Problem 36 - Prealgebra

Question

Mr. Farley is running a race at a speed of 3.5 miles per hour. Which equation can be used to find the number of miles \(m\) he can run in \(h\) hours? \(\mathbf{F} h=3.5+m\) \(\mathbf{G} m=3.5 h\) \(\mathbf{H} h=3.5 m\) \(\mathrm{J} \quad m=3.5+h\)

Step-by-Step Solution

Answer

The equation is \( m = 3.5h \), which corresponds to Option G.

Step 1: Understanding the Relationships

To solve this problem, we need to establish the relationship between speed, distance, and time. The formula that relates these three variables is: \[ \text{Distance} = \text{Speed} \times \text{Time} \]Here, Mr. Farley's speed is 3.5 miles per hour, distance is represented by \(m\), and time is \(h\).

Step 2: Substitute Given Values Into Equation

Substitute the given values into the standard speed formula:\[ m = 3.5 \times h \]This indicates that the distance \(m\) Mr. Farley runs is equal to his speed (3.5 miles per hour) times the number of hours \(h\) he runs.

Step 3: Identify the Correct Equation

Now, compare the derived equation \( m = 3.5 \times h \) with the given options:- **Option F**: \( h = 3.5 + m \) does not match.- **Option G**: \( m = 3.5 h \) is a direct match to our equation.- **Option H**: \( h = 3.5 m \) does not match.- **Option J**: \( m = 3.5 + h \) does not match.The correct choice is **Option G**.

Key Concepts

Distance-Rate-Time Relationship

Every journey can be described using three primary elements: distance, rate (or speed), and time. Understanding the connection between these elements allows you to solve many real-world problems, like determining travel times or distances. The core relationship is expressed through the formula:

• Distance = Rate × Time

This formula tells us that if you know any two of these elements, you can calculate the third. For example, if you know how fast you are traveling (your rate) and how long you have traveled (your time), you can find out how far you've gone (the distance).
In the context of our problem, Mr. Farley is running at a speed (rate) of 3.5 miles per hour, and we want to know how far he can travel (distance) in a given amount of time (hours). This relationship forms the basis of setting up our equation to solve for the unknown variable.

Basic Algebra

Algebra is the foundation of solving mathematical equations. It involves using symbols and letters to represent numbers and express relationships between these numbers. In our exercise, we encounter simple algebra through the formula:

• m = 3.5 h

Here, the variable m represents the distance Mr. Farley runs, and h represents the time in hours. Algebra allows us to rearrange this formula to find any variable when the others are known.
For example, if you know the distance and want to find the time, you can rearrange the formula to:

• h = \(\frac{m}{3.5}\)

Understanding these basics of algebraic manipulation is crucial as it allows you to solve for different parts of the equation depending on the information you have or need.

Problem Solving

Solving mathematical problems often involves a systematic approach, where defining the problem clearly is the first step. For Mr. Farley's running exercise:
1. **Define the Variables**: Identify the known and unknown quantities. Here, speed and time are known, while the distance is unknown. 2. **Use the Right Formula**: Look for the formula that connects these variables:

• Distance = Rate × Time

3. **Substitute the Values**: Replace the symbols with actual values and calculate the final answer.
By following this structured problem-solving cycle, you gain precision in your calculations and confidence in your answers. This method can be applied to different problems, helping you tackle complex scenarios by breaking them down into simpler, more manageable pieces.

Mathematical Reasoning

Mathematical reasoning involves logical thinking and the ability to make connections between different concepts to arrive at a solution. It's about understanding why a solution works, not just how to get to the answer. In Mr. Farley's case, consider:

• The fact that multiplying 3.5 miles per hour by the time in hours gives the distance directly.
• The equation form m = 3.5h reflects this relationship neatly and simply.

This reasoning leads us to determine that the correct equation among the choices is m = 3.5h (Option G).
By improving your reasoning abilities, you not only solve problems effectively, but also develop a deeper understanding of mathematical concepts. This skill is invaluable across all areas of learning and during problem-solving in real-world applications.