A, B and C working together can finish certain piece of work in 40 days. If (A + B) work together then take `(1)/(3)` less time than that of C takes alone. If (A + C) work together then they take `(1)/(4)` less time than that of B takes alone. In how many days (B + C) working together will finish the 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 work done by A, B, and C
Let the work done by A in one day be \( a \), by B be \( b \), and by C be \( c \).

Given that A, B, and C together can finish the work in 40 days, we can express this as:
\[ a + b + c = \frac{1}{40} \quad \text{(work done in one day)} \]

### Step 2: Relationship between A + B and C
The problem states that A + B take \( \frac{1}{3} \) less time than C alone.
Let the time taken by C alone be \( x \) days. Then, the time taken by A + B is:
\[ x - \frac{x}{3} = \frac{2x}{3} \]
Thus, we can express the work done by A + B as:
\[ a + b = \frac{1}{\frac{2x}{3}} = \frac{3}{2x} \]

### Step 3: Relationship between A + C and B
Similarly, A + C take \( \frac{1}{4} \) less time than B alone.
Let the time taken by B alone be \( y \) days. Then, the time taken by A + C is:
\[ y - \frac{y}{4} = \frac{3y}{4} \]
Thus, we can express the work done by A + C as:
\[ a + c = \frac{1}{\frac{3y}{4}} = \frac{4}{3y} \]

### Step 4: Set up equations
Now we have three equations:
1. \( a + b + c = \frac{1}{40} \)
2. \( a + b = \frac{3}{2x} \)
3. \( a + c = \frac{4}{3y} \)

### Step 5: Express C's work in terms of A + B
From equation 1, we can express \( c \):
\[ c = \frac{1}{40} - (a + b) \]

Substituting \( a + b \) from equation 2:
\[ c = \frac{1}{40} - \frac{3}{2x} \]

### Step 6: Express A in terms of B and C
From equation 1, we can also express \( a \):
\[ a = \frac{1}{40} - b - c \]

### Step 7: Solve for efficiencies
Now, we can find the efficiencies of A, B, and C. We know:
- \( a + b = \frac{3}{2x} \)
- \( a + c = \frac{4}{3y} \)

Using the ratios of time taken, we can express the efficiencies in terms of a common factor.

### Step 8: Find the efficiency of B + C
Now we need to find the efficiency of B + C:
\[ b + c = (a + b + c) - a = \frac{1}{40} - a \]

### Step 9: Calculate total work done
The total work done can be calculated as:
\[ \text{Total Work} = \text{Efficiency} \times \text{Time} \]
Using the total efficiency of A, B, and C, we can find the total work done.

### Step 10: Calculate days taken by B + C
Finally, we can find the time taken by B + C to finish the work:
\[ \text{Time taken by } B + C = \frac{\text{Total Work}}{b + c} \]

### Final Calculation
Substituting the values will give us the final answer.