Problem 2 - Prealgebra

Question

Express each of the following rates as a ratio with the given units. Miles/Hour A train travels 360 miles in 5 hours. What is the rate of the train in miles per hour?

Step-by-Step Solution

Answer

The train's rate is 72 miles per hour.

Step 1: Identify Total Distance and Time

To find the rate, we need to identify how far the train travels and how long it takes. Here, the train travels a total distance of 360 miles. The time taken to cover this distance is 5 hours.

Step 2: Use the Formula for Rate

The rate is given by the formula: \[\text{Rate} = \frac{\text{Total Distance}}{\text{Total Time}}\]Substitute the total distance and time into the formula: \[\text{Rate} = \frac{360 \text{ miles}}{5 \text{ hours}}\]

Step 3: Calculate the Division

Perform the division:\[\frac{360}{5} = 72\]The train's rate is 72 miles per hour.

Key Concepts

Understanding Distance in Motion

When it comes to understanding the concept of distance, we often think about how far something travels in a given amount of time. In this exercise, distance refers to the number of miles the train covers along its journey. Key points to remember about distance:

• Distance is a measure of how far one place is from another. In this case, it’s 360 miles.
• It’s important to always keep the units in mind. Here, it helps to know that the distance is given in miles.

The concept of distance helps us understand how much ground is covered when moving from one point to another.

Grasping the Concept of Time

Time is an essential component when we try to determine how fast something moves. In our exercise, time is the duration it takes for the train to travel the given distance. Here are some crucial aspects of time in this context:

• Time tells us how long the journey lasts. The train takes 5 hours in this exercise.
• Units of time are usually in hours, minutes, or seconds, but here we use hours.
• Understanding time helps us link how fast or slow an event occurs over a set period.

To calculate speed or rate, it’s critical to know both the time and the distance.

Simplifying with Division

Division is the mathematical operation we use to divide one number by another. This exercise requires us to find the train's speed, or rate, which involves division. Consider these key points about division:

• Division helps us break down the total distance into equal parts distributed over the total time.
• In doing so, we get a unit rate that tells how many miles the train travels per hour.
• In this scenario, 360 miles divided by 5 hours simplifies to 72 miles per hour.

This operation not only tells us the average speed but also provides a useful ratio to describe the train's pace. Understanding division makes it easier to calculate and interpret unit rates like miles per hour, which are crucial for comparing different speeds or analyzing travel efficiency.