the tank alone. The efficiency of pipe B is 3 units.

Let the time taken by pipe B to fill the tank be T hours. The work done by pipe B in T hours is:
- Work = Efficiency × Time = 3 units × T hours

Since the total work to fill the tank is 100 units, we can set up the equation:
\[ 3T = 100 \]

### Step 5: Solve for T
Now, solving for T:
\[ T = \frac{100}{3} \]
\[ T = 33 \frac{1}{3} \text{ hours} \]

### Conclusion
Thus, pipe B will fill the tank alone in **33 hours and 20 minutes** or **33 1/3 hours**.

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