https://file.aiquickdraw.com/mathbot/file/1727690747089pcktif8q.jpg

Let's break down and solve the problem from the image step-by-step.

### Problem Summary:
- 10 men can complete a piece of work in 12 days.
- 12 children can complete the same work in 16 days.
- 8 women can complete the same work in 20 days.
- Initially, 5 men and 10 children started working on the task.
- After 4 days, 8 children were replaced by 5 women.

We need to find in how many days the remaining work was completed after the replacement.

### Step 1: Calculate the work rates of each group.
- **Work rate of men**: 10 men finish the work in 12 days. So, 1 man’s rate = \( \frac{1}{10 \times 12} = \frac{1}{120} \).
- **Work rate of children**: 12 children finish the work in 16 days. So, 1 child’s rate = \( \frac{1}{12 \times 16} = \frac{1}{192} \).
- **Work rate of women**: 8 women finish the work in 20 days. So, 1 woman’s rate = \( \frac{1}{8 \times 20} = \frac{1}{160} \).

### Step 2: Calculate the work done in the first 4 days.
- Initially, 5 men and 10 children are working.
  - Work rate of 5 men = \( 5 \times \frac{1}{120} = \frac{5}{120} = \frac{1}{24} \).
  - Work rate of 10 children = \( 10 \times \frac{1}{192} = \frac{10}{192} = \frac{5}{96} \).
  - Combined work rate for the first 4 days = \( \frac{1}{24} + \frac{5}{96} \).
  - Let's simplify:
    \[
    \frac{1}{24} + \frac{5}{96} = \frac{4}{96} + \frac{5}{96} = \frac{9}{96} = \frac{3}{32}
    \]
  - Work done in 4 days = \( 4 \times \frac{3}{32} = \frac{12}{32} = \frac{3}{8} \).

### Step 3: Remaining work after 4 days.
- Total work is 1 unit. After 4 days, \( \frac{3}{8} \) of the work is done.
- Remaining work = \( 1 - \frac{3}{8} = \frac{5}{8} \).

### Step 4: Calculate the combined rate after 4 days.
- After 4 days, 8 children are replaced by 5 women.
- New group:
  - Work rate of 5 men = \( \frac{1}{24} \) (same as before).
  - Work rate of 2 children = \( 2 \times \frac{1}{192} = \frac{2}{192} = \frac{1}{96} \).
  - Work rate of 5 women = \( 5 \times \frac{1}{160} = \frac{5}{160} = \frac{1}{32} \).
- Combined work rate after the replacement:
  \[
  \frac{1}{24} + \frac{1}{96} + \frac{1}{32}
  \]
  Let's simplify:
  \[
  \text{LCM of 24, 96, 32} = 96
  \]
  \[
  \frac{1}{24} = \frac{4}{96}, \quad \frac{1}{96} = \frac{1}{96}, \quad \frac{1}{32} = \frac{3}{96}
  \]
  \[
  \text{Combined rate} = \frac{4}{96} + \frac{1}{96} + \frac{3}{96} = \frac{8}{96} = \frac{1}{12}
  \]

### Step 5: Calculate the remaining days.
- Remaining work = \( \frac{5}{8} \).
- Work rate after replacement = \( \frac{1}{12} \).
- Time to complete the remaining work = \( \frac{\frac{5}{8}}{\frac{1}{12}} = \frac{5}{8} \times 12 = 7.5 \text{ days} \).

### Final Answer:
The remaining work will be completed in **7.5 days**.

The correct option is **(4) 7.5**.

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Would you like more details or explanations on any step?

Here are 5 related questions to practice:
1. If 6 men and 8 women work together, how long will they take to complete the task?
2. How long would it take for 20 children alone to complete the work?
3. If the number of days the work took was reduced by 2, how many additional men would be needed?
4. What would happen if the replacement of children with women occurred after 6 days instead of 4?
5. If only children worked from the beginning, how long would they take to finish the task?

**Tip:** When dealing with work problems, always convert the work rate to a "per day" basis for each group and combine them to calculate the total efficiency.

Ten men can complete a piece of work in 12 days; 12 children can complete the same work in 16 days; 8 women can complete the same work in 20 days. Five men and 10 children started working on the work. If after 4 days, 8 children were replaced by 5 women, then in how many days was the remaining work completed?

Solve a work-time problem involving men, children, and women completing a task. Learn to calculate the work rate, remaining work, and how long it takes for the task to be finished after a group change. Suitable for Grades 10-12.

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