Two men are walking towards each other alongside a railway line. They both start at opposite ends of a part of the

railway track. One man is walking faster than the other. They pass each other 720 yards from the nearer of the two

ends. When each man reaches the other end of the track, they stop for a cigarette. Both men take exactly the same

amount of time to smoke a cigarette. Then, they turn around and head back to the original starting position. This time

they meet 400 yards from the other end of the track. What is the length of the track?

Let's define the problem and work through it step by step.

### Key Variables:
1. Let the length of the track be \( L \) yards.
2. Let the speed of the first man be \( v_1 \) yards per minute.
3. Let the speed of the second man be \( v_2 \) yards per minute.

### First Meeting:
- They meet 720 yards from the nearer end (we assume this is the starting point of the first man).
- This means the first man walks 720 yards, and the second man walks \( L - 720 \) yards.
- The time they spend walking to this first meeting point is the same for both:
  \[
  \frac{720}{v_1} = \frac{L - 720}{v_2}
  \]
  From this, we can express the relationship between their speeds as:
  \[
  v_2 = \frac{L - 720}{720} v_1
  \]

### Second Meeting (after the cigarette break):
- They meet 400 yards from the other end of the track, meaning the first man has walked \( L - 400 \) yards, and the second man has walked 400 yards.
- The time they spend walking to this second meeting point is the same for both:
  \[
  \frac{L - 400}{v_1} = \frac{400}{v_2}
  \]
  Substituting the expression for \( v_2 \) from earlier:
  \[
  \frac{L - 400}{v_1} = \frac{400}{\frac{L - 720}{720} v_1}
  \]
  Simplifying:
  \[
  \frac{L - 400}{1} = \frac{400 \times 720}{L - 720}
  \]
  \[
  L - 400 = \frac{288000}{L - 720}
  \]
  Multiply both sides by \( (L - 720) \):
  \[
  (L - 400)(L - 720) = 288000
  \]
  Expand the left-hand side:
  \[
  L^2 - 1120L + 288000 = 288000
  \]
  Simplifying the equation:
  \[
  L^2 - 1120L = 0
  \]
  Factor the equation:
  \[
  L(L - 1120) = 0
  \]
  Thus, \( L = 1120 \) yards.

### Conclusion:
The length of the track is 1120 yards.

Would you like further details or clarifications?

Here are 5 related questions to extend this topic:
1. How does the concept of relative speed play a role in solving this problem?
2. How would the solution change if one man took a longer break than the other?
3. Can this problem be solved geometrically using distance-time graphs?
4. What other real-world applications involve similar concepts of meeting points and distances?
5. How does solving this algebraically help in understanding motion?

**Tip:** When solving problems involving meeting points, it's often helpful to establish relationships between time, speed, and distance, as these variables are linked

Two men are walking towards each other alongside a railway line. They both start at opposite ends of a part of the railway track. One man is walking faster than the other. They pass each other 720 yards from the nearer of the two ends. When each man reaches the other end of the track, they stop for a cigarette. Both men take exactly the same amount of time to smoke a cigarette. Then, they turn around and head back to the original starting position. This time they meet 400 yards from the other end of the track. What is the length of the track?

This problem involves two men walking towards each other on a railway track, meeting at different points, and solving for the track length using algebraic relationships and quadratic equations. Learn how to solve this advanced relative speed problem step by step. Suitable for Grades 10-12 students.

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