If 10 men or 20 women or 40 children can do a piece of work in 7 months, then 5 men, 5 women and 5 children together can do half of the work in :

Options

A. 6 months
B. 4 months
C. 5 months
D. 8 months

Answer

To solve the problem step by step, we need to determine how long it will take for 5 men, 5 women, and 5 children to complete half of the work.

### Step 1: Determine the work done by each group in one month

We know that:
- 10 men can complete the work in 7 months.
- 20 women can complete the work in 7 months.
- 40 children can complete the work in 7 months.

Let's calculate the work done by each group in one month.

1. **Work done by 10 men in one month:**
\[ \text{Work done by 10 men in 7 months} = 1 \text{ (whole work)} \]
\[ \text{Work done by 10 men in 1 month} = \frac{1}{7} \]
Therefore, the work done by 1 man in one month is:
\[ \text{Work done by 1 man in 1 month} = \frac{1}{7} \div 10 = \frac{1}{70} \]

2. **Work done by 20 women in one month:**
\[ \text{Work done by 20 women in 7 months} = 1 \text{ (whole work)} \]
\[ \text{Work done by 20 women in 1 month} = \frac{1}{7} \]
Therefore, the work done by 1 woman in one month is:
\[ \text{Work done by 1 woman in 1 month} = \frac{1}{7} \div 20 = \frac{1}{140} \]

3. **Work done by 40 children in one month:**
\[ \text{Work done by 40 children in 7 months} = 1 \text{ (whole work)} \]
\[ \text{Work done by 40 children in 1 month} = \frac{1}{7} \]
Therefore, the work done by 1 child in one month is:
\[ \text{Work done by 1 child in 1 month} = \frac{1}{7} \div 40 = \frac{1}{280} \]

### Step 2: Calculate the total work done by 5 men, 5 women, and 5 children in one month

Now, we can find the total work done by 5 men, 5 women, and 5 children in one month:
- Work done by 5 men in one month:
\[ 5 \times \frac{1}{70} = \frac{5}{70} = \frac{1}{14} \]

- Work done by 5 women in one month:
\[ 5 \times \frac{1}{140} = \frac{5}{140} = \frac{1}{28} \]

- Work done by 5 children in one month:
\[ 5 \times \frac{1}{280} = \frac{5}{280} = \frac{1}{56} \]

### Step 3: Combine the work done by all groups

Now, we add the work done by 5 men, 5 women, and 5 children in one month:
\[ \text{Total work done in one month} = \frac{1}{14} + \frac{1}{28} + \frac{1}{56} \]

To add these fractions, we need a common denominator. The least common multiple of 14, 28, and 56 is 56.
- Convert each fraction:
\[ \frac{1}{14} = \frac{4}{56}, \quad \frac{1}{28} = \frac{2}{56}, \quad \frac{1}{56} = \frac{1}{56} \]
- Now add them:
\[ \frac{4}{56} + \frac{2}{56} + \frac{1}{56} = \frac{7}{56} = \frac{1}{8} \]

### Step 4: Determine the time to complete half of the work

Since 5 men, 5 women, and 5 children together can complete \(\frac{1}{8}\) of the work in one month, we need to find out how long it will take them to complete half of the work (\(\frac{1}{2}\)):
\[ \text{Time to complete half of the work} = \frac{\frac{1}{2}}{\frac{1}{8}} = \frac{1}{2} \times 8 = 4 \text{ months} \]

### Final Answer:
Thus, 5 men, 5 women, and 5 children together can do half of the work in **4 months**.

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