10 women can do a piece of work In 6 days. 6 men can do 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?

Options

A. `4 : 6 : 3`
B. `4 : 5 : 3`
C. ` 2 : 4 : 3 `
D. ` 4 : 8 : 3`

Answer

To find the ratio of the efficiency of a woman, a man, and a child, we can follow these steps:

### Step 1: Determine the total work done in terms of 'K'
- We know that 10 women can complete the work in 6 days.
- Therefore, the total work, W, can be expressed as:
\[ W = \text{Number of women} \times \text{Days} = 10 \times 6 = 60 \text{ woman-days} \]
- We can denote this total work as \( K \), so \( K = 60 \).

### Step 2: Express work done by men and children in terms of 'K'
- For men, 6 men can complete the work in 5 days:
\[ W = \text{Number of men} \times \text{Days} = 6 \times 5 = 30 \text{ man-days} \]
Thus, we can express this as \( K = 30 \).

- For children, 8 children can complete the work in 10 days:
\[ W = \text{Number of children} \times \text{Days} = 8 \times 10 = 80 \text{ child-days} \]
Thus, we can express this as \( K = 80 \).

### Step 3: Set up the equations
From the above, we can set up the following relationships:
- \( K = 60 \) (for women)
- \( K = 30 \) (for men)
- \( K = 80 \) (for children)

### Step 4: Equate the work done by each group
Now we can equate the work done by women, men, and children:
\[ 60 \text{ women} = 30 \text{ men} = 80 \text{ children} \]

### Step 5: Express efficiencies in terms of K
From the relationships, we can express the efficiencies of women, men, and children:
- Efficiency of 1 woman = \( \frac{K}{60} \)
- Efficiency of 1 man = \( \frac{K}{30} \)
- Efficiency of 1 child = \( \frac{K}{80} \)

### Step 6: Find a common denominator
To find the ratio of efficiencies, we need a common denominator. The least common multiple (LCM) of 60, 30, and 80 is 240.

### Step 7: Convert efficiencies to the common denominator
Now we convert each efficiency to have a denominator of 240:
- For women:
\[ \frac{K}{60} = \frac{K \times 4}{240} = \frac{4K}{240} \]
- For men:
\[ \frac{K}{30} = \frac{K \times 8}{240} = \frac{8K}{240} \]
- For children:
\[ \frac{K}{80} = \frac{K \times 3}{240} = \frac{3K}{240} \]

### Step 8: Write the ratio
Now we can write the ratio of efficiencies:
\[ \text{Efficiency of women : Efficiency of men : Efficiency of children} = 4K : 8K : 3K \]
This simplifies to:
\[ 4 : 8 : 3 \]

### Final Answer
Thus, the ratio of the efficiency of a woman, a man, and a child respectively is:
\[ \boxed{4 : 8 : 3} \]