Determining Distance Traveled by Walking and Cycling: A Two-Variable Approach
Imagine a scenario where a traveler covers a distance of 72 kilometers in 12 hours. This traveler uses two distinct modes of transportation: walking at a speed of 5 kilometers per hour (km/h) and cycling at a speed of 10 km/h. The goal is to determine the specific distance covered on foot. This article will explore various mathematically derived methods to solve this problem, providing a detailed breakdown of the calculations and concepts involved.
Method 1: Application of Average Speed
Let's begin with a method that utilizes the concept of average speed. Given that the total distance covered is 72 kilometers and the total travel time is 12 hours, the average speed can be calculated as:
Average Speed Total Distance / Total Time
Substituting the given values:
Average Speed 72 km / 12 hours 6 km/h
Let ( t_f ) be the time spent walking and ( t_c ) be the time spent cycling. The distance covered by walking can be represented as 5 km/h × ( t_f ), and the distance covered by cycling as 10 km/h × ( t_c ). The total distance covered by both walking and cycling is equal to 72 km:
5 km/h × ( t_f ) 10 km/h × ( t_c ) 72 km
Given the total time of 12 hours, we can express the total time as:
12 ( t_f ) ( t_c )
To solve the system of equations, we substitute ( t_f ) with ( 12 - t_c ) in the distance equation:
5 × (12 - ( t_c )) 10 × ( t_c ) 72
60 - 5 × ( t_c ) 10 × ( t_c ) 72
5 × ( t_c ) 12
( t_c ) 12 / 5 2.4 hours
( t_f ) 12 - 2.4 9.6 hours
Distance walked 5 × 9.6 48 km (However, this result contradicts our initial problem statement, indicating the need for a re-evaluation of the application of the method.)
Method 2: Using the Given Solution Approach
Let's now consider the detailed solution provided:
The total distance is given as 61 kilometers, and the total time is 9 hours. Let ( t_c ) be the time spent cycling and ( t_f ) the time spent walking. We can form two equations:
Equation 1: ( t_c t_f 9 )
Equation 2: Average Speed (cycling) Average Speed (walking) 9 km/h
Substituting the speeds:
10 × ( t_c ) / 9 5 × ( t_f ) / 4 9
Multiplying through by 36 to eliminate the denominators:
40 × ( t_c ) 45 × ( t_f ) 324
From Equation 1, we can express ( t_f ) as:
( t_f 9 - t_c )
Substituting this into the speed equation:
40 × ( t_c ) 45 × (9 - ( t_c )) 324
40 × ( t_c ) 405 - 45 × ( t_c ) 324
5 × ( t_c ) 81
( t_c ) 81 / 5 16.2 hours (This again contradicts our initial problem parameters, indicating the need to use the provided steps for precise calculation.)
Multiplying the earlier result for walking by the correct time:
( t_f 9 - 5 4 ) hours
Distance covered by walking 4 × 4 km/h 16 km
Distance covered by cycling 5 × 9 km/h 45 km
Method 3: Simplified Problem-Solving Approach
Another method involves substituting specific values and checking the results. For instance:
If the person cycled for 6 hours and walked for 3 hours, the distances would be:
Cycling distance 9 km/h × 6 54 km
Walking distance 4 km/h × 3 12 km
Total distance 54 12 66 km (This is 5 km more than the actual distance of 61 km, indicating a need to adjust the cycling time.)
When we replace one hour of cycling with walking, the distance changes by 9 km - 4 km 5 km. Therefore, the cycling time should be reduced by one hour to match the actual distance of 61 km:
Cycling time 5 hours, Walking time 4 hours
Cycling distance 5 × 9 45 km
Walking distance 4 × 4 16 km
Conclusion
The correct distances are:
Distance traveled by cycling 45 km
Distance traveled by walking 16 km
By understanding and applying the given methods, we can accurately determine the distance traveled by foot and by bicycle, ensuring a comprehensive solution to the problem. This approach not only provides the numerical result but also illustrates the multi-step reasoning required to solve such distance and speed problems.