A man can do a work in 6 days, whereas a woman can do the same work in 12 days. In how many days will a man and a woman working together be able to do the same work?

Options

A. `4`
B. `3 1/2`
C. `5`
D. `3`

Answer

To solve the problem of how many days a man and a woman working together can complete a task, we can follow these steps:

### Step 1: Determine the work done by the man and the woman individually.
- A man can complete the work in 6 days. Therefore, his work rate (efficiency) is:
\[ \text{Efficiency of Man} = \frac{1 \text{ work}}{6 \text{ days}} = \frac{1}{6} \text{ work per day} \]

- A woman can complete the same work in 12 days. Therefore, her work rate (efficiency) is:
\[ \text{Efficiency of Woman} = \frac{1 \text{ work}}{12 \text{ days}} = \frac{1}{12} \text{ work per day} \]

### Step 2: Calculate the combined efficiency of the man and the woman.
- When both work together, their efficiencies add up:
\[ \text{Combined Efficiency} = \text{Efficiency of Man} + \text{Efficiency of Woman} = \frac{1}{6} + \frac{1}{12} \]

- To add these fractions, we need a common denominator. The least common multiple of 6 and 12 is 12:
\[ \frac{1}{6} = \frac{2}{12} \]
\[ \text{Combined Efficiency} = \frac{2}{12} + \frac{1}{12} = \frac{3}{12} = \frac{1}{4} \text{ work per day} \]

### Step 3: Calculate the time taken to complete the work together.
- The time taken to complete the work when both are working together can be calculated using the formula:
\[ \text{Time} = \frac{\text{Total Work}}{\text{Combined Efficiency}} \]
- Since the total work is considered as 1 unit:
\[ \text{Time} = \frac{1 \text{ work}}{\frac{1}{4} \text{ work per day}} = 4 \text{ days} \]

### Conclusion
Thus, a man and a woman working together can complete the work in **4 days**.

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