We are given the following information:

• 10 men and 15 women together can complete a work in 6 days.
• One man alone takes 100 days to complete the same work.

Let's define the following variables:

• $M$ = the amount of work one man can do in one day.
• $W$ = the amount of work one woman can do in one day.

Step 1: Work done by 10 men and 15 women together

In 6 days, 10 men and 15 women together complete the entire work. So, the work done by 10 men and 15 women in one day is:

$\text{Work done by 10 men and 15 women in one day} = \frac{1}{6}$

The work done by 10 men in one day is $10M$, and the work done by 15 women in one day is $15W$. Therefore, we have:

$10M + 15W = \frac{1}{6}$

Step 2: Work done by one man in one day

It is given that one man takes 100 days to complete the work. So, the work done by one man in one day is:

$M = \frac{1}{100}$

Step 3: Substitute the value of $M$

Substituting $M = \frac{1}{100}$ into the equation $10M + 15W = \frac{1}{6}$:

$10 \times \frac{1}{100} + 15W = \frac{1}{6}$

This simplifies to:

$\frac{1}{10} + 15W = \frac{1}{6}$

Step 4: Solve for $W$

Now, subtract $\frac{1}{10}$ from both sides:

$15W = \frac{1}{6} - \frac{1}{10}$

Find the common denominator and simplify:

$15W = \frac{5}{30} - \frac{3}{30} = \frac{2}{30} = \frac{1}{15}$

Now, divide both sides by 15:

$W = \frac{1}{15 \times 15} = \frac{1}{225}$

Step 5: Calculate the time taken by one woman

The amount of work done by one woman in one day is $W = \frac{1}{225}$. Therefore, the number of days required for one woman to complete the work is:

$\text{Time taken by one woman} = \frac{1}{W} = 225 \text{ days}$

Thus, one woman alone will take 225 days to complete the same work.

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Would you like further details or have any questions?

Here are 5 related questions to extend your understanding:

1. How would the solution change if one man took 50 days to complete the work?
2. What if 5 men and 10 women work together? How many days would they take to finish the work?
3. Can you calculate how many days 20 men and 30 women would take to complete the work together?
4. How can this problem be solved if only the ratio of men's and women's work efficiency is given?
5. What happens if we introduce a third group of workers, like children, and they have their own rate of work?

Tip: When solving work-related problems, breaking down the problem into "work done per day" simplifies calculations!