A jogger leaves a point on a fitness trail running at a rate of 7 miles per hour. Ten minutes later, a second jogger leaves from the same location running at 8 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

To solve this problem, we can break it down into two parts: 

1. **Finding the time it takes for the second jogger to overtake the first.**
2. **Calculating the distance each jogger has run at that point.**

### 1. Time to Overtake

Let's denote:
- \( t_1 \) as the time (in hours) the first jogger runs before being overtaken.
- \( t_2 \) as the time (in hours) the second jogger runs.

Given:
- The first jogger runs at 7 miles per hour.
- The second jogger runs at 8 miles per hour.
- The second jogger starts 10 minutes (or \(\frac{1}{6}\) hours) later.

Since the second jogger starts 10 minutes later, we have:
\[
t_2 = t_1 - \frac{1}{6}
\]

When the second jogger overtakes the first, both will have covered the same distance. The distance each jogger covers is given by the formula \( \text{distance} = \text{speed} \times \text{time} \).

So, for the first jogger:
\[
\text{Distance}_1 = 7 \times t_1
\]

And for the second jogger:
\[
\text{Distance}_2 = 8 \times t_2 = 8 \times \left(t_1 - \frac{1}{6}\right)
\]

Since both distances are equal when the second jogger overtakes the first:
\[
7t_1 = 8\left(t_1 - \frac{1}{6}\right)
\]

Now, let's solve for \( t_1 \).

\[
7t_1 = 8t_1 - \frac{8}{6}
\]
\[
7t_1 - 8t_1 = -\frac{4}{3}
\]
\[
-t_1 = -\frac{4}{3 \times 7}
\]
\[
t_1 = \frac{4}{21} \text{ hours}
\]

### 2. Distance Covered

Now that we have the time \( t_1 = \frac{4}{21} \) hours, let's calculate the distance each jogger runs.

- For the first jogger:
  \[
  \text{Distance}_1 = 7 \times \frac{4}{21} = \frac{28}{21} = \frac{4}{3} \text{ miles}
  \]

- For the second jogger:
  \[
  \text{Distance}_2 = 8 \times \left(\frac{4}{21} - \frac{1}{6}\right) 
  \]
  
Since \(\frac{4}{21} - \frac{1}{6} = \frac{4}{21} - \frac{7}{42} = \frac{8 - 7}{42} = \frac{1}{42} \):
  \[
  \text{Distance}_2 = 8 \times \frac{1}{42} = \frac{8}{42} \approx 0.19 \text{ miles} 
  \]

The solution shows the time it takes to overtake is approximately \( 11.43 \) minutes.

Learn how to solve a jogging problem involving two joggers starting at different times and speeds. Calculate the time it takes for the second jogger to overtake the first and the distances each has covered. Suitable for Grades 7-9.

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