If 3 men or 6 women can do a piece of work in 16 days, in how many days can 12 men and 8 women do the same piece of work?

Options

A. 4 days
B. 5 days
C. 3 days
D. 2 days

Answer

To solve the problem, we need to determine how many days it will take for 12 men and 8 women to complete the same piece of work that 3 men or 6 women can do in 16 days.

### Step-by-Step Solution:

1. **Determine the total work done by 3 men or 6 women in 16 days:**
- Let's denote the work done by 1 man in 1 day as \( m \) and the work done by 1 woman in 1 day as \( w \).
- The total work done by 3 men in 16 days can be calculated as:
\[ \text{Work} = \text{Number of men} \times \text{Days} \times \text{Efficiency of 1 man} = 3m \times 16 = 48m \]
- Similarly, the total work done by 6 women in 16 days is:
\[ \text{Work} = 6w \times 16 = 96w \]
- Since both groups complete the same work, we can set these equal:
\[ 48m = 96w \]

2. **Find the ratio of the efficiencies of men and women:**
- From the equation \( 48m = 96w \), we can simplify it to find the ratio of \( m \) to \( w \):
\[ \frac{m}{w} = \frac{96}{48} = 2 \]
- This means that the efficiency of one man is twice that of one woman:
\[ m = 2w \]

3. **Calculate the total work in terms of women’s efficiency:**
- We can express the total work in terms of women's efficiency:
\[ \text{Total work} = 96w \]

4. **Calculate the daily work done by 12 men and 8 women:**
- The daily work done by 12 men is:
\[ \text{Work by 12 men} = 12m = 12 \times 2w = 24w \]
- The daily work done by 8 women is:
\[ \text{Work by 8 women} = 8w \]
- Therefore, the total daily work done by 12 men and 8 women is:
\[ \text{Total daily work} = 24w + 8w = 32w \]

5. **Determine the number of days required to complete the work:**
- To find the number of days required to complete 96 units of work at a rate of 32 units per day, we use the formula:
\[ \text{Days} = \frac{\text{Total work}}{\text{Daily work}} = \frac{96w}{32w} = 3 \]

### Final Answer:
Thus, 12 men and 8 women can complete the work in **3 days**.