James and Olivia are working on a project together. They decide to divide the project into two parts: Part A and Part B. James can complete Part A in 3 hours and Part B in 5 hours if he works alone. Olivia can complete Part A in 4 hours and Part B in 6 hours if she works alone.

They start working on the project together but find that Olivia had to leave early. By the time Olivia left, they had completed Part A and were halfway through Part B. The entire project took them 6 hours to complete.

Questions:

How long did Olivia work on the project before leaving?
How long did James work alone after Olivia left?

To determine how long Olivia worked on the project before leaving and how long James worked alone, let's break down the problem using their individual work rates.

First, we calculate their combined work rates for both parts of the project:

### Part A
- James's work rate for Part A: \(\frac{1}{3}\) of Part A per hour.
- Olivia's work rate for Part A: \(\frac{1}{4}\) of Part A per hour.
- Combined work rate for Part A: \(\frac{1}{3} + \frac{1}{4} = \frac{4}{12} + \frac{3}{12} = \frac{7}{12}\) of Part A per hour.

### Part B
- James's work rate for Part B: \(\frac{1}{5}\) of Part B per hour.
- Olivia's work rate for Part B: \(\frac{1}{6}\) of Part B per hour.
- Combined work rate for Part B: \(\frac{1}{5} + \frac{1}{6} = \frac{6}{30} + \frac{5}{30} = \frac{11}{30}\) of Part B per hour.

Since they completed Part A together, we can calculate the time taken to complete Part A:
\[
\text{Time to complete Part A} = \frac{1 \text{ Part A}}{\frac{7}{12} \text{ of Part A per hour}} = \frac{12}{7} \text{ hours}
\]

They were also halfway through Part B when Olivia left. Since Part B was halfway done when Olivia left, they had completed \( \frac{1}{2} \) of Part B together. To find the time taken to complete \( \frac{1}{2} \) of Part B together, we use their combined work rate for Part B:
\[
\text{Time to complete half of Part B} = \frac{\frac{1}{2} \text{ Part B}}{\frac{11}{30} \text{ of Part B per hour}} = \frac{1}{2} \times \frac{30}{11} = \frac{15}{11} \text{ hours}
\]

Adding the time for Part A and the half of Part B:
\[
\text{Total time Olivia worked} = \frac{12}{7} + \frac{15}{11} = \frac{132}{77} + \frac{105}{77} = \frac{237}{77} \approx 3.08 \text{ hours}
\]

Therefore, Olivia worked for approximately 3.08 hours before leaving.

Now, we know the entire project took 6 hours to complete. Thus, James worked alone for:
\[
\text{Time James worked alone} = 6 \text{ hours} - \frac{237}{77} \text{ hours} = 6 - \frac{237}{77} \approx 6 - 3.08 = 2.92 \text{ hours}
\]

So, James worked alone for approximately 2.92 hours after Olivia left.

### Summary
- Olivia worked on the project for approximately 3.08 hours before leaving.
- James worked alone for approximately 2.92 hours after Olivia left.

Would you like further details or have any questions?

### Additional Questions
1. What is the exact fraction form for the time Olivia worked before leaving?
2. How many hours would it have taken for Olivia to complete the entire project alone?
3. How many hours would it have taken for James to complete the entire project alone?
4. What is the total amount of work (in part units) done by Olivia?
5. What is the total amount of work (in part units) done by James?
6. How much time would it have taken James and Olivia to complete Part B if they worked together the entire time?
7. What is the total amount of work done by James after Olivia left?
8. How long would it take for Olivia to complete Part B alone?

### Tip
When solving work-rate problems, always convert individual work times to rates (work per unit of time) and sum the rates to find the combined work rate.

Explore the solution to a work rate problem involving James and Olivia working on a project together. Learn how to calculate the time Olivia worked before leaving and how long James worked alone. Suitable for high school students studying work rate problems and time and work concepts.

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