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The problem states:

- 10 women can complete a piece of work in 6 days.
- 6 men can complete the same work in 5 days.
- 8 children can complete the work in 10 days.

We are asked to find the ratio of the efficiency of a woman, a man, and a child.

### Step-by-Step Solution

1. **Work done by women:**
   - If 10 women can finish the work in 6 days, the total work done is \(10 \times 6 = 60 \) woman-days.
   - Therefore, 1 woman can complete the work in \(60\) days.
   - The work done by 1 woman per day is \(\frac{1}{60}\).

2. **Work done by men:**
   - If 6 men can finish the work in 5 days, the total work done is \(6 \times 5 = 30\) man-days.
   - Therefore, 1 man can complete the work in \(30\) days.
   - The work done by 1 man per day is \(\frac{1}{30}\).

3. **Work done by children:**
   - If 8 children can finish the work in 10 days, the total work done is \(8 \times 10 = 80\) child-days.
   - Therefore, 1 child can complete the work in \(80\) days.
   - The work done by 1 child per day is \(\frac{1}{80}\).

4. **Efficiency ratio:**
   - The efficiency of a worker is the work done per day. So, the ratio of efficiency of a woman, a man, and a child is:
   \[
   \text{Efficiency ratio} = \frac{1/60}{1/30} : \frac{1/60}{1/80} : 1 = 1:2: \frac{3}{4} = 4:8:3
   \]

Thus, the ratio of the efficiency of a woman, a man, and a child is **4:8:3**, which corresponds to option (4).

Let me know if you want further details or have any questions!  
Here are 5 related questions to consider:

1. How would the ratio change if 12 women could do the work in 6 days instead of 10?
2. What if the men took 7 days to complete the same task—how would that affect the ratio?
3. Can you derive the total work from the information provided in this problem?
4. How does the ratio change if two groups (women and men) work together on the same task?
5. How can you calculate the total time for all three groups (women, men, and children) to complete the task together?

**Tip:** Ratios of efficiency are directly related to the reciprocal of the time taken to complete the work individually.

10 women can do a piece of work in 6 days. 6 men can do the same work in 5 days and 8 children can do it in 10 days. What is the ratio of the efficiency of a woman, a man, and a child respectively?

Find the efficiency ratio of a woman, a man, and a child when 10 women, 6 men, and 8 children complete a task in different time frames. This solution covers the concepts of time efficiency and work ratio. Suitable for Grades 9-12.

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