A farmer travelled a distance of 61 km in 9 hours. He travelled partly on foot at a speed of 4 km/h and partly on a bicycle at a speed of 9 km/h. What distance (in km) did he travel on foot?

Introduction / Context:
This is a standard mixed-speed travel problem where a person covers part of the distance at one speed and the remaining distance at another speed. We are given the total distance, the total time, and both speeds, and we must find how much of the journey was covered at the slower speed (on foot). This problem tests your ability to set up and solve a linear equation involving two unknown partial distances or times.

Given Data / Assumptions:
- Total distance travelled = 61 km.
- Total time taken = 9 hours.
- Walking speed = 4 km/h.
- Cycling speed = 9 km/h.
- Let the distance travelled on foot be x km.
- Then the distance travelled by bicycle is (61 - x) km.
- Motion along each segment is at constant speed with no additional breaks.

Concept / Approach:
The total time is the sum of the time spent on foot plus the time spent on the bicycle. Time is distance divided by speed, so we can express each part of the total time in terms of x and then equate their sum to the given 9 hours. This yields a linear equation in x, which can be solved easily using basic algebraic steps. Once x is found, it directly gives the distance travelled on foot.

Step-by-Step Solution:
Step 1: Let distance on foot be x km.Then distance by bicycle = 61 - x km.Step 2: Express the total time using time = distance / speed.Time on foot = x / 4 hours.Time on bicycle = (61 - x) / 9 hours.Step 3: Total time is given as 9 hours, so:x / 4 + (61 - x) / 9 = 9.Step 4: Multiply through by the common denominator, 36, to clear fractions:36 * (x / 4) + 36 * ((61 - x) / 9) = 36 * 9.This simplifies to 9x + 4(61 - x) = 324.Step 5: Expand and solve: 9x + 244 - 4x = 324 ⇒ 5x + 244 = 324 ⇒ 5x = 80 ⇒ x = 16 km.

Verification / Alternative check:
Check the times using x = 16 km. Time on foot = 16 / 4 = 4 hours. Distance by bicycle = 61 - 16 = 45 km. Time on bicycle = 45 / 9 = 5 hours. Total time = 4 + 5 = 9 hours, exactly matching the problem statement. This confirms that 16 km on foot and 45 km by bicycle satisfy both the time and distance conditions.

Why Other Options Are Wrong:
- 14 km, 18 km, and 20 km, when used in the time equation, lead to total times that are not exactly 9 hours; either the total time becomes more than 9 hours or less than 9 hours.
- Only 16 km yields a total time of precisely 9 hours while still adding up to a total distance of 61 km.

Common Pitfalls:
Some candidates confuse time and distance and try to distribute 9 hours directly in proportion to 4 km/h and 9 km/h without setting up a proper equation. Another common issue is arithmetic mistakes when clearing fractions or solving the resulting linear equation. Writing the equation step by step and checking the solution by substituting it back is an effective way to avoid and catch such errors.

Final Answer:
The farmer travelled 16 km on foot.