B takes two times as long as (A + C) together to complete a work. C takes three times as much as (A + B) together to complete a work. If all the three, working together can complete the work in 36 days, then find the number of days A, B and C alone will take to complete this work.

Answer

To solve the problem step by step, we will define the efficiencies of A, B, and C based on the information provided and then calculate the time each would take to complete the work alone.

### Step 1: Define the efficiencies based on the given ratios
1. **B takes two times as long as (A + C) together**:
- Let the time taken by A + C together to complete the work be \( t \).
- Then, time taken by B = \( 2t \).
- Efficiency is inversely proportional to time. Therefore, if A + C's efficiency is \( 1 \), then B's efficiency is \( \frac{1}{2} \).
- Thus, the efficiency ratio of A + C to B is \( 1 : 2 \).

2. **C takes three times as long as (A + B) together**:
- Let the time taken by A + B together to complete the work be \( s \).
- Then, time taken by C = \( 3s \).
- Similarly, if A + B's efficiency is \( 1 \), then C's efficiency is \( \frac{1}{3} \).
- Thus, the efficiency ratio of A + B to C is \( 1 : 3 \).

### Step 2: Set up the equations based on the efficiencies
1. From the above, we can denote the efficiencies as:
- Efficiency of A + C = \( e_{AC} \)
- Efficiency of B = \( e_B = \frac{1}{2} e_{AC} \)
- Efficiency of A + B = \( e_{AB} \)
- Efficiency of C = \( e_C = \frac{1}{3} e_{AB} \)

2. Since A + C's efficiency and B's efficiency must be consistent, we can express:
- \( e_B = \frac{1}{2} e_{AC} \)
- \( e_C = \frac{1}{3} e_{AB} \)

### Step 3: Combine the efficiencies
1. Let’s denote the efficiencies of A, B, and C as \( e_A, e_B, e_C \):
- From the ratios, we can say:
- \( e_B = 4 \) (from the combined efficiency of A + C)
- \( e_C = 3 \) (from the combined efficiency of A + B)
- \( e_A + e_C = 8 \) (from A + C)
- \( e_A + e_B = 12 \) (from A + B)

2. Solving these equations:
- From \( e_A + e_C = 8 \) and \( e_C = 3 \):
- \( e_A + 3 = 8 \)
- \( e_A = 5 \)

### Step 4: Calculate total efficiency
1. Total efficiency of A, B, and C together:
- \( e_A + e_B + e_C = 5 + 4 + 3 = 12 \)

### Step 5: Calculate total work done
1. Since they can complete the work in 36 days:
- Total work = Total efficiency × Number of days
- Total work = \( 12 \times 36 = 432 \) units of work.

### Step 6: Calculate the time taken by A, B, and C to complete the work alone
1. Time taken by A:
- \( T_A = \frac{\text{Total work}}{e_A} = \frac{432}{5} = 86.4 \) days.

2. Time taken by B:
- \( T_B = \frac{\text{Total work}}{e_B} = \frac{432}{4} = 108 \) days.

3. Time taken by C:
- \( T_C = \frac{\text{Total work}}{e_C} = \frac{432}{3} = 144 \) days.

### Final Answer
- A takes 86.4 days,
- B takes 108 days,
- C takes 144 days to complete the work alone.