Jim took 2 1/4 h to travel from Town X to Town Y. On the way, he passed Ken who was travelling in the opposite direction at uniform speed of 52km/h. 1h 15 mins later, Jim reached Town Y while Ken was still 15km away from Town X. Find the distance between Town X and Town Y.

Let's break down the problem and solve it step by step:

### Key Information:
- Jim took \(2 \frac{1}{4}\) hours (which is \(2.25\) hours) to travel from Town X to Town Y.
- Ken was traveling in the opposite direction at a uniform speed of 52 km/h.
- After 1 hour 15 minutes (which is \(1.25\) hours) of passing Ken, Jim reached Town Y.
- Ken was still 15 km away from Town X when Jim reached Town Y.

### Step 1: Determine Jim's time to travel after meeting Ken.
Jim met Ken \(1.25\) hours before he reached Town Y. Since Jim took \(2.25\) hours for the entire journey, this means that:
\[
\text{Time from Town X to meeting Ken} = 2.25 - 1.25 = 1 \text{ hour}.
\]

### Step 2: Calculate how far Ken traveled after passing Jim.
Ken was traveling at 52 km/h. In the 1.25 hours after passing Jim, he would have traveled:
\[
\text{Distance traveled by Ken} = 52 \times 1.25 = 65 \text{ km}.
\]

Since Ken was still 15 km away from Town X at the time Jim reached Town Y, the total distance Ken would have had to travel from the point of passing Jim to reach Town X is:
\[
\text{Total distance} = 65 + 15 = 80 \text{ km}.
\]

### Step 3: Calculate the total distance between Town X and Town Y.
When Jim met Ken, they had both been traveling for 1 hour. In that 1 hour, Ken traveled 52 km (since his speed is 52 km/h), meaning Jim also covered 52 km in that hour.

Thus, the total distance between Town X and Town Y is the distance Jim traveled (52 km) plus the remaining distance Ken had to cover (80 km):
\[
\text{Distance between Town X and Town Y} = 52 + 80 = 132 \text{ km}.
\]

### Final Answer:
The distance between Town X and Town Y is **132 km**.

Would you like more details or have any questions? Here are some related questions for further exploration:

1. What was Jim's average speed during his journey?
2. How long would it have taken Ken to reach Town Y if he had continued without stopping?
3. If Jim increased his speed by 10 km/h, how long would it take him to travel from Town X to Town Y?
4. How far would Ken have traveled after 2 hours from leaving Town Y?
5. If Town Z is 50 km further from Town Y, how long would Jim take to reach Town Z at the same speed?

**Tip:** When dealing with time in fractional hours, it's helpful to convert mixed units like hours and minutes into decimal form for easier calculation.

Solve a real-world problem involving relative motion to find the distance between Town X and Town Y. Jim travels 2 1/4 hours, passing Ken who moves at 52 km/h. This solution involves using the Distance-Speed-Time formula and explores the relative speed theorem. Suitable for students in Grades 8-10.

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