A jogger leaves a point on a fitness trail running at a rate of 4 miles per hour. Twenty minutes later, a second jogger leaves from the same location running at 5 miles per hour. How long will it take the second jogger to overtake the first?
_____ min

How far will each have run at that point?
_____ mi

Let's break down the problem step by step.

### Step 1: Understand the situation
- **Jogger 1** starts first and runs at 4 miles per hour.
- **Jogger 2** starts 20 minutes later and runs at 5 miles per hour.
- We need to find out how long it will take Jogger 2 to overtake Jogger 1 and how far each will have run by that time.

### Step 2: Determine the time and distance
- Let \( t \) be the time (in hours) it takes for Jogger 2 to overtake Jogger 1 *after* Jogger 2 starts running.

In 20 minutes (which is \( \frac{1}{3} \) of an hour), Jogger 1 will have traveled:
\[
\text{Distance Jogger 1 covers in 20 minutes} = 4 \text{ miles/hour} \times \frac{1}{3} \text{ hours} = \frac{4}{3} \text{ miles} \approx 1.33 \text{ miles}
\]
Now, after Jogger 2 starts, the distance each jogger covers can be expressed as:
- **Jogger 1's distance**: \( 4 \times (t + \frac{1}{3}) \) miles (since they started 20 minutes earlier)
- **Jogger 2's distance**: \( 5 \times t \) miles

### Step 3: Set up the equation
To find when Jogger 2 overtakes Jogger 1, set the distances equal:
\[
5t = 4 \left(t + \frac{1}{3}\right)
\]
Expand and solve for \( t \):
\[
5t = 4t + \frac{4}{3}
\]
\[
t = \frac{4}{3} \text{ hours} \times \frac{60 \text{ minutes}}{1 \text{ hour}} = 80 \text{ minutes}
\]
This is the time it takes for Jogger 2 to overtake Jogger 1 after Jogger 2 starts running.

### Step 4: Calculate the total time and distances
- **Total time** for Jogger 2 to overtake Jogger 1 (including the 20-minute delay): 
  \[
  t + 20 \text{ minutes} = 80 \text{ minutes}
  \]
  
- **Distance each jogger will have run at the point of overtaking**:
  \[
  \text{Jogger 2's distance} = 5 \times \frac{4}{3} = \frac{20}{3} \text{ miles} \approx 6.67 \text{ miles}
  \]
  Since Jogger 1 is overtaken at the same point:
  \[
  \text{Jogger 1's distance} = 4 \times \frac{5}{3} = \frac{20}{3} \text{ miles} \approx 6.67 \text{ miles}
  \]

### Final Answers:
- **Time it takes for Jogger 2 to overtake Jogger 1**: \( 80 \) minutes
- **Distance each jogger will have run**: \( 6.67 \) miles

Would you like more details or have any questions? Here are 5 related questions:

1. What would the time be if Jogger 1 ran faster than Jogger 2?
2. How would the distances change if Jogger 2 started only 10 minutes later?
3. What if the speeds were reversed? How long would it take for Jogger 1 to overtake Jogger 2?
4. How would the solution change if the initial delay was in seconds instead of minutes?
5. Can you determine the distance covered by both joggers at any given time before overtaking?

**Tip:** When solving problems involving relative motion, it’s essential to set up equations that compare distances or speeds directly.

Learn how to solve a relative motion problem involving joggers. Understand how to calculate when one jogger overtakes another, considering their speeds and starting times. Includes step-by-step solution and key equations. Suitable for high school students.

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