8 men can finish a piece of work in 25 days. 15 women can finish the same piece of work in 16 days. 4 men and 8 women started working together and worked for 10 days. After that 6 more men joined them. How many days will they now take to finish the remaining work?

Options

A. `4 4/5`
B. `6 3/5`
C. `6 2/5`
D. `5 3/5`

Answer

To solve the problem step by step, we will first determine the work done by men and women, then calculate the total work, and finally find out how many days it will take to finish the remaining work after some work has already been done.

### Step 1: Calculate the total work done by men and women

1. **Work done by 8 men in 25 days:**
- If 8 men can finish the work in 25 days, the total work (W) can be calculated as:
\[ W = \text{Number of men} \times \text{Days} = 8 \times 25 = 200 \text{ man-days} \]

2. **Work done by 15 women in 16 days:**
- If 15 women can finish the work in 16 days, the total work (W) can also be calculated as:
\[ W = \text{Number of women} \times \text{Days} = 15 \times 16 = 240 \text{ woman-days} \]

### Step 2: Find the work done by one man and one woman

3. **Efficiency of one man (M):**
- Since 8 men can complete the work in 25 days:
\[ \text{Efficiency of one man} = \frac{W}{8 \times 25} = \frac{200}{200} = 1 \text{ unit/day} \]

4. **Efficiency of one woman (W):**
- Since 15 women can complete the work in 16 days:
\[ \text{Efficiency of one woman} = \frac{W}{15 \times 16} = \frac{240}{240} = 1 \text{ unit/day} \]

### Step 3: Calculate the work done by 4 men and 8 women in 10 days

5. **Total efficiency of 4 men and 8 women:**
- Work done by 4 men in one day:
\[ \text{Work by 4 men} = 4 \times 1 = 4 \text{ units/day} \]
- Work done by 8 women in one day:
\[ \text{Work by 8 women} = 8 \times 1 = 8 \text{ units/day} \]
- Total work done in one day by 4 men and 8 women:
\[ \text{Total work per day} = 4 + 8 = 12 \text{ units/day} \]

6. **Total work done in 10 days:**
\[ \text{Total work in 10 days} = 12 \times 10 = 120 \text{ units} \]

### Step 4: Calculate the remaining work

7. **Remaining work:**
- Total work is 200 units (from Step 1).
- Work remaining after 10 days:
\[ \text{Remaining work} = 200 - 120 = 80 \text{ units} \]

### Step 5: Calculate the new team of workers

8. **After 10 days, 6 more men join:**
- Now, the total number of men is \(4 + 6 = 10\).
- The efficiency of 10 men:
\[ \text{Work by 10 men} = 10 \times 1 = 10 \text{ units/day} \]
- The efficiency of 8 women remains the same:
\[ \text{Work by 8 women} = 8 \times 1 = 8 \text{ units/day} \]
- Total efficiency of 10 men and 8 women:
\[ \text{Total work per day} = 10 + 8 = 18 \text{ units/day} \]

### Step 6: Calculate the number of days to finish the remaining work

9. **Days required to finish the remaining work:**
\[ \text{Days required} = \frac{\text{Remaining work}}{\text{Total efficiency}} = \frac{80}{18} \approx 4.44 \text{ days} \]
- Rounding up, they will take 5 days to finish the remaining work.

### Final Answer:
The remaining work will take approximately **5 days** to finish.

---