A takes three times as long as (B + C) together to complete a work. B takes four times as much as (A + C) together to complete a work. If all the three, working together can complete the work in 22 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 first define the efficiencies of A, B, and C based on the information provided in the question.

### Step 1: Define the relationships between A, B, and C
1. Let the time taken by B and C together to complete the work be \( x \) days.
2. According to the problem, A takes three times as long as B and C together, so A takes \( 3x \) days.
3. The efficiency of A is \( \frac{1}{3x} \) and the efficiency of B + C is \( \frac{1}{x} \).

### Step 2: Define the time taken by B
1. Let the time taken by A + C together be \( y \) days.
2. According to the problem, B takes four times as long as A + C together, so B takes \( 4y \) days.
3. The efficiency of B is \( \frac{1}{4y} \) and the efficiency of A + C is \( \frac{1}{y} \).

### Step 3: Set up the equations
1. From the efficiencies, we have:
- Efficiency of A = \( \frac{1}{3x} \)
- Efficiency of B + C = \( \frac{1}{x} \)
- Efficiency of B = \( \frac{1}{4y} \)
- Efficiency of A + C = \( \frac{1}{y} \)

### Step 4: Relate the efficiencies
1. Since B + C = A + C + B, we can express the efficiencies:
- Efficiency of A + B + C = Efficiency of A + Efficiency of B + Efficiency of C
- From the given information, we know that A takes 3 times as long as B + C, and B takes 4 times as long as A + C.

### Step 5: Calculate the total efficiency
1. We know that A + B + C working together can complete the work in 22 days, which means their combined efficiency is \( \frac{1}{22} \).
2. Therefore, \( \text{Efficiency of A} + \text{Efficiency of B} + \text{Efficiency of C} = \frac{1}{22} \).

### Step 6: Solve for individual efficiencies
1. Let’s assume the efficiencies of A, B, and C are \( a, b, c \) respectively.
2. From the relationships, we can derive:
- \( a = \frac{1}{3}b \)
- \( b = \frac{1}{4}c \)
3. Substitute these into the equation for total efficiency:
- \( \frac{1}{3}b + b + c = \frac{1}{22} \)

### Step 7: Solve the equations
1. Substitute \( b = 4c \) into the equation:
- \( \frac{1}{3}(4c) + 4c + c = \frac{1}{22} \)
- This simplifies to \( \frac{4}{3}c + 4c + c = \frac{1}{22} \)
- Combine like terms: \( \frac{4}{3}c + \frac{12}{3}c = \frac{16}{3}c = \frac{1}{22} \)
- Solve for \( c \): \( c = \frac{1}{22} \times \frac{3}{16} = \frac{3}{352} \)

### Step 8: Calculate individual times
1. Now, using the efficiencies:
- For A: \( a = \frac{1}{3}b \) and \( b = 4c \)
- For B: \( b = 4c = 4 \times \frac{3}{352} = \frac{12}{352} = \frac{3}{88} \)
- For C: \( c = \frac{3}{352} \)

### Step 9: Calculate the time taken by each
1. Time taken by A = \( \frac{1}{a} = \frac{1}{\frac{3}{88}} = \frac{88}{3} \approx 29.33 \) days
2. Time taken by B = \( \frac{1}{b} = \frac{1}{\frac{3}{88}} = 88 \) days
3. Time taken by C = \( \frac{1}{c} = \frac{1}{\frac{3}{352}} = 352/3 \approx 117.33 \) days

### Final Answer:
- A will take approximately 29.33 days,
- B will take 88 days,
- C will take approximately 117.33 days.