Regan can do a job in 4 hours that brandon needs 5 hours to do. If Regan works for 1 hour before Brandon joins him, how long will it take them to finish the job?

Let's solve the problem step by step.

### Step 1: Calculate the work rate of Regan and Brandon individually.
- Regan can complete the job in 4 hours, so his work rate is:
  \[
  \text{Regan's rate} = \frac{1}{4} \text{ of the job per hour}
  \]
- Brandon can complete the job in 5 hours, so his work rate is:
  \[
  \text{Brandon's rate} = \frac{1}{5} \text{ of the job per hour}
  \]

### Step 2: Calculate the work done by Regan in the first hour.
- Regan works alone for 1 hour, so the work he completes in that hour is:
  \[
  \text{Work done by Regan in 1 hour} = \frac{1}{4} \text{ of the job}
  \]

### Step 3: Determine the remaining work.
- The total job is 1, so the remaining work after Regan works for 1 hour is:
  \[
  \text{Remaining work} = 1 - \frac{1}{4} = \frac{3}{4} \text{ of the job}
  \]

### Step 4: Calculate their combined work rate.
- When Regan and Brandon work together, their combined work rate is:
  \[
  \text{Combined rate} = \frac{1}{4} + \frac{1}{5} = \frac{5 + 4}{20} = \frac{9}{20} \text{ of the job per hour}
  \]

### Step 5: Calculate the time required to complete the remaining work.
- The remaining work is \(\frac{3}{4}\) of the job, so the time it takes them to complete it together is:
  \[
  \text{Time to complete remaining work} = \frac{\text{Remaining work}}{\text{Combined rate}} = \frac{\frac{3}{4}}{\frac{9}{20}} = \frac{3}{4} \times \frac{20}{9} = \frac{60}{36} = \frac{5}{3} \text{ hours}
  \]

### Step 6: Calculate the total time taken.
- Regan worked alone for 1 hour, and then they worked together for \(\frac{5}{3}\) hours, so the total time taken is:
  \[
  \text{Total time} = 1 + \frac{5}{3} = \frac{3}{3} + \frac{5}{3} = \frac{8}{3} \text{ hours}
  \]
  \[
  \frac{8}{3} \text{ hours} \approx 2.67 \text{ hours} = 2 \text{ hours and } 40 \text{ minutes}
  \]

So, it will take them approximately **2 hours and 40 minutes** to finish the job.

Would you like more details or have any other questions?

### Follow-up Questions:
1. What would be the total time if Regan and Brandon started working together from the beginning?
2. How much of the job would be completed after 2 hours if both started together?
3. How long would it take if Regan and Brandon both increased their speed by 50%?
4. What is the ratio of work done by Regan to the work done by Brandon in this scenario?
5. If the job is split into two equal parts, how long would each take to complete half of the job working together?

### Tip:
When solving work rate problems, always break down the task into smaller, more manageable parts and calculate each segment separately before combining them for the final answer.

Learn how to solve a work rate problem involving Regan and Brandon, where Regan works alone for an hour before Brandon joins. Understand the concept of work rates and time to complete tasks. Suitable for high school students.

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