Persons A, B and C can complete a task together in 81 days. A and B can complete the same task together in 97.2 days. B and C can complete the same task together in 162 days. In how many days can B alone complete the task?

Options

A. 243
B. 225
C. 261
D. 1

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Determine the efficiencies of A + B + C, A + B, and B + C

1. **A + B + C can complete the task in 81 days.**
- Efficiency of A + B + C = Total Work / Time = 1 / 81 tasks per day.

2. **A + B can complete the task in 97.2 days.**
- Efficiency of A + B = Total Work / Time = 1 / 97.2 tasks per day.

3. **B + C can complete the task in 162 days.**
- Efficiency of B + C = Total Work / Time = 1 / 162 tasks per day.

### Step 2: Find the Least Common Multiple (LCM)

To make calculations easier, we will find the LCM of the days taken by each group:

- LCM of 81, 97.2, and 162 is 486.

### Step 3: Calculate the individual efficiencies in terms of units of work

1. **Efficiency of A + B + C:**
- Total work = 486 units
- Efficiency = 486 / 81 = 6 units per day.

2. **Efficiency of A + B:**
- Efficiency = 486 / 97.2 = 5 units per day.

3. **Efficiency of B + C:**
- Efficiency = 486 / 162 = 3 units per day.

### Step 4: Set up equations based on efficiencies

Let:
- Efficiency of A = a units per day
- Efficiency of B = b units per day
- Efficiency of C = c units per day

From our calculations:
- a + b + c = 6 (from A + B + C)
- a + b = 5 (from A + B)
- b + c = 3 (from B + C)

### Step 5: Solve the equations

We can solve these equations step by step:

1. From the second equation, we can express A in terms of B:
- a = 5 - b

2. Substitute a into the first equation:
- (5 - b) + b + c = 6
- 5 + c = 6
- c = 1

3. Substitute c back into the third equation:
- b + 1 = 3
- b = 2

4. Now substitute b back into the equation for a:
- a = 5 - 2 = 3

### Step 6: Conclusion

Now we have:
- Efficiency of A = 3 units per day
- Efficiency of B = 2 units per day
- Efficiency of C = 1 unit per day

To find how many days B alone can complete the task:
- Total work = 486 units
- Efficiency of B = 2 units per day

Number of days for B to complete the task:
- Days = Total Work / Efficiency of B = 486 / 2 = 243 days.

### Final Answer:
B alone can complete the task in **243 days**.

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