If 40 men or 60 women or 80 children can do a piece of work in 6 months, then 10 men, 10 women and 10 children together do half of the work in how may months?

Options

A. `5(6)/(13)`
B. 6
C. `5(7)/(13)`
D. `11(1)/(13)`

Answer

To solve the problem, we will first determine the work done by men, women, and children in terms of a common unit. Then we will find out how long it takes for 10 men, 10 women, and 10 children to complete half of the work.

### Step 1: Determine the total work in terms of men, women, and children.

Given:
- 40 men can complete the work in 6 months.
- 60 women can complete the work in 6 months.
- 80 children can complete the work in 6 months.

Let's calculate the total work in terms of man-months.

Total work done by 40 men in 6 months:
\[ \text{Total Work} = 40 \text{ men} \times 6 \text{ months} = 240 \text{ man-months} \]

### Step 2: Calculate the work done by 1 man in 1 month.

Since the total work is 240 man-months, the work done by 1 man in 1 month is:
\[ \text{Work done by 1 man in 1 month} = \frac{240 \text{ man-months}}{40 \text{ men} \times 6 \text{ months}} = 1 \text{ unit of work} \]

### Step 3: Calculate the work done by women and children.

Now, let's find out how much work is done by women and children in terms of the same unit.

For women:
\[ \text{Total Work} = 60 \text{ women} \times 6 \text{ months} = 360 \text{ woman-months} \]
Thus, the work done by 1 woman in 1 month is:
\[ \text{Work done by 1 woman in 1 month} = \frac{360 \text{ woman-months}}{60 \text{ women} \times 6 \text{ months}} = 1 \text{ unit of work} \]

For children:
\[ \text{Total Work} = 80 \text{ children} \times 6 \text{ months} = 480 \text{ child-months} \]
Thus, the work done by 1 child in 1 month is:
\[ \text{Work done by 1 child in 1 month} = \frac{480 \text{ child-months}}{80 \text{ children} \times 6 \text{ months}} = 1 \text{ unit of work} \]

### Step 4: Calculate the combined work done by 10 men, 10 women, and 10 children in 1 month.

Now, we can find the work done by 10 men, 10 women, and 10 children together in 1 month:
- Work done by 10 men in 1 month = \(10 \times 1 = 10\) units
- Work done by 10 women in 1 month = \(10 \times 1 = 10\) units
- Work done by 10 children in 1 month = \(10 \times 1 = 10\) units

Total work done by 10 men, 10 women, and 10 children in 1 month:
\[ \text{Total Work} = 10 + 10 + 10 = 30 \text{ units} \]

### Step 5: Calculate the time taken to complete half of the work.

Since the total work is 240 man-months, half of the work is:
\[ \text{Half Work} = \frac{240}{2} = 120 \text{ units} \]

Now, we need to find how many months it will take for them to complete 120 units of work at the rate of 30 units per month:
\[ \text{Time} = \frac{120 \text{ units}}{30 \text{ units/month}} = 4 \text{ months} \]

### Conclusion

Therefore, 10 men, 10 women, and 10 children together can complete half of the work in **4 months**.

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