Hema takes a total of 9 hours 55 minutes to walk a certain distance and then cycle back to the same place from where she started. She could walk both ways in 12 hours 30 minutes. How much time will she take to cycle the same distance both ways?

Introduction / Context:
This is a classic time, speed, and distance problem involving two different modes of travel: walking and cycling. Hema travels along the same route, once in a mixed mode (walking one way and cycling back) and once hypothetically by walking both ways. From the total times given in these two scenarios, we need to determine how long she would take if she cycled both ways. This type of question tests your ability to translate word statements into equations and to use simple algebra to solve for unknown times.

Given Data / Assumptions:
- Let the one-way distance between the two points be D km.
- Let the time taken to walk one way be T_w hours.
- Let the time taken to cycle one way be T_c hours.
- Walking both ways takes 12 hours 30 minutes, i.e. 12.5 hours.
- Walking one way and cycling back takes 9 hours 55 minutes, i.e. 9 + 55/60 = 9.9167 hours (approximately).
- Hema walks and cycles at constant speeds on a straight route, and there are no breaks other than the mode change at the turning point.

Concept / Approach:
The key idea is that time for a journey is equal to distance divided by speed, but here we work directly with times. We express the two given total times in terms of T_w and T_c. From the walking-both-ways scenario, we get a direct value for T_w. From the mixed scenario, we then find T_c. Once T_c (time to cycle one way) is known, cycling both ways will simply take 2 * T_c. The actual distance D does not need to be calculated because it cancels out when we work with times only.

Step-by-Step Solution:
Step 1: From “she could walk both ways in 12 hours 30 minutes”, we have 2 * T_w = 12.5, so T_w = 12.5 / 2 = 6.25 hours.Step 2: From “walking one way and cycling back takes 9 hours 55 minutes”, we have T_w + T_c = 9.9167 hours.Step 3: Substitute T_w = 6.25 into the second equation: 6.25 + T_c = 9.9167.Step 4: Solve for T_c: T_c = 9.9167 - 6.25 = 3.6667 hours, which is 3 hours 40 minutes.Step 5: Time to cycle both ways is 2 * T_c = 2 * 3.6667 ≈ 7.3333 hours, which is 7 hours 20 minutes.

Verification / Alternative check:
Check the numbers logically. One-way walking time is 6.25 hours, so walking both ways indeed gives 12.5 hours. If cycling one way takes 3.6667 hours, then walking out (6.25 hours) and cycling back (3.6667 hours) gives approximately 9.9167 hours, which is exactly 9 hours 55 minutes. Doubling 3 hours 40 minutes clearly gives 7 hours 20 minutes, so the calculation is consistent with all the given conditions.

Why Other Options Are Wrong:
- 7 hours 15 minutes corresponds to a one-way cycling time slightly less than 3 hours 40 minutes and would not add up correctly with the given walking time to produce 9 hours 55 minutes.
- 7 hours 35 minutes and 7 hours 40 minutes correspond to longer cycling times that would make the mixed journey exceed 9 hours 55 minutes when combined with 6.25 hours of walking time.

Common Pitfalls:
Candidates sometimes try to find the actual walking and cycling speeds, which is unnecessary and can introduce rounding errors. Another typical mistake is to mishandle the time conversion between hours and minutes, especially for 9 hours 55 minutes and 12 hours 30 minutes. Always convert mixed times to decimal hours carefully and use them consistently in equations. Working symbolically with T_w and T_c before substituting numerical values reduces confusion.

Final Answer:
Hema will take 7 hours 20 minutes to cycle both ways.