A does half as much work as B and C does half as much work as A and B together. If C alone can finish the work in 40 days, then all together will finish the work in?

Answer

To solve the problem step by step, let's break down the information given and find the total time taken for A, B, and C to complete the work together.

### Step 1: Define the work done by A, B, and C
- Let the work done by B in one day be \( x \).
- Since A does half as much work as B, the work done by A in one day is \( \frac{x}{2} \).
- C does half as much work as A and B together. The work done by A and B together is \( \frac{x}{2} + x = \frac{3x}{2} \). Therefore, the work done by C in one day is \( \frac{1}{2} \times \frac{3x}{2} = \frac{3x}{4} \).

### Step 2: Determine C's total work
- We know that C alone can finish the work in 40 days. This means the total work can be calculated as:
\[ \text{Total Work} = \text{C's work per day} \times \text{Number of days} \]
\[ \text{Total Work} = \frac{3x}{4} \times 40 = 30x \]

### Step 3: Calculate the total work done by A, B, and C together
- Now, we will find the total work done by A, B, and C in one day.
- The total work done by A, B, and C together is:
\[ \text{Total Work per day} = A + B + C = \frac{x}{2} + x + \frac{3x}{4} \]
- To add these, we need a common denominator. The common denominator for 2, 1, and 4 is 4.
\[ = \frac{2x}{4} + \frac{4x}{4} + \frac{3x}{4} = \frac{2x + 4x + 3x}{4} = \frac{9x}{4} \]

### Step 4: Calculate the time taken for A, B, and C to finish the work together
- We know the total work is \( 30x \) and the work done together in one day is \( \frac{9x}{4} \).
- The time taken to complete the work together is:
\[ \text{Time} = \frac{\text{Total Work}}{\text{Total Work per day}} = \frac{30x}{\frac{9x}{4}} = 30x \times \frac{4}{9x} = \frac{120}{9} = \frac{40}{3} \text{ days} \]

### Final Answer
Thus, A, B, and C together will finish the work in \( \frac{40}{3} \) days, which is approximately 13.33 days.