Ramesh walked 35 km, partly at a speed of 5 km/hr and partly at 3 km/hr. If he…

Concept: if a journey has two segments walked at speeds s1 and s2 for durations t1 and t2, and a second hypothetical scenario swaps the speeds between the same two durations, the two resulting distances always satisfy D1 + D2 = (s1 + s2)(t1 + t2) — expanding the right-hand side regroups exactly into the two scenario distances. So adding the sum of speeds times the sum of durations gives the sum of the two distances directly.

1. Let t1 = number of hours Ramesh walked at 5 km/hr, and t2 = number of hours he walked at 3 km/hr.

2. The actual journey: 5t1 + 3t2 = 35.

3. The hypothetical speed-swapped journey: 3t1 + 5t2 = 37.

4. Adding the two equations: (5 + 3)t1 + (3 + 5)t2 = 35 + 37, i.e. 8t1 + 8t2 = 72.

5. Dividing by 8: t1 + t2 = 9.

6. To find each duration individually: multiplying the first equation by 5 and the second by 3, then subtracting, gives 16t1 = 64, so t1 = 4; substituting back into 5t1 + 3t2 = 35 gives t2 = 5.

Cross-check: 5(4) + 3(5) = 20 + 15 = 35, matching the actual journey; and 3(4) + 5(5) = 12 + 25 = 37, matching the hypothetical journey. Both conditions hold, confirming t1 = 4 hours and t2 = 5 hours.

So the total time Ramesh spent walking is t1 + t2 = 4 + 5 = 9 hours.