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To solve this problem, we need to analyze the scenario and calculate the longer distance between points \( X \) and \( Y \) along the circumference of the circular track.

### Step-by-Step Solution

1. **Given Information**:
   - The circular track has a circumference of 540 miles.
   - Tom's speed = 20 miles/hour, and Harry's speed = 4 miles/hour.
   - They meet for the 17th time at point \( X \) and for the 27th time at point \( Y \).

2. **Relative Speed**:
   Since they are running in opposite directions, their relative speed is the sum of their speeds:
   \[
   \text{Relative speed} = 20 + 4 = 24 \text{ miles/hour}
   \]

3. **Time Between Meetings**:
   The time taken for one meeting (or lap) around the track in terms of their relative speed is:
   \[
   \text{Time for one meeting} = \frac{\text{Circumference}}{\text{Relative speed}} = \frac{540}{24} = 22.5 \text{ hours}
   \]

4. **Distance Covered Between Meetings**:
   Since they meet every 22.5 hours, we calculate the distance they cover along the track for each meeting:
   \[
   \text{Distance covered in one meeting} = 22.5 \times 20 = 450 \text{ miles (for Tom)}
   \]
   This calculation matches Tom's speed multiplied by the time per meeting, considering he goes around the track in steps of 450 miles.

5. **Meeting at Points \( X \) and \( Y \)**:
   - The 17th meeting at point \( X \) means that Tom has covered \( 17 \times 450 \) miles along the track.
   - The 27th meeting at point \( Y \) means Tom has covered \( 27 \times 450 \) miles along the track.

6. **Difference in Distance**:
   To find the distance between points \( X \) and \( Y \), calculate the difference in the number of laps:
   \[
   \text{Difference in meetings} = (27 - 17) \times 450 = 10 \times 450 = 4500 \text{ miles}
   \]


Tom and Harry start running around a circular track of circumference 540 miles. They start running from the same point but in opposite directions. Their respective speeds are 20 miles/hour and 4 miles/hour. Their 17th meeting occurs at point X and the 27th meeting occurs at point Y. Find the longer distance (along the circumference) between points X and Y.

Learn how to determine the longer distance between two points on a circular track using speeds and meetings. This problem involves the concept of relative speed in circular motion, calculations for time intervals between meetings, and distance covered. Suitable for students in Grades 9-11.

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