Problem 34 - Elementary Algebra

Question

Kirk starts jogging at 5 miles per hour. One-half hour later, Nancy starts jogging on the same route at 7 miles per hour. How long will it take Nancy to catch Kirk?

Step-by-Step Solution

Answer

Nancy will take 1.25 hours to catch Kirk.

Step 1: Understand the Problem

Kirk starts jogging at 5 miles per hour, and Nancy starts 0.5 hours later at 7 miles per hour. We want to find out how long it will take Nancy to catch up to Kirk.

Step 2: Define Variables

Let \( t \) be the time in hours that Nancy jogs. In that time, Kirk will jog for \( t + 0.5 \) hours because he starts jogging half an hour before Nancy.

Step 3: Write Distance Equations

The distance Kirk jogs can be represented as \( D_K = 5(t + 0.5) \), and the distance Nancy jogs can be represented as \( D_N = 7t \). Nancy catches up to Kirk when these distances are equal: \( 5(t + 0.5) = 7t \).

Step 4: Solve the Equation

Expand and solve the equation: \[ 5t + 2.5 = 7t \]Subtract \( 5t \) from both sides:\[ 2.5 = 2t \]Divide both sides by 2:\[ t = 1.25 \]

Step 5: Answer the Question

Nancy will take 1.25 hours (or 1 hour and 15 minutes) to catch up to Kirk.

Key Concepts

Distance and Speed

When tackling real-world problems, understanding distance and speed is crucial. Distance measures how far an object travels, while speed indicates how quickly it moves. In this exercise, Kirk and Nancy's jogging speeds are given. Kirk jogs at 5 miles per hour and starts earlier. Nancy jogs faster at 7 miles per hour but begins half an hour later.
We want to find when they are at the same location on their jogging route. This means we need to calculate the time at which Nancy catches up with Kirk. Knowing their speeds allows us to formulate the problem.
Speed is a measure of how fast an object is moving and is often expressed in miles per hour (mph) in such problems. This forms the basis to establish equations that help in finding solutions. By setting up our knowledge of distances traveled, we can apply this understanding to solve the problem at hand.

Linear Equations

Linear equations are fundamental in algebra and are used to find unknown variables. In our exercise, the primary equation comes from setting Kirk and Nancy's jogging distances equal. This will reveal when Nancy catches up with Kirk.
To illustrate:

• Kirk's distance formula is: \( D_K = 5(t + 0.5) \)
• Nancy's distance formula is: \( D_N = 7t \)

Since the exercise seeks when these distances are equal, we set the equations equal: \( 5(t + 0.5) = 7t \).
These expressions show us that linear equations describe relationships where the change between variables is constant. It is why they are called linear—they can be represented as a straight line on a graph. Solving such equations is often straightforward and involves isolating variables and simplifying expressions.

Problem Solving Steps

Solving word problems effectively involves a strategic approach using clear problem solving steps. Here, each step builds towards a final solution.
The steps followed for our exercise were:

• Understanding the problem: Identify what you are solving. Here, finding the time until Nancy catches up with Kirk.
• Defining variables: Assign variables a meaning. Let \( t \) represent the time Nancy jogs.
• Writing equations: Translate the problem statements into mathematical equations.
• Solving the equations: Use algebraic techniques to find the value of unknowns.
• Answering the question: Interpret your solution in the context of the problem.

These strategic steps ensure clarity and completeness in solutions, turning complex word problems into manageable tasks.