10 men can complete project in 12 days 12 children can complete same in 16 days and 8 womwn can in 20 days 5 men and 12 children started project if after 4 days 8 children were replaced by 4 women in how many days remaining project was completed

Options

A. (4)2/5
B. (5)1/2
C. (7)1/2
D. (6)2/3

Answer

To solve the problem step by step, we need to calculate the work done by men, children, and women, and then determine how much work is left after certain days.

### Step 1: Calculate the total work in terms of man-days
- **10 men can complete the project in 12 days.**
- Total work = 10 men × 12 days = 120 man-days.

### Step 2: Calculate the work done by 5 men and 12 children in 4 days
- **12 children can complete the project in 16 days.**
- Total work = 12 children × 16 days = 192 child-days.
- Work done by 1 child in 1 day = 1/192 of the project.

- **Work done by 5 men in 1 day:**
- Work done by 1 man in 1 day = 1/120 of the project.
- Work done by 5 men in 1 day = 5 × (1/120) = 5/120 = 1/24 of the project.

- **Work done by 12 children in 1 day:**
- Work done by 12 children in 1 day = 12 × (1/192) = 12/192 = 1/16 of the project.

- **Total work done by 5 men and 12 children in 1 day:**
- Total work in 1 day = (1/24) + (1/16).
- To add these fractions, find a common denominator (48):
- (1/24) = 2/48
- (1/16) = 3/48
- Total work in 1 day = 2/48 + 3/48 = 5/48 of the project.

- **Work done in 4 days:**
- Work done in 4 days = 4 × (5/48) = 20/48 = 5/12 of the project.

### Step 3: Calculate the remaining work
- **Remaining work = Total work - Work done in 4 days**
- Remaining work = 1 - 5/12 = 7/12 of the project.

### Step 4: Calculate the new team composition after 4 days
- After 4 days, 8 children are replaced by 4 women.
- **Work done by 4 women:**
- **8 women can complete the project in 20 days.**
- Total work = 8 women × 20 days = 160 woman-days.
- Work done by 1 woman in 1 day = 1/160 of the project.
- Work done by 4 women in 1 day = 4 × (1/160) = 4/160 = 1/40 of the project.

- **Work done by 4 children:**
- Work done by 4 children in 1 day = 4 × (1/192) = 4/192 = 1/48 of the project.

- **Total work done by 4 women and 4 children in 1 day:**
- Total work in 1 day = (1/40) + (1/48).
- Find a common denominator (240):
- (1/40) = 6/240
- (1/48) = 5/240
- Total work in 1 day = 6/240 + 5/240 = 11/240 of the project.

### Step 5: Calculate the time required to complete the remaining work
- **Remaining work = 7/12 of the project.**
- **Time required = Remaining work / Work done in 1 day**
- Time required = (7/12) ÷ (11/240) = (7/12) × (240/11) = (7 × 20) / 11 = 140/11 days.

### Final Answer:
The remaining project was completed in approximately **12.73 days** (or 12 days and 16 hours).