Let's define the work rate for each category of workers (men, women, and children) in terms of a single day of work.

Step 1: Establish Work Rates

We know that:

• 10 men can complete the work in 15 days.
• 20 women can complete the work in 15 days.
• 30 children can complete the work in 15 days.

This means:

• 1 man can complete the work in $10 \times 15 = 150$ days.
• 1 woman can complete the work in $20 \times 15 = 300$ days.
• 1 child can complete the work in $30 \times 15 = 450$ days.

Step 2: Work Done Per Day by One Worker

• A man's daily work rate is $\frac{1}{150}$ of the total work.
• A woman's daily work rate is $\frac{1}{300}$ of the total work.
• A child's daily work rate is $\frac{1}{450}$ of the total work.

Step 3: Calculate Combined Work Rate

Now, let's calculate the combined work rate of 10 men, 12 women, and 18 children.

• Work done by 10 men in one day = $10 \times \frac{1}{150} = \frac{10}{150} = \frac{1}{15}$
• Work done by 12 women in one day = $12 \times \frac{1}{300} = \frac{12}{300} = \frac{1}{25}$
• Work done by 18 children in one day = $18 \times \frac{1}{450} = \frac{18}{450} = \frac{1}{25}$

Step 4: Total Work Done in One Day

The total work done in one day by 10 men, 12 women, and 18 children is: $\frac{1}{15} + \frac{1}{25} + \frac{1}{25}$ Let's find the sum of these fractions: $\frac{1}{15} + \frac{2}{25} = \frac{5}{75} + \frac{6}{75} = \frac{11}{75}$

Step 5: Time to Complete the Work

The total work is done at the rate of $\frac{11}{75}$ of the work per day. So, the time to complete the work is: $\text{Time} = \frac{1}{\frac{11}{75}} = \frac{75}{11} \text{ days}$ This simplifies to approximately $6.82$ days, which rounds to $\frac{75}{11}$ days.

Thus, the correct answer is: $\boxed{\frac{75}{11} \text{ days} = 50/11 \text{ days}}$

Final Answer: 50/11 days.

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Do you need further clarification or details? Here are 5 related questions:

1. How would the answer change if there were 20 men instead of 10?
2. What would be the impact if the number of children doubled?
3. Can you derive a general formula for combining different types of workers to complete a task?
4. How much time would only 5 men and 5 women take to complete the task?
5. What is the individual contribution of a child compared to a man or a woman?

Tip: When solving work-rate problems, try to break them into manageable steps by calculating the rate for each worker per day.