Problem 45 - Algebra 1: Concepts and Skills

Question

If you drive a car \(m\) miles in 2 hours, which expression will give the average speed of the car? F. \(m+2\) G. \(2 m\) H. \(\frac{m-2}{2}\) J. \(\frac{m}{2}\)

Step-by-Step Solution

Answer

The correct expression to calculate the average speed of the car is J: \(\frac{m}{2}\).

Step 1: Understand the Problem

Before solving the problem, we first need to understand it. The problem is asking to find the average speed of the car that could be represented by an algebraic expression.

Step 2: Apply the Average Speed Formula

The formula for average speed is: \[Average\, speed = \frac{total\, distance}{total\, time}\]In this problem, the total distance travelled is \(m\) miles and the total time taken is 2 hours. Replacing these values in the formula, we get: \[Average\, speed = \frac{m}{2}\]

Step 3: Find the Matching Expression

Now we need to find the same expression among the given options. Looking at the options (F, G, H, J), we see that the expression that matches our calculation is J: \(\frac{m}{2}\).

Key Concepts

Algebraic Expressions

Algebraic expressions are mathematical sentences made up of numbers, variables, and operations. They are used to represent situations in a compact form. In the context of our problem, finding an average speed involves creating an algebraic expression to represent a real-world scenario: driving a car a certain distance over a period of time.

• Variables: These are symbols, like \(m\), that are used to represent unknown or changeable values. In the exercise, \(m\) stands for miles driven.
• Operations: Operations like addition, subtraction, multiplication, and division present relationships between numbers and variables. Here, dividing \(m\) by 2 helps calculate average speed.

Understanding algebraic expressions is key in translating words into mathematical language. By applying the expression \(\frac{m}{2}\), we learn how to use algebra to solve for average speed in practical situations like driving.

Distance Formula

The distance formula isn't just about finding how far you've traveled—it’s about seeing the relationship between distance, time, and speed. In our given problem, we talk about how to find average speed using distance. The average speed formula is:\[Average\, speed = \frac{total\, distance}{total\, time}\]To find average speed accurately, we need to assess:

• Total Distance: This tells us how far the car went. In our problem, the distance is represented by \(m\), the total miles the car has traveled.


• Total Time: This is how long it took to go that distance. In our exercise, the car's journey lasted 2 hours.


• Average Speed Calculation: By substituting the known values into the formula, we get \(\frac{m}{2}\) to derive the car's speed over the period.

Understanding and applying this formula allows us to solve various motion-related problems seamlessly.

Problem Solving

Problem solving in math involves several key steps: understanding the problem, applying the right formulas, and finding the solution. Our exercise involves these steps by finding the average speed of a car.

• Understanding the Problem: We start by identifying what needs solving—here, it’s about knowing which expression represents the right scenario.


• Applying Correct Formulas: Recognizing that average speed is calculated as \(\frac{distance}{time}\), helps us select the appropriate mathematical approach.


• Finding Solutions: Look at choices provided, translate the math work into one of the available options, and verify the meaning. Here, option J, \(\frac{m}{2}\), matches the required solution.

Through consistent practice, breaking down problems step by step becomes second nature, greatly enhancing problem-solving skills in mathematics.