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Let's solve question 25:

The problem states that two men can complete a job together in 20 days. Working alone, it would take the slower man 9 days longer than it would take the faster man to complete the job.

Let:
- \( x \) = the number of days it takes the faster man to complete the job alone.
- \( x + 9 \) = the number of days it takes the slower man to complete the job alone.

The rates of work are as follows:
- Faster man’s work rate = \( \frac{1}{x} \) (jobs per day),
- Slower man’s work rate = \( \frac{1}{x + 9} \) (jobs per day).

Together, they complete the job in 20 days, so their combined rate is \( \frac{1}{20} \). The equation representing their combined work rate is:

\[
\frac{1}{x} + \frac{1}{x + 9} = \frac{1}{20}
\]

### Solving this equation:

1. First, find a common denominator on the left-hand side:
\[
\frac{(x + 9) + x}{x(x + 9)} = \frac{1}{20}
\]
\[
\frac{2x + 9}{x(x + 9)} = \frac{1}{20}
\]

2. Cross multiply to eliminate the fractions:
\[
20(2x + 9) = x(x + 9)
\]
\[
40x + 180 = x^2 + 9x
\]

3. Rearrange the equation to form a quadratic equation:
\[
x^2 - 31x - 180 = 0
\]

4. Solve this quadratic equation using the quadratic formula:
\[
x = \frac{-(-31) \pm \sqrt{(-31)^2 - 4(1)(-180)}}{2(1)}
\]
\[
x = \frac{31 \pm \sqrt{961 + 720}}{2}
\]
\[
x = \frac{31 \pm \sqrt{1681}}{2}
\]
\[
x = \frac{31 \pm 41}{2}
\]

Thus, \( x = \frac{31 + 41}{2} = 36 \) (we discard the negative solution).

So, the faster man takes 36 days, and the slower man takes \( 36 + 9 = 45 \) days to complete the job alone.

### Answer:
The number of days it takes the slower man to complete the job is **45 days**, so the correct answer is **C)**.

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Would you like further clarification or additional details? Here are five related questions for further practice:

1. If two workers can complete a task in 12 days and one takes 5 days longer than the other, how many days will the faster worker take?
2. How would the formula change if instead of taking 9 days longer, the slower worker took half as long as the faster one?
3. If the combined work rate was \( \frac{1}{30} \) instead of \( \frac{1}{20} \), how would that affect the solution?
4. How many days would it take both workers if one of them worked half the speed of the other?
5. If three people with different rates were involved, how would the equation extend?

**Tip:** Always look for opportunities to simplify the equation before solving it algebraically to save time.

Working together, two men can complete a job in 20 days. Working alone, however, it would take one man 9 days longer than it would take the other to complete the job. How many days would it take the slower man to complete the same job?

Learn how to solve a work rate problem involving two workers, where one takes 9 days longer than the other. This problem uses algebraic techniques and the quadratic formula to find how many days it would take the slower man to complete the task alone. Suitable for students in Grades 9-11.

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