Problem 292 - Elementary Algebra

Question

In the following exercises, solve. DaMarcus and Fabian live 23 miles apart and play soccer at a park between their homes. DaMarcus rode his bike for threequarters of an hour and Fabian rode his bike for half an hour to get to the park. Fabian's speed was six miles per hour faster than DaMarcus' speed. Find the speed of both soccer players.

Step-by-Step Solution

Answer

DaMarcus' speed is 16 mph, and Fabian's speed is 22 mph.

Step 1 - Define Variables

Let DaMarcus' speed be represented as \(s\) miles per hour. Thus, Fabian's speed will be taken as \(s + 6\) miles per hour.

Step 2 - Write the Distance Formula

The distance each of them travels to the park can be expressed using the formula \(distance = speed \times time\).

Step 3 - Calculate the Distance Each Rode

For DaMarcus: \( d_{DaMarcus} = s \times \frac{3}{4} \). For Fabian: \( d_{Fabian} = (s + 6) \times \frac{1}{2} \).

Step 4 - Setup the Equation

Since they meet at the park, their total distance combined is 23 miles. Therefore, \[ s \times \frac{3}{4} + (s + 6) \times \frac{1}{2} = 23 \].

Step 5 - Solve for \(s\)

Solve the equation: \[ \frac{3}{4}s + \frac{1}{2}(s + 6) = 23 \] \[ \frac{3}{4}s + \frac{1}{2}s + 3 = 23 \] Combine like terms: \[ \frac{5}{4}s + 3 = 23 \] Subtract 3 from both sides: \[ \frac{5}{4}s = 20 \] Multiply both sides by \(\frac{4}{5}\): \[\therefore s = 16 \text{ miles per hour} \].

Step 6 - Find Fabian's Speed

Since Fabian's speed is \(s + 6 \), substitute \( s = 16 \rightarrow 16 + 6 = 22 \text{ miles per hour} \). Hence, Fabian's speed is 22 miles per hour.

Key Concepts

Distance Formula

The distance formula is a critical concept in many algebra problems, especially those involving movement. It helps us determine how far something travels based on its speed and the time it moves. The formula is written as:
\[ \text{Distance} = \text{Speed} \times \text{Time} \]
By knowing any two of these variables, we can solve for the third. In our problem, we needed to find out how far DaMarcus and Fabian traveled to reach the park. We knew their travel times and defined their speeds as variables. Using the distance formula, we could express their distances in terms of these variables.
Understanding this formula is essential. It appears frequently in many applications, from physics to everyday problem-solving. When applying it, make sure your units for speed and time are compatible, like miles per hour for speed and hours for time.

Solving Equations

Solving equations involves finding the value of a variable that makes the equation true. In our exercise, we had an equation reflecting a real-world situation. Here's the step-by-step process used to solve it:
\[ \frac{3}{4}s + \frac{1}{2}(s + 6) = 23 \] Each term in the equation represents a part of the total scenario. DaMarcus's and Fabian's distances are combined to match the total distance to the park.
To solve, we:

• Combined like terms
• Isolated the variable terms on one side
• Simplified the equation step by step
• Used algebraic operations to solve for the variable

This systematic approach to solving equations ensures you accurately find the unknown value. It's a technique that lays the foundation for more advanced mathematics and practical problem-solving.

Rate and Time Calculations

Rate (speed) and time calculations are vital in determining distances, just like in our exercise. DaMarcus rode for three-quarters of an hour, and Fabian for half an hour. Let's break it down:
\[ \text{Rate} \times \text{Time} = \text{Distance} \]
For DaMarcus, this is:
\[ s \times \frac{3}{4} \]
For Fabian, whose speed is 6 miles per hour faster:
\[ (s + 6) \times \frac{1}{2} \]
By using their specific travel times and rates, we calculated how far each traveled. Understanding how to manipulate these basic calculations is extremely important, as they apply to countless real-world scenarios, from planning trips to determining deadlines.

Defining Variables

Defining variables is one of the first steps in solving word problems. It translates real-world scenarios into algebraic expressions. Here's how we did it in our problem:

• We let DaMarcus' speed be represented as \( s \) miles per hour.
• Fabian's speed was expressed as \( s + 6 \) miles per hour, since he was 6 miles per hour faster.

By defining these variables, we created a clear, mathematical representation of the problem.
Defining variables accurately is crucial for setting up a problem correctly. Ensuring the terms accurately represent the given information helps in formulating correct equations. Practice this through various problems to enhance your problem-solving efficiency.