A can complete a work in 'm' days and B can complete it in 'n' days. How many days will it take to complete the work if both A and B work together?

Options

A. `(m + n)` days
B. `(1/m xx 1/n)` days
C. `((m + n)/(mn)) ` days
D. `((mn)/(m + n))` days

Answer

To find out how many days it will take for A and B to complete the work together, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Work Done by A and B**:
- A can complete the work in \( m \) days.
- B can complete the work in \( n \) days.

2. **Calculate the Total Work**:
- We can assume the total work to be the Least Common Multiple (LCM) of \( m \) and \( n \). For simplicity, we can also consider the total work as \( MN \) units, where \( M \) is the number of days A takes and \( N \) is the number of days B takes.

3. **Calculate the Efficiency of A and B**:
- The efficiency of A (work done by A in one day) is given by:
\[ \text{Efficiency of A} = \frac{\text{Total Work}}{\text{Days taken by A}} = \frac{MN}{m} = \frac{N}{m} \text{ units per day} \]
- The efficiency of B (work done by B in one day) is given by:
\[ \text{Efficiency of B} = \frac{\text{Total Work}}{\text{Days taken by B}} = \frac{MN}{n} = \frac{M}{n} \text{ units per day} \]

4. **Combine the Efficiencies**:
- When A and B work together, their combined efficiency is:
\[ \text{Combined Efficiency} = \text{Efficiency of A} + \text{Efficiency of B} = \frac{N}{m} + \frac{M}{n} \]
- To combine these fractions, we can find a common denominator, which is \( mn \):
\[ \text{Combined Efficiency} = \frac{Nn + Mm}{mn} \text{ units per day} \]

5. **Calculate the Time Taken to Complete the Work Together**:
- The total time taken to complete the work when A and B work together is given by:
\[ \text{Time} = \frac{\text{Total Work}}{\text{Combined Efficiency}} = \frac{MN}{\frac{Nn + Mm}{mn}} = \frac{MN \cdot mn}{Nn + Mm} \]
- Simplifying this gives:
\[ \text{Time} = \frac{mn}{\frac{Nn + Mm}{M}} = \frac{mn}{\frac{Nn + Mm}{m+n}} = \frac{mn}{(m+n)} \]

### Final Answer:
The time taken for A and B to complete the work together is:
\[ \frac{mn}{m+n} \text{ days} \]