To solve the problem, we will first determine the work done by each group (men, women, and children) in one day, and then combine their efforts to find out how many days it will take for 10 men, 4 women, and 10 children to complete the work.
### Step 1: Calculate the work done by 8 men in one day.
- If 8 men can complete the work in 12 days, then the work done by 8 men in one day is:
\[
\text{Work done by 8 men in 1 day} = \frac{1}{12} \text{ (work units)}
\]
- Therefore, the work done by 1 man in one day is:
\[
\text{Work done by 1 man in 1 day} = \frac{1}{12} \div 8 = \frac{1}{96} \text{ (work units)}
\]
### Step 2: Calculate the work done by 4 women in one day.
- If 4 women can complete the work in 48 days, then the work done by 4 women in one day is:
\[
\text{Work done by 4 women in 1 day} = \frac{1}{48} \text{ (work units)}
\]
- Therefore, the work done by 1 woman in one day is:
\[
\text{Work done by 1 woman in 1 day} = \frac{1}{48} \div 4 = \frac{1}{192} \text{ (work units)}
\]
### Step 3: Calculate the work done by 10 children in one day.
- If 10 children can complete the work in 24 days, then the work done by 10 children in one day is:
\[
\text{Work done by 10 children in 1 day} = \frac{1}{24} \text{ (work units)}
\]
- Therefore, the work done by 1 child in one day is:
\[
\text{Work done by 1 child in 1 day} = \frac{1}{24} \div 10 = \frac{1}{240} \text{ (work units)}
\]
### Step 4: Calculate the total work done by 10 men, 4 women, and 10 children in one day.
- The work done by 10 men in one day is:
\[
\text{Work done by 10 men in 1 day} = 10 \times \frac{1}{96} = \frac{10}{96} = \frac{5}{48} \text{ (work units)}
\]
- The work done by 4 women in one day is:
\[
\text{Work done by 4 women in 1 day} = 4 \times \frac{1}{192} = \frac{4}{192} = \frac{1}{48} \text{ (work units)}
\]
- The work done by 10 children in one day is:
\[
\text{Work done by 10 children in 1 day} = 10 \times \frac{1}{240} = \frac{10}{240} = \frac{1}{24} \text{ (work units)}
\]
### Step 5: Combine the work done by all groups in one day.
- Total work done in one day by 10 men, 4 women, and 10 children:
\[
\text{Total work done in 1 day} = \frac{5}{48} + \frac{1}{48} + \frac{1}{24}
\]
- To add these fractions, we need a common denominator. The least common multiple of 48 and 24 is 48.
\[
\frac{1}{24} = \frac{2}{48}
\]
- Now, adding:
\[
\text{Total work done in 1 day} = \frac{5}{48} + \frac{1}{48} + \frac{2}{48} = \frac{8}{48} = \frac{1}{6} \text{ (work units)}
\]
### Step 6: Calculate the number of days to complete the work.
- If they can complete \(\frac{1}{6}\) of the work in one day, then the total number of days to complete the entire work is:
\[
\text{Total days} = \frac{1}{\frac{1}{6}} = 6 \text{ days}
\]
Thus, **10 men, 4 women, and 10 children can complete the work in 6 days.**
