Introduction / Context:
This question involves two different modes of travel, walking and riding, over the same distance between two points. By comparing the total times taken under different travel combinations, we can deduce the time associated with each mode and then find how long a full round trip would take using only one mode. This is a standard application of time, speed and distance relations.
Given Data / Assumptions:
- Time to walk from starting point to destination and ride back = 6 hours 15 minutes.
- Time to walk both ways (to and fro) = 7 hours 45 minutes.
- We must find the time to ride both ways.
- Distance between the two points is the same in all scenarios.
Concept / Approach:
Let d be the one way distance, v_w be the walking speed, and v_r be the riding speed. Time to walk one way is t_w = d / v_w, and time to ride one way is t_r = d / v_r. From the information given, we can form two equations involving t_w and t_r. Once we find t_r, the time to ride both ways is simply 2 * t_r.
Step-by-Step Solution:
Step 1: Convert mixed times to hours. 6 hours 15 minutes = 6.25 hours, 7 hours 45 minutes = 7.75 hours.
Step 2: Time to walk both ways: 2t_w = 7.75, so t_w = 7.75 / 2 = 3.875 hours.
Step 3: Time to walk one way and ride back: t_w + t_r = 6.25.
Step 4: Substitute t_w = 3.875 into t_w + t_r = 6.25, giving 3.875 + t_r = 6.25.
Step 5: Solve for t_r: t_r = 6.25 - 3.875 = 2.375 hours.
Step 6: Time to ride both ways = 2 * t_r = 2 * 2.375 = 4.75 hours, which is 4 hours 45 minutes.
Verification / Alternative check:
We can check consistency. Walking both ways takes 7.75 hours. Walking one way and riding back takes 3.875 + 2.375 = 6.25 hours, matching the given value. Riding both ways is 4.75 hours, which is shorter than both previous totals, as expected since riding is faster than walking.
Why Other Options Are Wrong:
- 4 hours: Implies each ride one way is 2 hours, which contradicts the mixed trip equation.
- 4 hours 30 minutes: Corresponds to t_r = 2.25 hours, inconsistent with the walking both ways data.
- 5 hours: Would give t_r = 2.5 hours per trip, again not matching any of the given combined times.
Common Pitfalls:
- Not converting minutes into decimal hours correctly (for example, using 6.15 instead of 6.25).
- Assuming distance is different in different scenarios, which is not true.
- Forgetting to double the one way riding time to get the total round trip time.
Final Answer:
The time taken by the man to ride both ways is
4 hours 45 minutes.
Discussion & Comments