Problem 46 - Prealgebra

Question

Track and Field. A person who runs a mile in 8 minutes has an average speed of 0.125 miles/minute. Convert 0.125 miles/minute to miles/hour.

Step-by-Step Solution

Answer

The speed is 7.5 miles/hour.

Step 1: Understanding the Conversion Factor

To convert from miles per minute to miles per hour, we need to know the conversion factor between minutes and hours. There are 60 minutes in an hour, so we can multiply the speed in miles per minute by 60 to get the speed in miles per hour.

Step 2: Setting Up the Conversion

We have the person's speed as 0.125 miles/minute. To find the equivalent speed in miles/hour, we will use the conversion factor derived from the previous step.

Step 3: Calculating the Speed in Miles per Hour

Multiply the speed in miles per minute by the number of minutes in an hour: \[ 0.125 ext{ miles/minute} imes 60 ext{ minutes/hour} = 7.5 ext{ miles/hour} \]

Step 4: Conclusion

After performing the multiplication, we find that the person's speed is 7.5 miles/hour.

Key Concepts

Understanding Average Speed

Average speed is a measure of how quickly an object moves over a specific distance. Unlike instantaneous speed, which considers the speed at a particular moment, average speed is focused on the overall movement over time. It is calculated by dividing the total distance traveled by the total time spent traveling.
For example, if a runner covers a mile in 8 minutes, their average speed can be calculated as:

• Distance: 1 mile
• Time: 8 minutes
• Average Speed = \( \frac{1 \text{ mile}}{8 \text{ minutes}} = 0.125 \text{ miles per minute} \)

Average speed is an essential concept in physics and everyday life, allowing us to understand how fast something is moving by looking at the overall journey rather than individual moments.

The Role of Conversion Factors

Conversion factors are tools that help translate measurements from one unit to another. These factors are essential for comparing speeds, distances, or any other measurements accurately. When converting between units, knowing the right conversion factor ensures that the measurement remains consistent and correct.
For instance, to convert miles per minute to miles per hour, you need to understand that there are 60 minutes in an hour. So, the conversion factor between minutes and hours is 60. This means you multiply the miles per minute by 60 to get miles per hour:

• Conversion factor: 60 minutes = 1 hour
• Example: \( 0.125 \text{ miles/minute} \times 60 \text{ minutes/hour} = 7.5 \text{ miles/hour} \)

By using conversion factors correctly, you ensure that your calculations are both accurate and meaningful, maintaining consistency across different types of measurements.

Understanding Miles per Hour (mph)

Miles per hour (mph) is a unit that measures speed, commonly used in the United States for road travel and other activities involving movement over distances. The unit tells us how many miles are traveled in one hour.
Using mph allows for easy comparisons and understanding of how fast something is moving. If a car travels at 60 mph, it means it covers 60 miles in an hour. Similarly, if a runner's speed is calculated to be 7.5 mph, they would cover 7.5 miles if they kept that speed for an entire hour.

• Mile: A unit of length equal to 5,280 feet
• Hour: A unit of time equal to 60 minutes
• mph: A measure of speed, calculated by \( \frac{\text{miles}}{\text{hour}} \)

This unit is not only useful in measuring speed but also essential for time and distance planning in travel and logistics.