A man travels from place A to place B in 26 hours. He covers one third of the distance at a speed of 4 km per hour and the remaining distance at a speed of 5 km per hour. What is the distance between A and B in kilometres?

Introduction / Context:
This problem belongs to the time and distance category, where the journey is divided into segments with different speeds. The total travel time is given and we are asked to determine the total distance between two points. The important idea is that time equals distance divided by speed, and we must sum the times of each segment to match the total time.

Given Data / Assumptions:

• Total time taken from A to B is 26 hours.
• One third of the distance is travelled at 4 km per hour.
• The remaining two thirds of the distance is travelled at 5 km per hour.
• Speeds are constant on each segment of the trip.

Concept / Approach:
Let the total distance between A and B be D kilometres. Then the first segment distance is D / 3 and the second segment distance is 2D / 3. Time taken for each segment is segment distance divided by segment speed. The sum of these two times must equal 26 hours. This gives a linear equation in D, which we can solve to obtain the total distance.

Step-by-Step Solution:
Step 1: Let total distance between A and B be D kilometres.Step 2: Distance of the first part = D / 3 kilometres, speed = 4 km per hour.Step 3: Time for first part = (D / 3) / 4 = D / 12 hours.Step 4: Distance of the second part = 2D / 3 kilometres, speed = 5 km per hour.Step 5: Time for second part = (2D / 3) / 5 = 2D / 15 hours.Step 6: Total time = time for first part + time for second part = D / 12 + 2D / 15.Step 7: This total time is given as 26 hours, so D / 12 + 2D / 15 = 26.Step 8: Use common denominator 60 to combine fractions: D / 12 = 5D / 60 and 2D / 15 = 8D / 60, total = 13D / 60.Step 9: So 13D / 60 = 26. Multiply both sides by 60: 13D = 26 * 60.Step 10: Compute 26 * 60 = 1560. Thus 13D = 1560, so D = 1560 / 13 = 120 kilometres.

Verification / Alternative check:
Check the total time with D = 120 kilometres. First segment distance = 120 / 3 = 40 km at 4 km per hour gives 40 / 4 = 10 hours. Second segment distance = 80 km at 5 km per hour gives 80 / 5 = 16 hours. Total time = 10 + 16 = 26 hours, exactly matching the given total. This confirms that the distance of 120 km is correct.

Why Other Options Are Wrong:

• 45 km: With this distance, the computed time would be much less than 26 hours.
• 80 km: This gives first segment time 80 / 3 divided by 4 and second segment at 5 km per hour, which does not sum to 26 hours.
• 90 km: This also results in a total time that does not equal 26 hours when calculated segment by segment.

Common Pitfalls:
Some students mistakenly average the two speeds without weighting by distance or time. Others may incorrectly assume half the distance is travelled at each speed. It is important to respect the stated fractions of the distance and always write expressions for time as distance divided by speed, then sum them to match the given total time.

Final Answer:
The distance between A and B is 120 km.