A can do a piece of work in 30 days while B can do it in 40 days. In how many days can A and B working together do it?

Options

A. 70 days
B. `42(3)/4` days
C. `27(1)/7`days
D. `17(1)/7` days

Answer

To solve the problem of how many days A and B can complete a piece of work together, we can follow these steps:

### Step 1: Determine the Work Done by A and B Individually
- A can complete the work in 30 days.
- B can complete the work in 40 days.

### Step 2: Calculate the Efficiency of A and B
- Efficiency is calculated as the amount of work done in one day.
- For A:
\[ \text{Efficiency of A} = \frac{1 \text{ work}}{30 \text{ days}} = \frac{1}{30} \]
- For B:
\[ \text{Efficiency of B} = \frac{1 \text{ work}}{40 \text{ days}} = \frac{1}{40} \]

### Step 3: Calculate the Combined Efficiency of A and B
- To find the combined efficiency, we add the efficiencies of A and B:
\[ \text{Combined Efficiency} = \text{Efficiency of A} + \text{Efficiency of B} = \frac{1}{30} + \frac{1}{40} \]
- To add these fractions, we need a common denominator. The least common multiple of 30 and 40 is 120.
\[ \frac{1}{30} = \frac{4}{120} \quad \text{and} \quad \frac{1}{40} = \frac{3}{120} \]
- Now, adding the fractions:
\[ \text{Combined Efficiency} = \frac{4}{120} + \frac{3}{120} = \frac{7}{120} \]

### Step 4: Calculate the Time Taken to Complete the Work Together
- The time taken to complete the work together can be calculated using the formula:
\[ \text{Time} = \frac{\text{Total Work}}{\text{Combined Efficiency}} \]
- Assuming the total work is 1 unit, we have:
\[ \text{Time} = \frac{1}{\frac{7}{120}} = \frac{120}{7} \approx 17.14 \text{ days} \]

### Conclusion
- A and B together can complete the work in approximately \( \frac{120}{7} \) days, which is about 17.14 days.