A person sets out to cycle from A to B and at the same time another person starts from B to A. After passingeach other,theycomplete their journeysin16hand25h,respectively. Findtheratio ofspeedsof the 1st man to that of the 2nd man.

Options

A. `5:4`
B. `5 :3`
C. `4:5`
D. `3:5`

Answer

To solve the problem step by step, we will follow the reasoning provided in the video transcript.

### Step 1: Understand the Problem
We have two cyclists starting from points A and B, and they meet at some point. After they meet, the first cyclist takes 16 hours to reach point B, and the second cyclist takes 25 hours to reach point A.

### Step 2: Define the Variables
Let:
- \( t_1 = 16 \) hours (time taken by the first man after meeting)
- \( t_2 = 25 \) hours (time taken by the second man after meeting)

### Step 3: Use the Inverse Relationship of Speed and Time
The distance each cyclist travels after they meet is the same. Therefore, the ratio of their speeds is inversely proportional to the ratio of the times they take to complete their journeys after meeting.

### Step 4: Calculate the Ratio of Speeds
The ratio of speeds \( S_1 \) (speed of the first man) to \( S_2 \) (speed of the second man) can be expressed as:
\[ \frac{S_1}{S_2} = \frac{t_2}{t_1} \]
Substituting the values:
\[ \frac{S_1}{S_2} = \frac{25}{16} \]

### Step 5: Simplify the Ratio
To express this ratio in a more standard form, we can write it as:
\[ S_1 : S_2 = 25 : 16 \]

### Step 6: Conclusion
Thus, the ratio of the speeds of the first man to that of the second man is \( 25 : 16 \).