Short Answer 
The relationship between distance, speed, and time helps calculate travel times for a bicyclist journeying 30 miles to a railroad station. By setting equations for each leg of the trip and solving for time, we find that the distance is confirmed to be 30 miles.
Step 1: Understand the Distance and Speed Relationship
The first concept to grasp is the fundamental relationship between distance, speed, and time. Speed is calculated as the ratio of distance to time, expressed mathematically as:
- Speed = Distance / Time
This means that if you know any two of these variables, you can find the third. In this case, we’re dealing with a distance of 30 miles from a village to a railroad station.
Step 2: Set Up the Equations
To analyze the bicyclist’s journey, set up two equations based on his speed when going to and returning from the station. When traveling to the station at 15 mph, the time taken can be expressed as:
- Distance = Speed ‚àöo Time ‚ÄöUi d = 15a
On the return trip, he travels at 10 mph, taking longer time:
- Distance = Speed ‚àöo Time ‚ÄöUi d = 10b
From these equations, we know that the return trip took 1 hour longer, which leads to the equation:
- b = a + 1
Step 3: Solve for Distance
Now, combine the equations to find the values of a and b. Start with:
- 15a = 10(a + 1)
Expanding this gives:
- 15a = 10a + 10
- 5a = 10
- a = 2
Lastly, substitute back to find the distance:
- Distance = 15a = 15(2) = 30 miles
This confirms that the distance from the village to the railroad station is indeed 30 miles.