How to Calculate Distance Using Speed and Time: A Real-World Problem

In everyday life, we often find ourselves in situations where we need to solve problems related to distance, speed, and time. For instance, a simple task like returning a friend's bicycle can turn into an interesting math problem. Let's dive into how you can solve such problems using basic mathematical principles.

Problem Statement

A girl decides to return her friend's bicycle. She rides to the friend's house at a speed of 9 miles per hour and walks back home at a speed of 3 miles per hour. If the total time taken for the round trip is 1.5 hours, how far is her friend's house?

Step-by-Step Solution

Step 1: Define Variables and Equations

We can define the distance to the friend's house as d miles. The girl rides to her friend's house at a speed of 9 miles per hour and walks back home at a speed of 3 miles per hour.

The time taken to ride to her friend's house can be calculated using the formula:

Time frac{Distance}{Speed}

Thus, the time taken to ride to her friend's house is:

Time_{riding} frac{d}{9}

The time taken to walk back home is:

Time_{walking} frac{d}{3}

Step 2: Set Up the Total Time Equation

According to the problem, the total time for the round trip is 1.5 hours. Therefore, we can write the equation:

frac{d}{9} frac{d}{3} 1.5

Step 3: Find a Common Denominator and Simplify

The common denominator for 9 and 3 is 9. We can rewrite the second term:

frac{d}{3} frac{3d}{9}

Now, substituting this into the equation gives:

frac{d}{9} frac{3d}{9} 1.5

Combining the fractions:

frac{4d}{9} 1.5

Step 4: Solve for d

Multiply both sides by 9 to eliminate the fraction:

4d 1.5 times 9

Calculating the right side:

4d 13.5

Now, divide by 4:

d frac{13.5}{4} 3.375

Conclusion: Thus, the distance to her friend's house is 3.375 miles.

Alternative Methods

Alternatively, we can solve this problem using a different set of equations. We know that it takes 3 times as long to walk than to bike, and the total time spent is 1.5 hours:

Total time frac{x}{9} frac{x}{3} 1.5

x 3x 13.5

4x 13.5

x frac{13.5}{4} 3.375

Real-World Applications

This type of math problem is not only useful in real-world scenarios but also in various professions such as transportation, logistics, and urban planning. Understanding the relationship between distance, speed, and time is crucial for optimizing travel routes and schedules.

Conclusion

By solving problems step-by-step and understanding the underlying mathematical principles, we can easily calculate distances and times. In this particular example, the girl’s round trip distance to her friend's house is 3.375 miles or 3 miles and 3 furlongs.

Remember, the key to solving such problems lies in breaking them down into simpler components and using the right formulas. Practice these types of problems to enhance your problem-solving skills and mathematical intuition.