Problem 304 - Elementary Algebra

Question

In the following exercises, solve. Lizette is training for a marathon. At 7:00 she left her house and ran until \(8: 15,\) then she walked until 11:15. She covered a total distance of 19 miles. Her running speed was five miles per hour faster than her walking speed. Find her running and walking speeds.

Step-by-Step Solution

Answer

Walking speed: 3 mph, Running speed: 8 mph.

Step 1: Define Variables

Let Lizette's walking speed be denoted as \( w \) (in miles per hour). Therefore, her running speed will be \( w + 5 \) (in miles per hour).

Step 2: Calculate Time Spent Running and Walking

Lizette ran from 7:00 to 8:15, which is \( 1.25 \) hours (1 hour and 15 minutes). She walked from 8:15 to 11:15, which is \( 3 \) hours.

Step 3: Write Distance Equations

The distance covered while running is \[ (w + 5) \times 1.25 \], and the distance covered while walking is \[ w \times 3 \]. These distances sum to 19 miles: \[ (w + 5) \times 1.25 + w \times 3 = 19 \].

Step 4: Simplify the Equation

Expand and simplify the equation: \[ 1.25w + 6.25 + 3w = 19 \], which simplifies to \[ 4.25w + 6.25 = 19 \].

Step 5: Solve for Walking Speed

Subtract 6.25 from both sides: \[ 4.25w = 12.75 \], and then divide by 4.25: \[ w = 3 \]. So, Lizette's walking speed is 3 miles per hour.

Step 6: Calculate Running Speed

Since her running speed is 5 miles per hour faster than her walking speed, the running speed is \[ w + 5 = 3 + 5 = 8 \] miles per hour.

Key Concepts

Defining Variables

In any problem-solving scenario, identifying and defining variables is crucial. Here, we need to solve for Lizette's running and walking speeds.

Let's start by defining Lizette's walking speed as a variable. We use the letter \( w \) to represent her walking speed in miles per hour.

Because Lizette's running speed is five miles per hour faster than her walking speed, we express her running speed using the same variable: her running speed is \( w + 5 \) miles per hour.

Clearly defining variables helps simplify the process and avoid confusion as we work through the calculations.

Calculating Time

Understanding and calculating the time Lizette spends on different activities is essential for solving the problem.

Lizette starts running at 7:00 AM and runs until 8:15 AM. This is a total of 1 hour and 15 minutes. To convert this to hours, divide 15 minutes by 60 to get 0.25 hours. Thus, 1 hour and 15 minutes is 1.25 hours.

Next, she walks from 8:15 AM to 11:15 AM, which is a total of 3 hours.

With these times calculated, we are better prepared to use the distance formula to create our linear equation.

Distance Formula

The distance formula is crucial for solving problems involving speed, time, and distance. The basic formula is:

\[ \text{Distance} = \text{Speed} \times \text{Time} \]

To apply this formula, we need to calculate the distances Lizette covered while running and walking.

Her running speed is \( w + 5 \) miles per hour, and she ran for 1.25 hours, so the distance she ran is: \[ (\text{running speed}) \times (\text{time spent running}) = (w + 5) \times 1.25 \]

Her walking speed is \( w \) miles per hour, and she walked for 3 hours, so the distance she walked is:
\[ (\text{walking speed}) \times (\text{time spent walking}) = w \times 3 \]

Since the total distance covered equals 19 miles, we set up the equation:
\[ (w + 5) \times 1.25 + w \times 3 = 19 \]

Simplifying Equations

Simplifying equations involves breaking down complex expressions into more manageable parts.

Let's simplify the equation \[ (w + 5) \times 1.25 + w \times 3 = 19 \]:

First, distribute 1.25 across \[ (w + 5) eutral\]:

\[ 1.25w + 6.25 \]

Next, combine it with the second term in the equation: \[ 1.25w + 6.25 + 3w = 19 \]

Combine the terms with \( w \): \[ 4.25w + 6.25 = 19 \]

Solve this simplified equation by isolating \( w \). First, subtract 6.25 from both sides: \[ 4.25w = 12.75 \]

Then divide both sides by 4.25:
\frac{w = 12.75}{4.25} = 3

So, Lizette's walking speed is 3 miles per hour.

Since her running speed is 5 miles per hour faster, it is: \[ w + 5 = 3 + 5 = 8 \] miles per hour.