Problem 30 - Elementary Algebra

Question

Suppose that Karen, riding her bicycle at 15 miles per hour, rode 10 miles farther than Michelle, who was riding her bicycle at 14 miles per hour. Karen rode for 30 minutes longer than Michelle. How long did Michelle and Karen each ride their bicycles?

Step-by-Step Solution

Answer

Michelle rode for 2.5 hours, and Karen rode for 3 hours.

Step 1: Understanding the Problem

Let's define the variables first. Let the time that Michelle rode be denoted as \( t \) in hours. Since Karen rode for 30 minutes longer, her riding time is \( t + 0.5 \) hours. We also know that Karen rode 10 miles more than Michelle.

Step 2: Setting up the Equations

The distance each person rides can be expressed as speed multiplied by time. For Michelle, the distance is \( 14t \), while for Karen, it is \( 15(t + 0.5) \). We know that Karen's distance is Michelle's distance plus 10 miles. Thus, the equation is \( 15(t + 0.5) = 14t + 10 \).

Step 3: Solving the Equation

First, distribute the 15 in the equation: \( 15t + 7.5 = 14t + 10 \). Next, subtract \( 14t \) from both sides to get: \( 15t - 14t + 7.5 = 10 \), resulting in \( t + 7.5 = 10 \). Finally, subtract 7.5 from both sides to solve for \( t \): \( t = 2.5 \).

Step 4: Calculating Riding Times

With \( t = 2.5 \), we can find Michelle's riding time as 2.5 hours. Since Karen rode for an additional 0.5 hours, her riding time is \( 2.5 + 0.5 = 3 \) hours.

Step 5: Verification

Confirm the solution by checking distances: Michelle rode \( 14 imes 2.5 = 35 \) miles. Karen rode \( 15 imes 3 = 45 \) miles. The difference is indeed 10 miles, as required.

Key Concepts

Distance-Speed-Time Relationship

The distance-speed-time relationship is a fundamental concept in algebra, especially for solving word problems dealing with motion. In simple terms, the distance traveled is the product of speed and time. This relationship can be used to describe the motion of objects, as in the problem where Karen and Michelle ride their bicycles.
For Michelle, who rides at a speed of 14 miles per hour, if she rides for time \(t\) in hours, her distance can be represented mathematically as \(14t\).
In contrast, Karen rides her bicycle at a slightly faster speed of 15 miles per hour and for a time of \(t + 0.5\) hours. Her distance can be calculated as \(15(t + 0.5)\).

• The formula \( \text{Distance} = \text{Speed} \times \text{Time} \) allows us to express complex real-world scenarios in a simplified mathematical model.
• Understanding each variable and its real-world meaning helps in setting up and solving these kinds of equations accurately.

Linear Equations

Linear equations play a crucial role in solving algebraic problems where relationships are proportional. In the given problem, we use linear equations to determine the unknown riding times of two cyclists.
To find these times, we start by setting up an equation based on the distance each cyclist traveled. Given the distance-speed-time relationship, we create expressions for both Michelle and Karen's distances: \(14t\) and \(15(t + 0.5)\), respectively.Let’s walk through the steps:

• Write the equation: \(15(t + 0.5) = 14t + 10\). This expresses that Karen's distance is Michelle's distance plus an additional 10 miles.
• Distribute the terms: \(15t + 7.5\).
• Simplify: Subtract \(14t\) from both sides to isolate \(t\).
• Solve for \(t\): Find the value that satisfies the equation.

Understanding linear equations helps in transitioning from a word problem to a mathematical solution, essentially turning words into numbers and operations.

Problem Solving Steps

Approaching algebra word problems requires a clear and methodical problem-solving process. Let's break down the steps used in the given exercise:

• **Understand the Problem:** Clearly identify what is being asked. Here, the goal is to determine the time each person spent riding.
• **Define Variables:** Assign letters, like \(t\), to unknown quantities. This step helps in formulating equations.
• **Set Up Equations:** Use known relationships to express unknown quantities in terms of variables. Focus on how distances relate to each other through speeds and times.
• **Solve the Equations:** Use algebraic methods, like simplifying and rearranging terms, to find the values of variables.
• **Verify the Solution:** Plug the values back into the situation to ensure they satisfy the conditions of the problem.

Following these problem-solving steps can alleviate confusion and provide a structured approach, ensuring all relevant information is accounted for and leading to a successful solution.