A walks from Jammu to Delhi and at the same time B starts walking from delhi to jammu. After passing each other, they complete their journey in 361 hours and 289 hours. Find the ratio of speed of A to that of B ?

Let's denote the speed of A as \( S_A \) and the speed of B as \( S_B \).

The time taken by A to reach Delhi after passing B is 361 hours, and the time taken by B to reach Jammu after passing A is 289 hours. 

According to the problem, the distance traveled by each person after passing each other is the same because they are walking towards each other and meet at some point. The relationship between speed, distance, and time tells us that:

\[
\text{Distance} = \text{Speed} \times \text{Time}
\]

Let the distance from the meeting point to Delhi be \( D_A \) and from the meeting point to Jammu be \( D_B \). Then:

\[
D_A = S_A \times 361
\]
\[
D_B = S_B \times 289
\]

Since \( D_A = D_B \) (because they meet at the same point):

\[
S_A \times 361 = S_B \times 289
\]

To find the ratio of \( S_A \) to \( S_B \), we divide both sides by \( S_B \times 361 \):

\[
\frac{S_A}{S_B} = \frac{289}{361}
\]

Next, simplify the fraction by finding the greatest common divisor (GCD) of 289 and 361. Notice that:

\[
289 = 17 \times 17 = 17^2
\]
\[
361 = 19 \times 19 = 19^2
\]

Thus:

\[
\frac{S_A}{S_B} = \frac{17^2}{19^2} = \left(\frac{17}{19}\right)^2
\]

Therefore, the ratio of the speed of A to the speed of B is:

\[
\boxed{\frac{17}{19}}
\]

### Final Answer:
- The ratio of the speed of A to the speed of B is **17:19**.

Would you like further details on the steps or have any other questions?

### Related Questions:

1. How would the problem change if A took 400 hours and B took 250 hours after passing each other?
2. What if the distance between Jammu and Delhi is known? How can we find the actual speeds of A and B?
3. How would the problem change if A walked 10% faster than he currently does?
4. Can this method be applied if A and B use different modes of transport (e.g., A walks and B cycles)?
5. How does the ratio change if they meet at a different point other than halfway?
6. What if B had a head start of 50 km before A started walking?
7. How can we solve this problem if the time taken by A and B to complete their journey is given in different units?
8. How does the speed ratio affect the time at which A and B meet? 

**Tip:** Ratios often simplify problems significantly by eliminating the need to know the actual values, focusing instead on the relative differences.

Find the ratio of speed between A and B based on their journey times after passing each other. Detailed step-by-step solution provided, focusing on speed, distance, and time relationships.

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