A man travelled a distance of 50 km in 8 hours. He travelled partly on foot at 5 km/h and partly on a bicycle at 7 km/h. What is the distance (in km) that he travelled on foot?

Introduction / Context:
This is a mixed-mode travel problem where a man covers a certain total distance using two different modes of transport with different speeds: walking and cycling. We are given the total distance and total time, along with both speeds, and we must determine how much of the distance was covered on foot. This type of question tests the ability to set up and solve a simple system involving distances and times.

Given Data / Assumptions:
- Total distance = 50 km.
- Total time = 8 hours.
- Walking speed = 5 km/h.
- Bicycling speed = 7 km/h.
- Let the distance travelled on foot be x km.
- Then the distance travelled by bicycle is (50 - x) km.
- The man travels at constant speeds on a straight route, with no extra waiting or resting time included in the 8 hours.

Concept / Approach:
The key idea is that the total time is the sum of the time spent walking and the time spent bicycling. Time is distance divided by speed, so we express the two components of the total time as x / 5 and (50 - x) / 7. Equating their sum to 8 hours yields a linear equation in x. Solving that equation gives the distance walked. We then verify that this value is reasonable and satisfies both total distance and total time constraints.

Step-by-Step Solution:
Step 1: Let distance on foot be x km.Then distance by bicycle is 50 - x km.Step 2: Express the time spent in each mode.Time on foot = x / 5 hours.Time on bicycle = (50 - x) / 7 hours.Step 3: Total time is given as 8 hours.So x / 5 + (50 - x) / 7 = 8.Step 4: Clear denominators by multiplying through by 35 (the least common multiple of 5 and 7):35 * (x / 5) + 35 * ((50 - x) / 7) = 35 * 8.This simplifies to 7x + 5(50 - x) = 280.Step 5: Expand and simplify:7x + 250 - 5x = 280 ⇒ 2x + 250 = 280 ⇒ 2x = 30 ⇒ x = 15 km.

Verification / Alternative check:
With x = 15 km walked, the bicycling distance is 50 - 15 = 35 km. Time on foot = 15 / 5 = 3 hours. Time on bicycle = 35 / 7 = 5 hours. Total time = 3 + 5 = 8 hours, exactly as given. The distance sum is 15 + 35 = 50 km, confirming that both total distance and total time constraints are satisfied.

Why Other Options Are Wrong:
- 20 km on foot would lead to total time 20/5 + 30/7 = 4 + 30/7 ≈ 8.286 hours, which is more than 8 hours.
- 25 km on foot gives 25/5 + 25/7 = 5 + 25/7 ≈ 8.571 hours, again larger than 8 hours.
- 30 km on foot gives 30/5 + 20/7 = 6 + 20/7 ≈ 8.857 hours, still not equal to 8 hours.

Common Pitfalls:
Some students attempt to directly proportion the distances based on speeds without using the time information correctly, or they treat 8 hours as an average instead of a total. Another common mistake is arithmetic error when clearing fractions or solving the linear equation. Writing each step clearly and verifying by substitution helps to avoid these issues.

Final Answer:
The man travelled 15 km on foot.