A 10 hectare field is reaped by 2 men, 3 women and 4 children together in 10 days. If working capabilities of a man, a woman and a child are in the ratio 5:4:2, then a 16 hectare field will be reaped by 6 men, 4 women and 7 children in

Options

A. 5 days
B. 6 days
C. 7 days
D. 8 days

Answer

To solve the problem step by step, we will use the information provided about the work rates of men, women, and children, as well as the total work done.

### Step 1: Determine the work rates based on the efficiency ratio
The working capabilities of a man, a woman, and a child are in the ratio of 5:4:2. We can assign:
- Efficiency of 1 man = 5 units of work
- Efficiency of 1 woman = 4 units of work
- Efficiency of 1 child = 2 units of work

### Step 2: Calculate the total efficiency of the first group (2 men, 3 women, and 4 children)
Now, we calculate the total efficiency of the group that reaped the 10-hectare field:
- Total efficiency of 2 men = 2 × 5 = 10 units
- Total efficiency of 3 women = 3 × 4 = 12 units
- Total efficiency of 4 children = 4 × 2 = 8 units

Adding these together:
\[ \text{Total efficiency} = 10 + 12 + 8 = 30 \text{ units of work per day} \]

### Step 3: Calculate the total work done in 10 days
Since they completed the 10-hectare field in 10 days, we can calculate the total work done:
\[ \text{Total work} = \text{Total efficiency} \times \text{Number of days} \]
\[ \text{Total work} = 30 \text{ units/day} \times 10 \text{ days} = 300 \text{ units of work} \]

### Step 4: Determine the work rate for the second group (6 men, 4 women, and 7 children)
Now, we calculate the efficiency of the second group:
- Total efficiency of 6 men = 6 × 5 = 30 units
- Total efficiency of 4 women = 4 × 4 = 16 units
- Total efficiency of 7 children = 7 × 2 = 14 units

Adding these together:
\[ \text{Total efficiency} = 30 + 16 + 14 = 60 \text{ units of work per day} \]

### Step 5: Calculate the time required to reap the 16-hectare field
Now, we need to find out how many days it will take for this group to reap a 16-hectare field. The total work for a 16-hectare field is:
\[ \text{Total work} = 16 \text{ hectares} \times 30 \text{ units/hectare} = 480 \text{ units of work} \]

Using the formula:
\[ \text{Time} = \frac{\text{Total work}}{\text{Total efficiency}} \]
\[ \text{Time} = \frac{480 \text{ units}}{60 \text{ units/day}} = 8 \text{ days} \]

### Conclusion
Thus, the time required for 6 men, 4 women, and 7 children to reap the 16-hectare field is **8 days**.

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