A man travels a total distance of 61 km in 9 hours, walking part of the way at 4 km/h and cycling the remaining distance at 9 km/h. How many kilometres does he cover by walking?

Introduction / Context:
This problem is a mixture of two constant speeds over different segments of a journey: walking and cycling. It tests your ability to set up and solve simultaneous equations based on both total distance and total time. Such questions are very frequent in aptitude tests and strengthen algebraic reasoning connected to real life travel situations.

Given Data / Assumptions:
Total distance traveled = 61 km.
Walking speed = 4 km/h.
Cycling speed = 9 km/h.
Total time taken for the full journey = 9 hours.
The man walks some distance and cycles the remaining distance in a straight journey.
We must find the distance covered while walking.

Concept / Approach:
Let the walking distance be W km and the cycling distance be C km. We know W + C = 61. Time taken for each part is distance divided by speed, so total time is W / 4 + C / 9 = 9 hours. This gives a system of two linear equations in two variables. Solving this system yields W, the walking distance. Substitution is usually the easiest method in such straightforward linear setups.

Step-by-Step Solution:
Step 1: Let W be the distance walked and C be the distance cycled.Step 2: Total distance equation: W + C = 61.Step 3: Total time equation: W / 4 + C / 9 = 9 hours.Step 4: From W + C = 61, express C as C = 61 − W.Step 5: Substitute C into the time equation: W / 4 + (61 − W) / 9 = 9.Step 6: Take the least common multiple of 4 and 9, which is 36, and multiply both sides by 36.Step 7: This gives 9W + 4(61 − W) = 9 * 36.Step 8: Expand: 9W + 244 − 4W = 324.Step 9: Combine like terms: 5W + 244 = 324.Step 10: Subtract 244 from both sides: 5W = 80.Step 11: So W = 80 / 5 = 16 km.

Verification / Alternative check:
Using W = 16 km, we get C = 61 − 16 = 45 km. Check the total time: time walking = 16 / 4 = 4 hours; time cycling = 45 / 9 = 5 hours. Total time = 4 + 5 = 9 hours, which matches the given information. The distances also add to 61 km, so both conditions are satisfied perfectly.

Why Other Options Are Wrong:
12 km or 15 km would make the cycling distance too large and the total time would no longer equal 9 hours when you compute with the given speeds.
17 km and 20 km also fail when substituted back, because either the total distance or total time comes out different from the given values in the question.

Common Pitfalls:
Some learners mistakenly add the speeds or average them without using the given time condition. Others confuse which variable stands for which distance and mix up the equations. A systematic approach with clearly defined variables and careful substitution prevents these errors. Always verify your solution by checking both the distance and the time conditions at the end.

Final Answer:
The man covers 16 km of the journey by walking.