Introduction / Context:
This problem involves mixed-mode travel with two different speeds: walking and cycling. The total distance and total time are given, and we know the speeds for each mode. We must determine how much of the journey is done on foot. These questions test your ability to set up and solve a system of equations derived from distance = speed * time relationships for different segments of a trip.
Given Data / Assumptions:
- Total distance = 99 km.
- Total time = 9 hours.
- Walking speed = 9 km/h.
- Cycling speed = 18 km/h.
- Let time spent walking be t1 hours and time spent cycling be t2 hours.
- We must find the distance walked, which is 9 * t1 km.
Concept / Approach:
We form two equations. One equation uses the total time, and the other uses the total distance. Specifically:
t1 + t2 = 9
9 * t1 + 18 * t2 = 99
Solving this system gives us t1 and t2. Then distance walked is speed multiplied by t1.
Step-by-Step Solution:
Step 1: Write the time equation.
t1 + t2 = 9.
Step 2: Write the distance equation.
Distance walked = 9 * t1.
Distance cycled = 18 * t2.
Total distance: 9 * t1 + 18 * t2 = 99.
Step 3: Use substitution from the time equation.
t2 = 9 - t1.
Substitute into the distance equation:
9 * t1 + 18 * (9 - t1) = 99.
Step 4: Expand and simplify.
9 * t1 + 18 * 9 - 18 * t1 = 99.
9 * t1 - 18 * t1 = -9 * t1.
So -9 * t1 + 162 = 99.
Step 5: Solve for t1.
-9 * t1 = 99 - 162 = -63.
t1 = -63 / -9 = 7 hours.
Step 6: Compute distance walked.
Distance walked = 9 * t1 = 9 * 7 = 63 km.
Therefore, the man walks 63 km and cycles the remaining part.
Verification / Alternative check:
If t1 = 7 hours, then t2 = 9 - 7 = 2 hours. Distance walked = 9 * 7 = 63 km. Distance cycled = 18 * 2 = 36 km. Total distance = 63 + 36 = 99 km, which matches the given total. Total time = 7 + 2 = 9 hours, so all conditions are satisfied.
Why Other Options Are Wrong:
36 km, 45 km, 54 km and 27 km do not allow both the time and distance equations to be satisfied simultaneously when converted back to walking and cycling times at the given speeds. For example, if 54 km were walked, the remaining 45 km at 18 km/h would require 2.5 hours, giving a total time of more than 9 hours. Only 63 km gives times and distances consistent with all the data.
Common Pitfalls:
Some learners confuse time and distance, equating 9 * t1 + 18 * t2 with total time instead of total distance. Others miscalculate when substituting t2 = 9 - t1, especially during distribution and simplification. Another common error is to assume equal times or equal distances for walking and cycling, which is not supported by the problem statement. Carefully forming the two equations from first principles and solving methodically avoids these errors.
Final Answer:
The man travels
63 km on foot and the remainder by bicycle.
Discussion & Comments