A man covers a distance of 1200 km in 70 days when he rests for 9 hours each day and walks at a constant speed during the remaining hours. If instead he rests for 10 hours each day but walks with a speed that is one and a half times his previous walking speed, in how many days will he cover a distance of 840 km?

Introduction / Context:
This problem involves work rate and time over several days, with changes in both rest duration and walking speed. First, the man covers a known distance under one schedule. Then his rest time per day increases and speed also increases. We must compute how many days he now needs to cover a different distance.

Given Data / Assumptions:

• Initial journey: distance = 1200 km.
• Number of days for initial journey = 70 days.
• Rest per day initially = 9 hours, so walking time per day = 24 - 9 = 15 hours.
• Walking speed initially is constant, say v km/h.
• New plan: rest per day = 10 hours, so walking time per day = 14 hours.
• New walking speed = 1.5 times old speed = 3v/2.
• Distance to be covered under the new plan = 840 km.

Concept / Approach:
First, we find the original walking speed using distance = speed * time over the original 70 day schedule. Then we compute the new effective daily distance using the increased speed and changed walking hours. Finally, the required days for 840 km is distance divided by new daily distance. This uses proportional reasoning over many days.

Step-by-Step Solution:
Initially, total walking hours = 70 days * 15 hours/day = 1050 hours. The man walks 1200 km in 1050 hours, so speed v = 1200 / 1050 km/h. Simplify v = 1200 / 1050 = 8 / 7 km/h. Under the new plan, walking time per day = 14 hours. New speed = 1.5 * v = (3/2) * (8/7) = 24 / 14 = 12 / 7 km/h. Distance covered per day in the new plan = new speed * 14 hours = (12 / 7) * 14 = 24 km per day. Now, required distance = 840 km. Number of days needed = 840 / 24. Compute 840 / 24 = 35 days.

Verification / Alternative check:
Multiply the new daily distance by the number of days: 24 km/day * 35 days = 840 km, matching the required distance. The figures are consistent and use the same speed ratio and altered rest pattern described in the problem, confirming that 35 days is correct.

Why Other Options Are Wrong:
If we choose 39, 37, 33 or 30 days, the total distance would be 24 * 39 = 936 km, 24 * 37 = 888 km, 24 * 33 = 792 km or 24 * 30 = 720 km respectively, each differing from the required 840 km. These options contradict the computed daily distance and therefore are incorrect.

Common Pitfalls:
Common mistakes include treating days as if walking continues 24 hours, ignoring rest, or forgetting that new speed and new walking hours both change together. Another issue is leaving speed as a fraction and miscomputing daily distance. Carefully computing original speed and then daily distance under each schedule avoids these errors.

Final Answer:
The man will cover 840 km in 35 days under the new schedule.