24 men can complete a piece of work in 16 days and 18 women can complete the same work in 32 days. 12 men and 6 women work together for 16 days. If the remaining work was to be completed in 2 days, how many additional men would be required besisdes 12 men and 6 women ?

Answer

To solve the problem step by step, we will first determine the work done by men and women, then calculate how much work is left after 16 days, and finally find out how many additional men are needed to complete the remaining work in 2 days.

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

1. **Work done by 24 men in 16 days:**
- Total work = Number of men × Number of days = 24 men × 16 days = 384 man-days.
- Therefore, 1 man can complete \( \frac{1}{384} \) of the work in one day.

2. **Work done by 18 women in 32 days:**
- Total work = Number of women × Number of days = 18 women × 32 days = 576 woman-days.
- Therefore, 1 woman can complete \( \frac{1}{576} \) of the work in one day.

### Step 2: Calculate the work done by 12 men and 6 women in 16 days

1. **Work done by 12 men in 1 day:**
- Work done by 12 men in 1 day = \( 12 \times \frac{1}{384} = \frac{12}{384} = \frac{1}{32} \).

2. **Work done by 6 women in 1 day:**
- Work done by 6 women in 1 day = \( 6 \times \frac{1}{576} = \frac{6}{576} = \frac{1}{96} \).

3. **Total work done by 12 men and 6 women in 1 day:**
- Total work in 1 day = Work done by men + Work done by women = \( \frac{1}{32} + \frac{1}{96} \).
- To add these fractions, we find a common denominator, which is 96:
- \( \frac{1}{32} = \frac{3}{96} \).
- Therefore, \( \frac{1}{32} + \frac{1}{96} = \frac{3}{96} + \frac{1}{96} = \frac{4}{96} = \frac{1}{24} \).

4. **Work done in 16 days:**
- Work done in 16 days = \( 16 \times \frac{1}{24} = \frac{16}{24} = \frac{2}{3} \).

### Step 3: Calculate the remaining work

1. **Remaining work:**
- Total work = 1 (whole work).
- Work done = \( \frac{2}{3} \).
- Remaining work = \( 1 - \frac{2}{3} = \frac{1}{3} \).

### Step 4: Calculate how many additional men are required to complete the remaining work in 2 days

1. **Work done by 12 men and 6 women in 2 days:**
- Work done by 12 men in 2 days = \( 2 \times \frac{1}{32} = \frac{2}{32} = \frac{1}{16} \).
- Work done by 6 women in 2 days = \( 2 \times \frac{1}{96} = \frac{2}{96} = \frac{1}{48} \).
- Total work done in 2 days = \( \frac{1}{16} + \frac{1}{48} \).
- To add these fractions, we find a common denominator, which is 48:
- \( \frac{1}{16} = \frac{3}{48} \).
- Therefore, \( \frac{1}{16} + \frac{1}{48} = \frac{3}{48} + \frac{1}{48} = \frac{4}{48} = \frac{1}{12} \).

2. **Remaining work after 2 days:**
- Remaining work = \( \frac{1}{3} - \frac{1}{12} \).
- To subtract these fractions, we find a common denominator, which is 12:
- \( \frac{1}{3} = \frac{4}{12} \).
- Therefore, \( \frac{1}{3} - \frac{1}{12} = \frac{4}{12} - \frac{1}{12} = \frac{3}{12} = \frac{1}{4} \).

### Step 5: Calculate how many additional men are required to complete the remaining work in 2 days

1. **Let n be the number of additional men required.**
- Work done by \( 12 + n \) men in 2 days = \( (12 + n) \times \frac{1}{384} \times 2 \).
- This should equal the remaining work \( \frac{1}{4} \):
\[ (12 + n) \times \frac{2}{384} = \frac{1}{4} \]
- Simplifying gives:
\[ (12 + n) \times \frac{1}{192} = \frac{1}{4} \]
- Cross-multiplying:
\[ 12 + n = 48 \]
- Therefore, \( n = 48 - 12 = 36 \).

### Final Answer
**36 additional men are required.**