Zip left the house at noon and rode their bike at a speed of 9 miles per hour. Ten minutes later, Mimi left the same house at a speed of 14 miles per hour. At a certain point in time, Mimi caught up with Zip, and so they both stopped biking.How much total time did Zip bike?

To find out how much total time Zip biked, we need to calculate the point at which Mimi caught up with Zip. Let's break it down step by step.

### Step 1: Define Variables
Let:
- \( t \) = the total time Zip biked (in hours).
- Since Mimi started 10 minutes later, the time Mimi biked will be \( t - \frac{1}{6} \) (since 10 minutes is \( \frac{10}{60} = \frac{1}{6} \) hours).

### Step 2: Use the Distance Formula
The formula for distance is:
\[
\text{Distance} = \text{Speed} \times \text{Time}
\]
For both Zip and Mimi, the distances they biked must be equal at the point where Mimi catches up. 

- Distance Zip biked = \( 9 \times t \)
- Distance Mimi biked = \( 14 \times \left( t - \frac{1}{6} \right) \)

At the point when Mimi catches up, the distances are equal, so:
\[
9 \times t = 14 \times \left( t - \frac{1}{6} \right)
\]

### Step 3: Solve the Equation
Now, let's solve for \( t \).

First, expand the right-hand side:
\[
9t = 14 \times t - \frac{14}{6}
\]
\[
9t = 14t - \frac{7}{3}
\]

Next, subtract \( 9t \) from both sides:
\[
0 = 5t - \frac{7}{3}
\]

Add \( \frac{7}{3} \) to both sides:
\[
\frac{7}{3} = 5t
\]

Now, solve for \( t \) by dividing both sides by 5:
\[
t = \frac{7}{3} \times \frac{1}{5} = \frac{7}{15}
\]

### Step 4: Convert Time to Minutes
Since \( t = \frac{7}{15} \) hours, we can convert this into minutes by multiplying by 60:
\[
t = \frac{7}{15} \times 60 = 28 \text{ minutes}
\]

### Conclusion
Zip biked for a total of **28 minutes** before Mimi caught up.

Would you like more details on any part of the process? Here are some related questions to explore:

1. How far did Zip travel before being caught by Mimi?
2. How long did Mimi bike before catching up with Zip?
3. What if Mimi had biked at a slower speed? How would that affect the outcome?
4. What is the significance of the difference in their speeds for solving this problem?
5. How could this problem change if they both left at the same time?

**Tip:** When working with time, always make sure to convert minutes into hours if needed for consistency in calculations.

Zip left the house at noon and rode their bike at a speed of 9 miles per hour. Ten minutes later, Mimi left the same house at a speed of 14 miles per hour. At a certain point in time, Mimi caught up with Zip, and so they both stopped biking. How much total time did Zip bike?

Learn how to solve a biking distance problem where Zip and Mimi ride at different speeds. Mimi catches up with Zip after a time. This solution uses the distance-speed-time relationship and algebra to determine how long Zip biked before being caught. Suitable for students in Grades 6-8.

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