Problem 34 - Prealgebra

Question

Use the rate equation \(r=\frac{d}{t}\) to solve. At 8: 30 A.M. Ethan leaves his house and drives 220 miles to his brother's house. He arrives at 12: 30 P.M. a. How many hours was the drive to his brother's house? b. What was Ethan's average speed?

Step-by-Step Solution

Answer

a. 4 hours; b. 55 mph

Step 1: Identify Total Travel Time

Ethan leaves at 8:30 A.M. and arrives at 12:30 P.M. To find the duration of his drive, calculate the difference between these two times. Between 8:30 A.M. and 12:30 P.M., Ethan travels for 4 hours.

Step 2: Calculate Average Speed

Using the rate equation \( r = \frac{d}{t} \), where \( d \) is the distance traveled and \( t \) is the time taken, we can find Ethan's average speed. Plug in the values: \( d = 220 \) miles, \( t = 4 \) hours. Therefore, \( r = \frac{220}{4} = 55 \) mph.

Key Concepts

Time Calculation

Time calculation is essential for planning trips and estimating arrival times. In this exercise, Ethan starts his journey at 8:30 A.M. and arrives at his destination at 12:30 P.M. To determine how long he traveled, we calculate the difference between these two times. Here’s how we can do it:

• Start by writing down the start time: 8:30 A.M.
• Write down the arrival time: 12:30 P.M.
• Subtract the start time from the arrival time: 12:30 P.M. minus 8:30 A.M. equals a total of 4 hours.

Understanding how to calculate time differences can help you make better plans and manage your schedule effectively. It’s a simple subtraction problem, but important for organizing activities and events.

Average Speed

When you need to find out how fast someone or something is moving, average speed gives you a clear picture. It tells you the rate at which distance is covered. In Ethan's journey, we use the formula for average speed: \[ r = \frac{d}{t}\]where:

• \( r \) is the average speed,
• \( d \) is the total distance traveled, and
• \( t \) is the total time taken.

For Ethan:

• Distance traveled, \( d \), is 220 miles.
• Time taken, \( t \), is 4 hours.

Plug these values into the formula to find the average speed:\[ r = \frac{220}{4} = 55\text{ mph}\]This calculation helps you understand how fast Ethan must have been driving, on average, throughout his journey from his house to his brother's house.

Distance and Time Relationship

The relationship between distance and time is rooted in the rate equation, which connects these two variables with speed. It’s an essential part of understanding motion and travel calculations. Let’s break it down:

• Distance \( (d) \) is how far the traveler moves or the length of the trip.
• Time \( (t) \) refers to the duration the traveler has been moving.
• Speed \( (r) \), or rate, expresses how quickly the distance is covered per unit of time.

For Ethan’s trip:

• The distance between Ethan’s house and his brother’s place is 220 miles.
• He spent 4 hours on the road.
• From previous calculations, his average speed was 55 mph.

Understanding the interplay of these elements allows you to solve various travel problems, such as predicting arrival times or estimating travel duration. It enables smart decisions during trip planning and offers a clear picture of any journey's dynamics.