A and b start from two towns X and Y towards Y and X respectively. The meet on the way at Z. If A and B take 36 and 25 minutes respectively after meeting, to reach their destinations, what is the ration of the speeds of A and B?

To solve this problem, we can use the concept of relative speed and the relationship between distance, speed, and time.

Let:
- \( v_A \) = speed of A
- \( v_B \) = speed of B
- \( d_A \) = distance from Z to Y (which A covers after meeting)
- \( d_B \) = distance from Z to X (which B covers after meeting)

From the problem, we know:
- A takes 36 minutes to reach Y from Z.
- B takes 25 minutes to reach X from Z.

We can express the distances in terms of speed and time:
1. For A: 
   \[
   d_A = v_A \times \frac{36}{60} = v_A \times 0.6 \text{ (in hours)}
   \]
2. For B:
   \[
   d_B = v_B \times \frac{25}{60} = v_B \times \frac{5}{12} \text{ (in hours)}
   \]

Since A and B meet at point Z, the distances they travel after meeting are proportional to their speeds. The ratio of the distances covered after meeting is equal to the inverse of the ratio of their speeds:

\[
\frac{d_A}{d_B} = \frac{v_B}{v_A}
\]

Substituting the expressions for \( d_A \) and \( d_B \):

\[
\frac{v_A \times 0.6}{v_B \times \frac{5}{12}} = \frac{v_B}{v_A}
\]

Cross-multiplying gives:

\[
v_A^2 \times 0.6 = v_B^2 \times \frac{5}{12}
\]

Rearranging this gives:

\[
\frac{v_A^2}{v_B^2} = \frac{5}{12 \times 0.6} = \frac{5}{7.2} = \frac{25}{36}
\]

Taking the square root of both sides, we find:

\[
\frac{v_A}{v_B} = \frac{5}{6}
\]

Thus, the ratio of the speeds of A and B is:

\[
\boxed{\frac{5}{6}}
\]

### Questions for Further Exploration:
1. What assumptions are made about the paths taken by A and B?
2. How would the problem change if A took longer than B after meeting?
3. Can you derive a formula to find speeds if the meeting time is known?
4. What would the speeds be if the distances to their destinations were provided?
5. How does the ratio of speeds relate to the time taken to meet?

### Tip:
When dealing with speed, distance, and time problems, remember that the ratios can often be simplified by looking at how distances relate to each other, especially in meeting or collision scenarios.

A and b start from two towns X and Y towards Y and X respectively. The meet on the way at Z. If A and B take 36 and 25 minutes respectively after meeting, to reach their destinations, what is the ratio of the speeds of A and B?

Explore how to calculate the speed ratio between two travelers, A and B, based on their meeting point and subsequent travel times. A takes 36 minutes and B takes 25 minutes after they meet. Understand the concepts of speed, distance, and ratios in motion. Suitable for Grades 8-10.

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