10 men and 15 women finish a work in 6 days. One man alone finishes that work in 100 days. In how many days will a women finish the work ?

Options

A. 125 days
B. 150 days
C. 90 days
D. 225 days

Answer

To solve the problem step by step, we will follow the reasoning provided in the video transcript.

### Step-by-Step Solution:

1. **Understanding the Work of One Man**:
- It is given that one man can finish the work in 100 days.
- Therefore, the work done by one man in one day is:
\[ \text{Work done by 1 man in 1 day} = \frac{1}{100} \]

2. **Calculating the Work Done by 10 Men in One Day**:
- If one man does \(\frac{1}{100}\) of the work in one day, then 10 men will do:
\[ \text{Work done by 10 men in 1 day} = 10 \times \frac{1}{100} = \frac{10}{100} = \frac{1}{10} \]

3. **Setting Up the Equation for Women**:
- Let \(x\) be the number of days taken by one woman to finish the work.
- Therefore, the work done by one woman in one day is:
\[ \text{Work done by 1 woman in 1 day} = \frac{1}{x} \]
- For 15 women, the work done in one day is:
\[ \text{Work done by 15 women in 1 day} = 15 \times \frac{1}{x} = \frac{15}{x} \]

4. **Total Work Done by Men and Women Together**:
- It is given that 10 men and 15 women together finish the work in 6 days.
- Therefore, the work done by both in one day is:
\[ \text{Total work done in 1 day} = \frac{1}{6} \]

5. **Setting Up the Equation**:
- Now, we can set up the equation:
\[ \frac{1}{10} + \frac{15}{x} = \frac{1}{6} \]

6. **Finding a Common Denominator**:
- To solve for \(x\), we need to find a common denominator. The least common multiple (LCM) of 10 and 6 is 30.
- Rewriting the equation:
\[ \frac{3}{30} + \frac{15}{x} = \frac{5}{30} \]

7. **Isolating the Fraction with \(x\)**:
- Rearranging gives:
\[ \frac{15}{x} = \frac{5}{30} - \frac{3}{30} = \frac{2}{30} \]
- Simplifying:
\[ \frac{15}{x} = \frac{1}{15} \]

8. **Cross-Multiplying to Solve for \(x\)**:
- Cross-multiplying gives:
\[ 15 \cdot 15 = 1 \cdot x \implies x = 225 \]

### Final Answer:
- A woman alone will finish the work in **225 days**.