An old man takes 45 minutes and a young man takes 30 minutes to walk from…

Concept: When two people cover the same fixed distance in different times, their speeds are in the inverse ratio of those times. If a faster traveller starts later from the same point along the same route, the moment they draw level (meet) is found by equating the distance each has independently covered, measured from each person's own start time.

1. Old man's time for the full apartment-to-office distance is 45 minutes and the young man's is 30 minutes, so their speed ratio (old : young) is 1/45 : 1/30 = 2 : 3 (inverse of the time ratio).

2. The old man starts at 9:25 AM and the young man starts 10 minutes later, at 9:35 AM.

3. Let t be the number of minutes after 9:25 AM at which they meet. In that time the old man has walked for t minutes, while the young man, who joined 10 minutes later, has walked for (t - 10) minutes.

4. Since they meet at the same point, the distances covered are equal: with speeds in ratio 2 : 3, this gives 2t = 3(t - 10).

5. Solving: 2t = 3t - 30, so t = 30 minutes.

6. 30 minutes after 9:25 AM is 9:55 AM - the moment the young man catches up with the old man.

Cross-check: At t = 30 minutes the old man has walked only 30 of his 45 minutes and the young man only 20 of his 30 minutes (elapsed since 9:35 AM), so both are still short of the office - confirming the meeting genuinely happens en route, not at either endpoint. (Their individual arrival times at the office - 10:10 AM for the old man and 10:05 AM for the young man - both come after the 9:55 AM meeting time, as they must.)