Tribeni has to travel 175 km in 3.5 hours. She actually travels at 58 km/h for 2.5 hours, then rests for 20 minutes, and then completes the remaining journey. If she still manages to reach exactly on time, what must have been her speed during the last stretch (in km/h)?

Introduction / Context:
This is a time and distance planning problem involving different phases of a journey: travelling at one speed, resting, and then travelling at another speed, with a fixed overall time limit. It tests the ability to break the journey into segments, compute distance and time for each part, and then deduce the required speed in the last segment so that the total schedule is met. Such questions are common in aptitude tests and mirror real life travel planning where delays must be compensated by faster travel later.

Given Data / Assumptions:

Total distance required = 175 km.
Total time available = 3.5 hours = 3 hours 30 minutes.
First segment: speed = 58 km/h for 2.5 hours.
Rest period: 20 minutes = 1 / 3 hour, during which she covers zero distance.
Last segment: unknown speed v km/h for the remaining distance and time.
She reaches exactly on time, so the sum of all segment times equals 3.5 hours.

Concept / Approach:
We first calculate the distance covered during the first segment at 58 km/h and subtract this from the total required distance to get the remaining distance. Next, we determine how much time is left after the first segment and the rest period are accounted for. The remaining distance must be covered in this remaining time at a constant speed v. We use v = remaining_distance / remaining_time to find the required speed in the last stretch.

Step-by-Step Solution:
Step 1: Compute distance during the first segment. distance_first = 58 * 2.5 km. 58 * 2.5 = 145 km. Step 2: Compute remaining distance. remaining_distance = total_distance - distance_first. remaining_distance = 175 - 145 = 30 km. Step 3: Compute time used in first segment and rest. time_first = 2.5 hours. rest_time = 20 minutes = 20 / 60 = 1 / 3 hour. total_used_time = 2.5 + 1 / 3 hours. 2.5 = 7 / 2, 1 / 3 = 1 / 3, so total_used_time = 7 / 2 + 1 / 3 = (21 + 2) / 6 = 23 / 6 hours ≈ 3.833 / 6 * 6. Numerically, 2.5 + 0.333... = 2.833... hours. Step 4: Compute remaining time. total_time = 3.5 hours = 3.5. remaining_time = 3.5 - 2.833... = 0.666... hours. So remaining_time = 2 / 3 hour (which is 40 minutes). Step 5: Compute required speed for the last stretch. v = remaining_distance / remaining_time. v = 30 / (2 / 3) = 30 * (3 / 2) = 45 km/h.

Verification / Alternative check:
At 45 km/h, travelling for 2 / 3 hour covers 45 * 2 / 3 = 30 km, which is exactly the remaining distance. The total time spent is 2.5 hours in the first segment, 1 / 3 hour resting, and 2 / 3 hour in the last segment. Adding these gives 2.5 + 1 / 3 + 2 / 3 = 2.5 + 1 = 3.5 hours, which matches the total time allowed. Thus the journey is completed exactly on schedule at this speed.

Why Other Options Are Wrong:
At 36 km/h, in 2 / 3 hour Tribeni would cover only 24 km, falling short of the required 30 km. At 30 km/h, she would cover only 20 km, which is even further from the target. At 42.5 km/h, in 2 / 3 hour she would cover 42.5 * 2 / 3 ≈ 28.33 km, still not enough. Only 45 km/h allows her to cover exactly 30 km in the remaining 40 minutes and arrive on time.

Common Pitfalls:
One common mistake is to forget to convert 20 minutes into hours and attempt to add minutes directly to hours. Another is to subtract the rest time from the total time without first including the travel time, which mixes steps. It is also easy to compute the distance in the first part incorrectly by mishandling 2.5 hours. Writing all times in hours and handling them as fractions or decimals consistently helps avoid these issues.

Final Answer:
Tribeni's speed during the last stretch must have been 45 km/hr.