One person can complete a typing project in 3 hours, and another can complete the same project in 8 hours. How long will it take the two people working together to complete the project?

To determine how long it will take for two people working together to complete the typing project, we can use the concept of work rates.

### Step 1: Determine the individual work rates.
- The first person can complete the project in 3 hours, so their work rate is \( \frac{1}{3} \) of the project per hour.
- The second person can complete the project in 8 hours, so their work rate is \( \frac{1}{8} \) of the project per hour.

### Step 2: Add the work rates together.
When both people work together, their combined work rate is the sum of their individual work rates:
\[
\text{Combined work rate} = \frac{1}{3} + \frac{1}{8}
\]

### Step 3: Find the common denominator and add the fractions.
To add \( \frac{1}{3} \) and \( \frac{1}{8} \), find the common denominator, which is 24:
\[
\frac{1}{3} = \frac{8}{24}, \quad \frac{1}{8} = \frac{3}{24}
\]
\[
\text{Combined work rate} = \frac{8}{24} + \frac{3}{24} = \frac{11}{24}
\]

### Step 4: Calculate the time to complete the project.
The combined work rate of \( \frac{11}{24} \) means that together, they complete \( \frac{11}{24} \) of the project per hour. The time \( t \) required to complete the entire project is the reciprocal of the combined work rate:
\[
t = \frac{24}{11} \text{ hours}
\]
\[
t \approx 2.18 \text{ hours}
\]

So, it will take approximately **2 hours and 11 minutes** for the two people working together to complete the project.

Would you like further details or have any other questions?

### Related Questions:
1. If the first person worked for 1 hour alone before the second person joined, how much time would they take together to finish the remaining work?
2. If both start working at the same time but the first person leaves after 1 hour, how much time would the second person need to finish the remaining work?
3. How long would it take if a third person, who completes the project in 5 hours, joined them?
4. What is the total work rate if only the second person and a third person (who completes the project in 6 hours) work together?
5. If the first person works twice as fast as before, how long would it take them both to complete the project together?

### Tip:
When solving problems involving work rates, remember that rates can be added together directly if the tasks are done simultaneously.

Discover how to calculate the time required for two people to complete a project with different work rates. This problem involves determining the combined work rate using fractions and finding the reciprocal to calculate the completion time. Suitable for Grades 9-12.

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