Ten men and fifteen women together can complete a work in 6 days. One man alone takes 100 days to complete the same work. In how many days will one woman alone complete the work?

Introduction / Context:
This question involves comparing the efficiencies of men and women in a work problem. You are given the combined time for a group of 10 men and 15 women, and the individual time for one man. Using this, you must determine the time required for one woman alone to complete the same work. The problem is a direct application of linear equations based on group work rates.

Given Data / Assumptions:
Ten men and fifteen women together can complete the work in 6 days. One man alone takes 100 days to complete the work. We assume that all men have the same work rate and all women have the same work rate, with work rates remaining constant. The total work is considered as one complete job.

Concept / Approach:
We treat the total work as 1 unit and define m as the daily work rate of one man and w as the daily work rate of one woman. The given data then leads to an equation for the group rate of 10 men and 15 women and another equation for the rate of one man. Solving these allows us to find w, and the time taken by one woman alone is the reciprocal of w. The method is straightforward algebra using total work and daily rates.

Step-by-Step Solution:
Step 1: Let the total work be 1 unit. Step 2: Let the rate of one man be m work per day, and the rate of one woman be w work per day. Step 3: One man alone takes 100 days, so his rate is m = 1 / 100 work per day. Step 4: Ten men and fifteen women working together have a combined rate of 10m + 15w work per day. Step 5: They complete the job in 6 days, so 6(10m + 15w) = 1. Step 6: Simplify this equation: 60m + 90w = 1. Step 7: Substitute m = 1 / 100 into the equation: 60 * (1 / 100) + 90w = 1. Step 8: Compute 60 * (1 / 100) = 0.6, so 0.6 + 90w = 1. Step 9: Rearrange to find w: 90w = 1 - 0.6 = 0.4. Step 10: So w = 0.4 / 90 = 4 / 900 = 1 / 225 work per day. Step 11: Time taken by one woman alone to complete the work is the reciprocal of w: time = 1 / (1 / 225) = 225 days.

Verification / Alternative check:
We can verify the result by checking the combined work. One woman completes 1 / 225 of the job per day. One man completes 1 / 100 of the job per day. For 10 men and 15 women, total daily work = 10 * (1 / 100) + 15 * (1 / 225) = 10 / 100 + 15 / 225 = 0.1 + 0.0666... = 0.1666... which is 1 / 6. Over 6 days, they complete 6 * (1 / 6) = 1 job, matching the original statement. This confirms that the result of 225 days for one woman is correct.

Why Other Options Are Wrong:
215, 235, 240, and 250 days correspond to work rates slightly larger or smaller than 1 / 225 per day, and do not satisfy the equation 60m + 90w = 1 when combined with the known rate of one man. Only 225 days yields a rate that makes the combined total daily work equal to 1 / 6, consistent with the 6 days needed by the group.

Common Pitfalls:
Some learners incorrectly assume that the time for a woman can be found by simple proportional scaling without setting up the equation for the combined group. Others may mix up the roles of men and women or mis-handle the substitution of m into the group equation. Always express the total work and daily rates clearly, substitute known values carefully, and solve step by step.

Final Answer:
One woman alone will complete the work in 225 days.