Karthik can cover a distance of 200 km in 22 days while resting for 2 hours each day. How many days (approximately) will he take to cover a distance of 250 km if he still rests for 2 hours each day but moves at two thirds of his previous speed?

Introduction / Context:
This question explores the relationship between speed, distance, and time in a multi-day travel context. The person travels a certain distance in a given number of days, and then we consider a new situation where his speed is reduced but his daily rest pattern remains the same. The problem asks for the approximate number of days needed to cover a larger distance under the slower speed condition. It focuses on proportional reasoning rather than exact daily scheduling.

Given Data / Assumptions:
- Karthik covers 200 km in 22 days with 2 hours of rest per day.
- In the new situation, he still rests for 2 hours per day, so his effective daily travel time pattern is similar.
- His new speed is two thirds of his previous effective travel speed.
- We want to know how many days he will take to cover 250 km at the reduced speed with the same daily rest habit.
- We treat his effective travel hours per day as the same in both scenarios, so daily distance is directly proportional to speed.

Concept / Approach:
Because Karthik rests for the same amount of time each day in both scenarios, his actual travel hours per day are the same. Thus, the distance he covers per day is proportional to his speed. We first calculate his original average distance per day from the 200 km in 22 days. Then, we reduce that by the factor two thirds to reflect the slower speed and compute how many days would be required to cover 250 km at the new daily distance rate.

Step-by-Step Solution:
Step 1: Compute Karthik's original average distance per day.Average distance per day originally = 200 km / 22 days ≈ 9.09 km per day.Step 2: His new speed is two thirds of the old speed, so his new daily distance is also two thirds of the old daily distance (same travel hours per day).New daily distance ≈ (2/3) * 9.09 ≈ 6.06 km per day.Step 3: We now need 250 km to be covered at approximately 6.06 km per day.Number of days ≈ 250 / 6.06.Step 4: Compute the division: 250 / 6.06 is approximately 41.25 days.

Verification / Alternative check:
Another way to think about it is to use the ratio of speeds and distances. Original speed is some value S, new speed is (2/3)S. To travel 200 km at speed S takes 22 days. To travel 250 km at speed (2/3)S would multiply the time by both 250/200 (for increased distance) and 3/2 (for the slower speed). So new days = 22 * (250/200) * (3/2) = 22 * (5/4) * (3/2) = 22 * (15/8) = (22 * 15) / 8 = 330 / 8 = 41.25 days, which matches our previous calculation exactly.

Why Other Options Are Wrong:
- 37.5 days and 39.75 days are too small and would imply a higher effective speed than two thirds of the original when checked via proportional reasoning.
- 40 days is also smaller than the correctly computed 41.25 days and does not satisfy the precise ratio calculations.

Common Pitfalls:
Some candidates try to handle the daily rest hours explicitly and overcomplicate the problem. However, because the rest period is unchanged in both scenarios, it cancels out when comparing daily effective distances. Another mistake is to apply the two thirds factor to the number of days instead of to the speed or daily distance. Always apply the speed change factor to the distance per day, not directly to time, unless you systematically invert it.

Final Answer:
Karthik will take approximately 41.25 days to cover 250 km at two thirds of his original speed.