Kathy left home and drove toward the recycling plant at an average speed of 24 km/h Heather left three hours later and drove in the same direction but with an average speed of 60km/h.Find the number of hours Kathryn drove before Heather caught

To solve this problem, we can use the concept of relative motion and set up equations based on distance, speed, and time.

### Step-by-step solution:

Let:
- \( t \) be the number of hours Kathy drove before Heather caught up.
- Kathy's speed \( v_K = 24 \) km/h.
- Heather's speed \( v_H = 60 \) km/h.
- The time Heather drove is \( t - 3 \), since she started 3 hours later.

**Step 1: Set up the equation for distance traveled by both.**

Since they both travel the same distance by the time Heather catches up with Kathy, we can write:

\[
\text{Distance traveled by Kathy} = \text{Distance traveled by Heather}
\]

Using the formula for distance \( d = v \times t \), we have:

\[
24t = 60(t - 3)
\]

**Step 2: Solve for \( t \).**

Expand the right-hand side:

\[
24t = 60t - 180
\]

Now, subtract \( 60t \) from both sides:

\[
24t - 60t = -180
\]

Simplify:

\[
-36t = -180
\]

Divide both sides by \(-36\):

\[
t = \frac{180}{36} = 5
\]

So, Kathy drove for **5 hours** before Heather caught up.

---

Would you like more details on any step?

Here are some related questions to expand your understanding:
1. What if Heather had driven at 80 km/h? How would that change the result?
2. How far did both Kathy and Heather travel when they met?
3. How would the solution change if Heather left 2 hours later instead of 3?
4. What would be the time difference if Kathy had driven at 30 km/h instead?
5. How can we graph the motion of both Kathy and Heather to visualize this scenario?

**Tip:** When two objects move in the same direction, the relative speed helps in determining how fast one catches up with the other.

Kathy left home and drove toward the recycling plant at an average speed of 24 km/h. Heather left three hours later and drove in the same direction but with an average speed of 60 km/h. Find the number of hours Kathy drove before Heather caught up.

Learn how to solve a relative motion problem involving two drivers, Kathy and Heather. Kathy drives at 24 km/h, and Heather at 60 km/h, but Heather starts 3 hours later. This solution walks through the step-by-step process of finding when Heather catches up to Kathy, using distance-speed-time relationships. Suitable for Grades 8-10.

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