Problem 284 - Elementary Algebra

Question

In the following exercises, solve. Kathy and Cheryl are walking in a fundraiser. Kathy completes the course in 4.8 hours and Cheryl completes the course in 8 hours. Kathy walks two miles per hour faster than Cheryl. Find Kathy's speed and Cheryl's speed

Step-by-Step Solution

Answer

Cheryl's speed: 3 mph, Kathy's speed: 5 mph

Step 1: Define Variables

Let Cheryl's speed be denoted as \( x \) (in miles per hour).

Step 2: Express Kathy's Speed

Since Kathy walks two miles per hour faster than Cheryl, Kathy's speed is \( x + 2 \) miles per hour.

Step 3: Write Distance Formulas

Use the formula distance = speed \( \times \) time. For Cheryl, the distance is \( 8x \) (since she walks for 8 hours). For Kathy, the distance is \( 4.8(x + 2) \) (since she walks for 4.8 hours).

Step 4: Set up the Equation

Since both walked the same distance, set the distance equations equal to each other: \[ 8x = 4.8(x + 2) \]

Step 5: Solve for Cheryl's Speed

Solve the equation for \( x \): \[ 8x = 4.8x + 9.6 \] Rearrange to isolate \( x \): \[ 8x - 4.8x = 9.6 \] \[ 3.2x = 9.6 \] \[ x = 3 \] Thus, Cheryl's speed is 3 miles per hour.

Step 6: Find Kathy's Speed

Since Kathy's speed is \( x + 2 \): \[ x + 2 = 3 + 2 = 5 \] Thus, Kathy's speed is 5 miles per hour.

Key Concepts

distance formula

When solving word problems related to motion, it's important to understand the distance formula. The distance formula connects the distance traveled, speed, and time. It is expressed as:\[ \text{Distance} = \text{Speed} \times \text{Time} \]
In the given problem, we know Kathy and Cheryl both walked the same distance, but at different speeds and times. Using the distance formula, we can express Cheryl's distance as \( 8x \) since she walked for 8 hours at a speed of \( x \) miles per hour. Similarly, Kathy's distance can be written as \( 4.8(x + 2) \) because she walked for 4.8 hours at a speed of \( x + 2 \) miles per hour. By setting these two distances equal, we can create an equation that allows us to find their individual speeds.

solving equations

To find the solution to this problem, we need to set up and solve an algebraic equation. This is how we can determine the speeds of Kathy and Cheryl.
First, recall Cheryl's distance formula is \( 8x \), and Kathy's is \( 4.8(x + 2) \). Since both cover the same distance, set the two expressions equal to each other: \( 8x = 4.8(x + 2) \).
Next, solve for \( x \). First, expand and arrange the equation: \( 8x = 4.8x + 9.6 \).
Subtract \( 4.8x \) from both sides: \( 8x - 4.8x = 9.6 \)
Simplify the terms: \( 3.2x = 9.6 \).
Finally, isolate \( x \) by dividing both sides by 3.2: \( x = 3 \).
This tells us that Cheryl's speed is 3 miles per hour. To find Kathy's speed, add 2 miles per hour to Cheryl's speed, giving us \( 3 + 2 = 5 \), so Kathy's speed is 5 miles per hour.

speed and distance

Understanding the relationship between speed, distance, and time is crucial in solving word problems involving travel. In these problems, the concepts of speed and distance are often intertwined.
Speed refers to how fast someone or something is moving. It's usually given in units like miles per hour (mph) or kilometers per hour (km/h).
Distance is a measure of how far someone or something travels. It's calculated by multiplying speed by the time traveled, as given by the distance formula.
In this problem, we knew the walking times for Kathy and Cheryl and we expressed their speeds and distances using variables. Finding these critical variables required setting their distance equations equal, solving for the unknown, and then applying those results to understand their speeds and distances more accurately.
This method is applicable to many real-life situations where you need to determine how far or fast someone or something is moving over a period.