Problem 290 - Elementary Algebra

Question

In the following exercises, solve. Ethan and Leo start riding their bikes at the opposite ends of a 65-mile bike path. After Ethan has ridden 1.5 hours and Leo has ridden 2 hours, they meet on the path. Ethan's speed is six miles per hour faster than Leo's speed. Find the speed of the two bikers.

Step-by-Step Solution

Answer

Leo's speed is 16 mph. Ethan's speed is 22 mph.

Step 1: Define Variables

Let the speed of Leo be denoted as \(v\) miles per hour. Therefore, the speed of Ethan is \(v + 6\) miles per hour.

Step 2: Express Distances

The distance covered by Ethan in 1.5 hours is \((v + 6) \times 1.5\) miles. The distance covered by Leo in 2 hours is \(v \times 2\) miles.

Step 3: Set Up Equation

Since they meet on a 65-mile path, the sum of the distances they covered equals 65 miles. Hence, we write the equation: \[(v + 6) \times 1.5 + v \times 2 = 65\].

Step 4: Simplify the Equation

Expand and simplify the equation: \(1.5(v + 6) + 2v = 65\). This becomes \(1.5v + 9 + 2v = 65\). Combine like terms to get \(3.5v + 9 = 65\).

Step 5: Solve for \(v\)

Isolate \(v\) by subtracting 9 from both sides: \(3.5v = 56\). Divide both sides by 3.5 to get \(v = 16\).

Step 6: Find Ethan's Speed

Since Ethan's speed is 6 miles per hour faster than Leo's, and Leo's speed is 16 miles per hour, Ethan's speed is \(16 + 6 = 22\) miles per hour.

Key Concepts

Linear equations

A linear equation is an equation that makes a straight line when it is graphed. It has the form ax + b = c, which could be simplified to solve for x. In our problem, we created a linear equation to model the total distance traveled by Ethan and Leo. Linear equations often emerge from real-world problems, like figuring out distances, times, and speeds. They are powerful tools because they represent relationships between different quantities. We will break down the equation step-by-step to make it more understandable.

Speed and distance calculations

Speed and distance problems often use the formula \( \text{Distance} = \text{Speed} \times \text{Time} \). Here, we calculate the distances each biker covers by multiplying their speed by their respective times.

• Ethan's distance: \( (v + 6) \times 1.5 \ miles \).
• Leo's distance: \( v \times 2 \ miles \).

Combining these gives the total distance they travel when they meet, which is 65 miles. This formula helps us set up the relationship between speed, time, and distance, and is crucial for solving these types of word problems.

Variable substitution

Variable substitution is a method used to simplify equations or expressions by replacing a variable with its equivalent value.
In this problem, let \( v \) be Leo's speed. Then, Ethan's speed will be \( v + 6 \). We substitute these into our distance equations. By substituting, we can create one equation that is easier to solve because it only has one variable, \( v \).
This technique is especially helpful when you have to deal with multiple relationships and you need to isolate variables to find their values.

Equation solving steps

To solve the equation, follow these steps:

• Combine like terms: \( (1.5v + 9) + (2v) = 65 \) simplifies to \( 3.5v + 9 = 65 \).
• Isolate the variable: Subtract 9 from both sides to get \( 3.5v = 56 \).
• Solve for \( v \): Divide by 3.5 to find \( v = 16 \).

Now, substitute \( v \) back to find Ethan's speed: \( v + 6 = 22 \) miles per hour. Remember to double-check each step to catch any mistakes. Breaking down the problem into smaller steps can make it easier to understand and solve.