One person can complete a task 8 hours sooner than another person. Working together, both people can perform the task in 3 hours. How many hours does it take each person to complete the task working alone?

Let's denote the time it takes the slower person to complete the task working alone as \( x \) hours. Then, the faster person can complete the task in \( x - 8 \) hours.

Since both people are working together, their combined rate of work is the sum of their individual rates. The rate of work for each person is given by:

- The slower person's rate: \( \frac{1}{x} \) (since they complete the task in \( x \) hours)
- The faster person's rate: \( \frac{1}{x - 8} \) (since they complete the task in \( x - 8 \) hours)

Working together, their combined rate is \( \frac{1}{x} + \frac{1}{x - 8} \). We know they can complete the task together in 3 hours, so their combined rate is \( \frac{1}{3} \).

Thus, we can set up the equation:

\[
\frac{1}{x} + \frac{1}{x - 8} = \frac{1}{3}
\]

### Solving the equation:

1. Multiply the entire equation by \( 3x(x - 8) \) to eliminate the denominators:

\[
3(x - 8) + 3x = x(x - 8)
\]

2. Expand both sides:

\[
3x - 24 + 3x = x^2 - 8x
\]

3. Combine like terms:

\[
6x - 24 = x^2 - 8x
\]

4. Rearrange the equation to form a quadratic equation:

\[
x^2 - 14x + 24 = 0
\]

5. Solve the quadratic equation using the quadratic formula:

\[
x = \frac{-(-14) \pm \sqrt{(-14)^2 - 4(1)(24)}}{2(1)}
\]

\[
x = \frac{14 \pm \sqrt{196 - 96}}{2}
\]

\[
x = \frac{14 \pm \sqrt{100}}{2}
\]

\[
x = \frac{14 \pm 10}{2}
\]

So, \( x = \frac{14 + 10}{2} = 12 \) or \( x = \frac{14 - 10}{2} = 2 \).

Since \( x - 8 \) must be positive (because time cannot be negative), we reject \( x = 2 \). Therefore, \( x = 12 \).

### Conclusion:

- The slower person takes 12 hours to complete the task alone.
- The faster person takes \( 12 - 8 = 4 \) hours to complete the task alone.

Would you like further details or explanations?

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Here are 5 follow-up questions to deepen your understanding:

1. How would the equation change if the slower person took 10 hours more than the faster person?
2. What would happen to the combined time if both people worked at the same speed?
3. How can you use the concept of rates to solve problems involving machines or systems?
4. What if three people were working together instead of two? How would you set up that equation?
5. How would the problem change if both people were working at different rates but didn't complete the task together?

**Tip:** When working on combined rate problems, always think in terms of how much of the task each person completes per unit of time (i.e., their rate of work). This makes setting up the equation easier.

Learn how to solve a work rate problem involving two people completing a task together. One person finishes 8 hours sooner than the other, and together they finish in 3 hours. This step-by-step solution covers key algebraic concepts, including setting up and solving quadratic equations. Suitable for students in Grades 9-11.

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