Solving Bicycle Speed Problems Using Algebra: Steps and Examples

Bicycle speed problems are a classic type of word problem involving algebra that often tests a student's understanding of the relationship between distance, speed, and time. This article will guide you through the steps to solve such problems, using multiple examples to illustrate the process.

Understanding the Relationship Between Distance, Speed, and Time

The fundamental relationship between distance, speed, and time is given by the formula:

Distance Speed times; Time

This simple formula can be rearranged to solve for any of the three variables, depending on the information given in the problem. For instance, if we know the distance and want to find the time, we can use the formula:

Time Distance / Speed

Let's apply this to the problem presented. We will use algebra to solve the problem step-by-step.

Solving the Problem: Jodie and Robert's Bicycling Speeds

In this problem, we are given that Jodie bicycles 5 km/h faster than Robert. Additionally, in the same amount of time, Jodie can bicycle 54 km while Robert can bicycle 39 km. Let's use algebra to determine the speed of each bicyclist.

Step-by-Step Solution

Step 1: Define Variables

First, we need to define our variables:

Let r be the speed of Robert in km/h. Then Jodie's speed is r 5 km/h.

Step 2: Set Up Equations

Using the formula Time Distance / Speed, we can express the time it takes each bicyclist to travel their respective distances:

Time for Robert 39/r

Time for Jodie 54/(r 5)

Step 3: Set the Times Equal

Since they travel for the same amount of time, we can set the two times equal to each other:

39/r 54/(r 5)

Step 4: Cross-Multiply

Cross-multiplying gives us:

39(r 5) 54r

Expanding the left side:

39r 195 54r

Step 5: Solve for r

Rearranging the equation:

195 54r - 39r

195 15r

r 195 / 15

r 13 km/h

Step 6: Find Jodie's Speed

Now that we have Robert's speed, we can find Jodie's speed:

Jodie's speed r 5 13 5 18 km/h

Conclusion

The speed of Robert is 13 km/h, and the speed of Jodie is 18 km/h.

Additional Example: Praedup and Kathy's Bicycling Speeds

Let's consider another example involving Praedup and Kathy's speeds:

Step-by-Step Solution

Let Praedup's Speed be “a” km/h

According to the question, Kathy's speed is a 8 km/h.

Using the Distance-Speed-Time Relationship

Since the time taken is constant for both cyclists, we can set up the following equation:

57/a 81/(a 8)

Solving for a

Cross-multiplying gives us:

57(a 8) 81a

57a 456 81a

456 81a - 57a

456 24a

a 456 / 24

a 19 km/h

Thus, Praedup's speed is 19 km/h, and Kathy's speed is 19 8 27 km/h.

Conclusion

In this article, we have solved two bicycle speed problems step-by-step using algebra. By understanding the relationship between distance, speed, and time, you can tackle similar problems with confidence and ease.

Frequently Asked Questions (FAQ)

Q: How do you determine the speed of two individuals based on their travel distances?

A: Start by defining the speed of one individual as a variable. Use the distance-speed-time relationship to set up an equation for each individual. Then, equate the times taken by both individuals as they travel their respective distances to solve for the unknown variable.

Q: What is the formula that connects distance, speed, and time?

A: The formula is Distance Speed times; Time. This can be rearranged to solve for any of the three variables depending on the information given in the problem.

Q: How do you solve an algebraic equation involving fractions?

A: To eliminate fractions, cross-multiply the equation to create a standard algebraic equation. Then, solve the equation by performing the necessary operations to isolate the variable.

Summary: Understanding the relationship between distance, speed, and time using algebra can help solve a variety of word problems, including those about bicycles.