Cyclist's Hill Ride: A Distance and Speed Problem Solved with Math

When a cyclist faces a challenge like going up and down a hill with different speeds, the journey becomes both a physical and mathematical test. This article delves into solving a real-life problem using distance, speed, and time relationships. Let's explore how the cyclist managed to tackle these challenges using some simple algebraic techniques.

Problem Description

A cyclist went up a hill at a speed of 12 km/h and went down the hill at a speed of 20 km/h. The ride up the hill took 16 minutes longer than the ride down the hill. The question is, how many minutes did the cyclist take to go down the hill?

Mathematical Approach

To solve this problem, we need to use the fundamental relationship between distance, speed, and time: [ text{Distance} text{Speed} times text{Time} ]

Step 1: Define Variables

Let's denote:

d: the distance up and down the hill (in kilometers) t_d: the time taken to go down the hill (in hours) t_u: the time taken to go up the hill (in hours)

Given the speeds:

Speed going up the hill 12 km/h Speed going down the hill 20 km/h

We know that the time taken to go up the hill is 16 minutes longer than the time taken to go down the hill. Since 16 minutes is (frac{16}{60} frac{4}{15}) hours, we can express this relationship as:

[ t_u t_d frac{4}{15} ]

Step 2: Express Times in Terms of Distance and Speed

Using the relationship between distance, speed, and time:

[ t_u frac{d}{12} ] (time to go up) [ t_d frac{d}{20} ] (time to go down)

Substituting these into the equation for the difference in time:

[ frac{d}{12} frac{d}{20} frac{4}{15} ]

Step 3: Solve for Distance

To solve for d, we first find a common denominator for the fractions. The least common multiple of 12, 20, and 15 is 60. We can rewrite the equation by multiplying everything by 60:

[ 60 cdot frac{d}{12} 60 cdot frac{d}{20} 60 cdot frac{4}{15} ]

This simplifies to:

[ 5d 3d 16 ]

Now, subtract 3d from both sides:

[ 2d 16 ]

Dividing both sides by 2 gives:

[ d 8 text{ km} ]

Step 4: Calculate Time to Go Down

Now that we have the distance, we can find the time taken to go down the hill:

[ t_d frac{d}{20} frac{8}{20} frac{2}{5} text{ hours} ]

To convert this into minutes, we multiply by 60:

[ t_d frac{2}{5} times 60 24 text{ minutes} ]

Thus, the cyclist took 24 minutes to go down the hill.

Conclusion

The cyclist faced a mathematical challenge but successfully calculated the time taken to descend the hill using basic algebraic techniques. This problem not only showcases the practical application of mathematics but also highlights the importance of understanding distance, speed, and time in real-life scenarios.