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Let's break down the question:

**Given:**
- 10 men and 15 women can complete a task in 6 days.
- 1 man can complete the task in 100 days.
  
We need to find how many days 1 woman will take to complete the same task.

### Step-by-step solution:

1. **Work done by one man in 1 day:**
   - 1 man takes 100 days to complete the work. Therefore, the work done by 1 man in 1 day is:
   \[
   \frac{1}{100} \text{ of the total work.}
   \]

2. **Work done by 10 men in 1 day:**
   - Since the work done by 1 man in 1 day is \( \frac{1}{100} \), the work done by 10 men in 1 day is:
   \[
   \frac{10}{100} = \frac{1}{10} \text{ of the total work.}
   \]

3. **Total work done by 10 men and 15 women in 1 day:**
   - 10 men and 15 women together can complete the work in 6 days, so the work done by them in 1 day is:
   \[
   \frac{1}{6} \text{ of the total work.}
   \]
   
4. **Work done by 15 women in 1 day:**
   - The total work done by 10 men and 15 women in 1 day is \( \frac{1}{6} \). We already calculated the work done by 10 men in 1 day as \( \frac{1}{10} \).
   - So, the work done by 15 women in 1 day is:
   \[
   \frac{1}{6} - \frac{1}{10}
   \]
   To subtract these fractions, find a common denominator:
   \[
   \frac{1}{6} = \frac{5}{30}, \quad \frac{1}{10} = \frac{3}{30}
   \]
   \[
   \frac{1}{6} - \frac{1}{10} = \frac{5}{30} - \frac{3}{30} = \frac{2}{30} = \frac{1}{15}
   \]
   Therefore, 15 women together do \( \frac{1}{15} \) of the total work in 1 day.

5. **Work done by 1 woman in 1 day:**
   - Since 15 women together do \( \frac{1}{15} \) of the total work in 1 day, the work done by 1 woman in 1 day is:
   \[
   \frac{1}{15} \div 15 = \frac{1}{15 \times 15} = \frac{1}{225}
   \]

6. **Number of days 1 woman will take to complete the work:**
   - If 1 woman does \( \frac{1}{225} \) of the work in 1 day, she will take 225 days to complete the work.

### Final Answer:
1 woman will take **225 days** to complete the work, which corresponds to option (D).

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Let me know if you'd like more details! Here are five questions related to this problem:

1. If 5 men and 10 women worked together, how many days would it take them to complete the work?
2. What if 1 woman and 1 man worked together? How many days would they take to finish the task?
3. How much work can 1 man and 1 woman do together in 1 day?
4. If the number of men and women is doubled, how does that affect the completion time of the work?
5. How long would it take for 20 men and 30 women to finish the same task?

**Tip:** Understanding how to manipulate ratios and proportions is essential for solving work and time-related problems efficiently!

10 men and 15 women can do a work in 6 days. If 1 man can do this work in 100 days, then find in how many days can 1 woman do this work?

This problem explores how long it will take for one woman to complete a task when 10 men and 15 women can finish it together in 6 days. Learn to solve this using ratios, work distribution, and time concepts. Suitable for students in Grades 8-10.

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