Problem 39 - Elementary Algebra

Question

Jeff leaves home on his bicycle and rides out into the country for 3 hours. On his return trip, along the same route, it takes him three-quarters of an hour longer. If his rate on the return trip was 2 miles per hour slower than on the trip out into the country, find the total roundtrip distance.

Step-by-Step Solution

Answer

The total roundtrip distance is 60 miles.

Step 1: Define Variables

Let \( r \) represent Jeff's speed in miles per hour on his trip out, and corresponding time \( t = 3 \) hours. His speed on the return trip will be \( r - 2 \) miles per hour, and it takes him \( 3 + \frac{3}{4} = \frac{15}{4} \) hours to return.

Step 2: Write Distance Equations

For the trip out, the distance \( D \) is given by \( D = r \times 3 \). For the return trip, the distance remains the same, hence \( D = (r - 2) \times \frac{15}{4} \).

Step 3: Setup Equation

Since both distances are the same, set the two equations equal to each other: \[ 3r = \left( r - 2 \right) \times \frac{15}{4} \].

Step 4: Solve for Speed \( r \)

Multiply through to simplify: \[ 12r = 15r - 30 \]. Rearrange to find \( r \): \[ 3r = 30 \] thus \( r = 10 \).

Step 5: Calculate Distance

Substitute \( r = 10 \) back into the distance equation: \( D = 3 \times 10 = 30 \) miles one way. Hence, the total roundtrip distance is \( 2 \times 30 = 60 \) miles.

Key Concepts

Distance-Rate-Time Problems

Distance-rate-time problems are a fundamental concept in algebra that help us calculate different quantities when an object is moving. These problems revolve around the core formula:

• Distance = Rate × Time

This formula breaks down into three key components:

• Distance: The length of the path traveled by a moving object. Measured in units like miles or kilometers.
• Rate: The speed at which the object is moving, usually in miles per hour (mph) or kilometers per hour (km/h).
• Time: How long the object has been moving, often expressed in hours or minutes.

In the exercise, Jeff's roundtrip experience involved different rates and times for each part of his journey. His journey out to the countryside took 3 hours at a certain speed (rate), while his return with a 2 mph slower speed and a longer time of 15/4 hours. We use the core formula to establish equations that allow us to find out the distance traveled based on the variables defined. This setup showcases how algebra assists in solving real-world problems through systematic analysis and equation solving.

Problem Solving Steps

Problem-solving steps in algebra word problems make these challenges manageable. By following a clear, systematic process, anyone can tackle even the trickiest tasks. Here's how the exercise achieved this:

• Step 1: Define the variables involved in the problem.
• Step 2: Write distance equations using the defined variables and given information.
• Step 3: Set up an equation to express the relationship (equality) of the distances calculated.
• Step 4: Solve the equation to find the unknown, which is often a rate or time.
• Step 5: Calculate the distance using the known values found through solving.

Each step is crucial and ensures nothing is left to guesswork. It also bolsters understanding by piecing together the problem logically. For Jeff's journey, this structured approach led to finding both his rates of travel and the total roundtrip distance. Remember that each part in this sequence provides the foundation for the following step, leading to a coherent solution.

Variable Definition in Algebra

In algebra, understanding and defining variables is the cornerstone of solving word problems. Variables act as placeholders for quantities we need to find. In this exercise, using variables helped simplify Jeff's biking problem:- We let \( r \) represent Jeff's speed on the way out, and connected it with specific quantities.- This allowed us to express his return speed as \( r - 2 \), indicating a 2 mph slower pace.When defining variables:

• Think of the problem's context. What are you trying to find? In Jeff's case, it was his speed.
• Use consistent symbols like \( r \) for rate or \( t \) for time; these keep your equations clear and organized.
• Connect your variables to the known quantities given in the problem. This ties them to real values.

Building equations from well-defined variables provides a structured way to investigate the unknowns. By doing so, you transform complex real-life questions into manageable mathematical tasks – leading to precise and accurate solutions.