Segmented uphill timing — finding the first-third time: In hiking to Purnagiri, a man takes half as long again to climb the second third as the first third, and a quarter as long again for the last third as for the second third. If the total time is 5 h 50 min, how long did the first third take?

Introduction / Context:
When a trip is split into equal-distance thirds with specified time ratios per segment, one can express all times in terms of the first segment and sum to match the given total.

Given Data / Assumptions:

• Let T be time for the first third.
• Second third time = 1.5T (half as long again).
• Third third time = 1.25 * (second third) = 1.25 * 1.5T = 1.875T.
• Total time = 5 h 50 min = 350 min.

Concept / Approach:
Sum the three segment times: T + 1.5T + 1.875T = 4.375T. Solve 4.375T = 350 min for T.

Step-by-Step Solution:

Verification / Alternative check:
Segment times: 80, 120, 150 min; sum = 350 min = 5 h 50 min, matching the condition.

Why Other Options Are Wrong:
72, 81, 88, 75 min do not satisfy the stated ratios when summed to the total.

Common Pitfalls:
Interpreting “half as long again” incorrectly as 0.5T instead of 1.5T, or mixing time ratios with distance ratios.

Final Answer:
80 min