A man sets out to cycle from points P to Q and at the same time another man starts to cycle from points Q to P. After passing each other, they complete their journeys in 9 h and 4 h, respectively. Find the ratio of speeds of 1st man to that of 2nd man.

Answer

To find the ratio of the speeds of the two cyclists, we can use the relationship between speed and time. The key concept here is that if two objects travel the same distance, the ratio of their speeds is inversely proportional to the ratio of the times taken to complete the journey after they meet.

### Step-by-Step Solution:

1. **Identify the times taken after they meet:**
- The first man takes 9 hours to complete his journey after they meet.
- The second man takes 4 hours to complete his journey after they meet.

2. **Set up the ratio of the times:**
- Let \( T_1 \) be the time taken by the first man (9 hours).
- Let \( T_2 \) be the time taken by the second man (4 hours).
- The ratio of the times is \( T_1 : T_2 = 9 : 4 \).

3. **Use the formula for the ratio of speeds:**
- The formula for the ratio of speeds \( S_1 : S_2 \) is given by:
\[ S_1 : S_2 = \sqrt{T_2} : \sqrt{T_1} \]
- Substituting the values:
\[ S_1 : S_2 = \sqrt{4} : \sqrt{9} \]

4. **Calculate the square roots:**
- \( \sqrt{4} = 2 \)
- \( \sqrt{9} = 3 \)

5. **Write the final ratio:**
- Therefore, the ratio of the speeds of the first man to the second man is:
\[ S_1 : S_2 = 2 : 3 \]

### Final Answer:
The ratio of the speeds of the first man to that of the second man is \( 2 : 3 \).