20 men or 24 women can complete a piece of work in 20 days. If 30 men and 112 women under take to complete the work, the work will be completed in:

Options

A. 10 days
B. 12 days
C. 15 days
D. 16 days

Answer

To solve the problem step-by-step, we will first determine the work done by men and women, and then calculate how long it will take for 30 men and 12 women to complete the work.

### Step 1: Determine the work done by men and women
The problem states that 20 men or 24 women can complete the work in 20 days.

- Work done by 20 men in 20 days = 1 unit of work
- Therefore, the work done by 1 man in 20 days = 1/20 units of work
- Hence, the work done by 1 man in 1 day = 1/400 units of work (since 20 days are considered)

For women:
- Work done by 24 women in 20 days = 1 unit of work
- Therefore, the work done by 1 woman in 20 days = 1/24 units of work
- Hence, the work done by 1 woman in 1 day = 1/480 units of work (since 20 days are considered)

### Step 2: Establish the relationship between men and women
From the information provided:
- 20 men = 24 women
- This can be simplified to find the relationship between the work done by men and women:

\( 5 \text{ men} = 6 \text{ women} \)

Thus, the efficiency of 1 man can be expressed in terms of women:
- 1 man = (6/5) women

### Step 3: Calculate the efficiency of 30 men and 12 women
Now, we need to calculate the total efficiency of 30 men and 12 women.

1. **Efficiency of 30 men**:
- 30 men = 30 * (1/400) = 30/400 = 3/40 units of work per day

2. **Efficiency of 12 women**:
- 12 women = 12 * (1/480) = 12/480 = 1/40 units of work per day

3. **Total efficiency**:
- Total efficiency = Efficiency of men + Efficiency of women
- Total efficiency = (3/40) + (1/40) = 4/40 = 1/10 units of work per day

### Step 4: Calculate the time taken to complete the work
Since the total efficiency of 30 men and 12 women is 1/10 units of work per day, we can find the time taken to complete 1 unit of work:

- Time taken = Total work / Total efficiency
- Time taken = 1 / (1/10) = 10 days

### Conclusion
Thus, 30 men and 12 women can complete the work in **10 days**.