Problem 26 - Prealgebra

Question

At the beginning of a trip, the odometer on a car read \(32,567.2\) miles. At the end of the trip, it read \(32,741.8\) miles. If the trip took \(4 \frac{1}{4}\) hours, what was the rate of the car in miles per hour to the nearest tenth?

Step-by-Step Solution

Answer

The car's rate was 41.0 miles per hour.

Step 1: Determine the total distance traveled

To find the total distance traveled, subtract the initial odometer reading from the final odometer reading. The initial reading was 32,567.2 miles and the final was 32,741.8 miles. So, the distance traveled is:\[ 32,741.8 - 32,567.2 = 174.6 \]Therefore, the car traveled a total of 174.6 miles.

Step 2: Convert the time into a decimal

The trip took \(4 \frac{1}{4}\) hours. Convert this mixed number into a decimal for easier calculation.\[4 \frac{1}{4} = 4 + \frac{1}{4} = 4 + 0.25 = 4.25 \]So, the trip took 4.25 hours.

Step 3: Calculate the rate of the car in miles per hour

To calculate the rate in miles per hour, divide the total distance by the total time in hours. Using the values from earlier steps, the calculation is:\[ \text{Rate} = \frac{174.6}{4.25} \]Performing the division gives:\[ \text{Rate} \approx 41.035 \]Rounding to the nearest tenth, the rate is 41.0 miles per hour.

Key Concepts

Distance Traveled

Understanding how to calculate distance traveled is fairly simple once you have the necessary initial and final readings from an odometer. First, check the initial odometer reading before a trip commences. Let's say it was 32,567.2 miles. Next, at the end of your journey, take note of the final odometer reading, which was 32,741.8 miles in this case. Subtraction is the key here. You simply subtract the initial odometer reading from the final odometer reading:

• Final reading: 32,741.8 miles
• Initial reading: 32,567.2 miles
• Distance Traveled: 32,741.8 - 32,567.2 = 174.6 miles

Thus, the total distance traveled is 174.6 miles. This simple subtraction helps you track how much ground you've actually covered on your journey.

Time Conversion

When you're working with time, it's often easier to convert any mixed numbers into decimals. For example, the total time for this trip was given as 4 hours and 15 minutes, which is written as \( 4 \frac{1}{4} \) hours. Here's how you can convert that into a decimal:

• Understand that 15 minutes is one-quarter of an hour, which is written as \( \frac{1}{4} \).
• Convert \( \frac{1}{4} \) into a decimal, which is 0.25.
• Add this decimal to the whole number of hours: 4 + 0.25 = 4.25 hours.

So, the entire trip took 4.25 hours. This simple conversion ensures calculations involving time are more straightforward, especially when determining rates like miles per hour.

Miles Per Hour

Calculating miles per hour (mph) involves finding out how many miles are covered in one hour. To do this, you divide the total distance traveled by the total time taken in hours. With our earlier steps in mind, let's calculate the rate:

• Distance Traveled: 174.6 miles
• Time Taken: 4.25 hours
• Rate Calculation: \( \text{Rate} = \frac{174.6}{4.25} \approx 41.035 \) mph

By dividing the distance by the time, you obtain a rate of roughly 41.035 mph. When rounding to the nearest tenth, the car was traveling at around 41.0 miles per hour. Calculating rate in this manner not only gives you an idea of speed but also allows you to understand and compare how efficiently you can cover distances.