Problem 32 - Elementary Algebra

Question

Two cars start from the same place traveling in opposite directions. One car travels 4 miles per hour faster than the other car. Find their speeds if after 5 hours they are 520 miles apart.

Step-by-Step Solution

Answer

The slower car travels at 50 mph, and the faster car at 54 mph.

Step 1: Define Variables

Let the speed of the slower car be \( x \) miles per hour. Therefore, the speed of the faster car will be \( x + 4 \) miles per hour.

Step 2: Calculate Distance Traveled by Each Car

The distance traveled by the slower car in 5 hours can be represented as \( 5x \) miles. The distance traveled by the faster car in 5 hours is \( 5(x + 4) \) miles.

Step 3: Set up the Equation Based on Total Distance

Since the two cars are traveling in opposite directions, we add their distances: \( 5x + 5(x + 4) = 520 \).

Step 4: Simplify the Equation

Expand and simplify the equation: \( 5x + 5x + 20 = 520 \) which simplifies to \( 10x + 20 = 520 \).

Step 5: Solve for x

Move the constant term to the other side: \( 10x = 500 \). Then divide by 10 to find \( x \): \( x = 50 \).

Step 6: Find the Speed of Each Car

The slower car travels at \( 50 \) miles per hour, and the faster car travels at \( x + 4 = 50 + 4 = 54 \) miles per hour.

Key Concepts

Distance and Speed Calculations

When dealing with problems involving cars, trains, or other moving objects, understanding how to calculate distance and speed is crucial. These calculations help us determine how far something has moved over a period of time. The basic formula that connects distance, speed, and time is:\[ \text{Distance} = \text{Speed} \times \text{Time} \]This formula indicates that distance is the product of speed and time. If you know any two of these quantities, you can calculate the third one. In our scenario, two cars are traveling for the same duration, namely 5 hours.

• For the slower car: The distance is calculated as \( 5x \), where \( x \) is the speed in miles per hour.
• For the faster car: The distance is calculated as \( 5(x + 4) \), since its speed is 4 miles per hour faster than the slower car.

As they move in opposite directions, their total distance apart adds up, and this value is given as 520 miles. This understanding allows us to form an important step in solving this problem.

Equation Solving

Equation solving is at the heart of algebraic problems like this one. Once we establish the relationship between distance, speed, and time, we set up an equation to find unknown values. From our problem, we want to find the speed of each car.
Setting up the equation involves translating the word problem into math language. We know:- After 5 hours, the total distance is 520 miles.- The combined distances of both cars form the equation: \[ 5x + 5(x + 4) = 520 \]Simplifying this equation step by step:

• First, expand the equation: \(5x + 5x + 20 = 520 \)
• Combine like terms: \(10x + 20 = 520\)
• Isolate the variable term by subtracting 20 from both sides: \(10x = 500\)
• Finally, divide by 10 to solve for \(x\): \(x = 50\)

By solving this equation, we find the speed of one car, and indirectly, the speed of the second car.

Defining Variables

Defining variables is an essential precursor to solving algebraic problems. It involves choosing symbols to represent unknown quantities that we aim to determine. This step simplifies complex information, allowing us to work with it mathematically.

In our problem about two cars:- We designate \(x\) as the speed of the slower car in miles per hour.- The speed of the faster car is expressed in terms of \(x\) as \(x + 4\). This correctly represents the given condition that one car is 4 miles per hour faster than the other.
Choosing clear and meaningful variables aids greatly in setting up the problem for solution. It bridges the gap between understanding the word problem and converting it into a mathematical model. This precise definition ensures that every calculation leads logically towards finding the solution.