12 men can complete a painting work in 8 days. However, 16 women can complete the same painting work in 12 days 8 men started painting the house. After 6 days of painting. 2 men were replaced by 4 women. Now how many days will they take to complete the remaining painting?

Options

A. 4
B. 6
C. 8
D. 5

Answer

To solve the problem step by step, we will first calculate the total work done in terms of "man-days" and "woman-days," then determine how much work has been completed, and finally find out how many more days are needed to finish the remaining work.

### Step 1: Calculate the total work in man-days
12 men can complete the painting work in 8 days. Therefore, the total work can be calculated as:
\[ \text{Total work} = \text{Number of men} \times \text{Number of days} = 12 \text{ men} \times 8 \text{ days} = 96 \text{ man-days} \]

### Step 2: Calculate the work done by 8 men in 6 days
Now, we need to find out how much work 8 men can do in 6 days:
\[ \text{Work done by 8 men in 6 days} = 8 \text{ men} \times 6 \text{ days} = 48 \text{ man-days} \]

### Step 3: Calculate the remaining work
Now, we subtract the work done from the total work:
\[ \text{Remaining work} = \text{Total work} - \text{Work done} = 96 \text{ man-days} - 48 \text{ man-days} = 48 \text{ man-days} \]

### Step 4: Calculate the work rate after replacing 2 men with 4 women
After 6 days, 2 men are replaced by 4 women. Now we have:
- 6 men remaining
- 4 women added

First, we need to find the work rate of 1 woman. We know that 16 women can complete the work in 12 days:
\[ \text{Total work in woman-days} = 16 \text{ women} \times 12 \text{ days} = 192 \text{ woman-days} \]
Thus, the work done by 1 woman in 1 day is:
\[ \text{Work rate of 1 woman} = \frac{192 \text{ woman-days}}{16 \text{ women} \times 12 \text{ days}} = \frac{192}{192} = 1 \text{ man-day} \]
This means that 1 woman does \(\frac{1}{16}\) of the work done by 1 man in a day.

Now, we can find the work done by 6 men and 4 women in one day:
- Work done by 6 men in one day:
\[ 6 \text{ men} = 6 \text{ man-days} \]
- Work done by 4 women in one day:
\[ 4 \text{ women} = 4 \times \frac{1}{16} \text{ man-days} = \frac{4}{16} = \frac{1}{4} \text{ man-days} \]
Thus, the total work done by 6 men and 4 women in one day is:
\[ \text{Total work per day} = 6 + \frac{1}{4} = 6.25 \text{ man-days} \]

### Step 5: Calculate the number of days to complete the remaining work
Now, we need to find out how many days it will take to complete the remaining 48 man-days of work:
\[ \text{Days required} = \frac{\text{Remaining work}}{\text{Total work per day}} = \frac{48 \text{ man-days}}{6.25 \text{ man-days/day}} = 7.68 \text{ days} \]
Since we typically round up in such scenarios, we can say it will take approximately 8 days to complete the remaining work.

### Final Answer
Thus, the total number of days required to complete the remaining painting work after the changes is approximately **8 days**.