Problem 28 - 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 traveled at an average speed of 41.1 miles per hour.

Step 1: Determine Total Distance Traveled

Subtract the initial reading from the final reading of the odometer to find the total distance traveled during the trip. \[ 32,741.8 - 32,567.2 = 174.6 \] The total distance traveled is 174.6 miles.

Step 2: Convert Time to Hours

Convert the time of the trip, 4 \( \frac{1}{4} \) hours, to a decimal. 4 \( \frac{1}{4} \) hours is equal to 4.25 hours.

Step 3: Calculate the Rate in Miles per Hour

Use the formula for rate, which is distance divided by time. \[ \text{Rate} = \frac{\text{Distance}}{\text{Time}} = \frac{174.6}{4.25} \]Calculate the result: \[ \text{Rate} = 41.082\overline{3} \]Thus, the rate is approximately 41.1 miles per hour when rounded to the nearest tenth.

Key Concepts

Understanding Distance

Distance, in most simple terms, refers to how far one thing is from another. It's often measured in units such as miles or kilometers. In our problem, we need to find out how far a car traveled during a trip. To do this, we look at the car’s odometer readings at the start and end of the journey.

For example:

• Starting odometer reading: 32,567.2 miles
• Ending odometer reading: 32,741.8 miles

Subtract the starting reading from the ending reading to get the total distance traveled. This gives us a clear idea of the journey's length.

In math terms, the distance is computed as follows: \[ \text{Distance} = 32,741.8 - 32,567.2 = 174.6 \text{ miles} \]

This tells us that the car traveled a total of 174.6 miles during the trip.

Understanding Time Conversion

Converting time is a crucial step when calculating speed. Time is often given in hours and minutes, and sometimes, in fractions.

In this exercise, we deal with mixed numbers, like 4 \( \frac{1}{4} \) hours. This might seem complex, but it’s manageable once broken down.

The key is knowing that 1 hour equals 60 minutes. Thus, \( \frac{1}{4} \) hour means a quarter of 60 minutes, which is 15 minutes.

For clarity:

• 4 hours remain 4 hours
• \( \frac{1}{4} \) of an hour is 15 minutes

Therefore, 4 \( \frac{1}{4} \) hours can be converted into a decimal representation: \[4 + 0.25 = 4.25 \text{ hours}\]
Using decimals helps us in the calculation of speed later on.

Understanding Miles per Hour

Miles per hour (mph) is a unit used to express speed. It tells us how many miles a vehicle travels in one hour.

In this problem, we know how far the car traveled and how long it took. To find the speed in miles per hour, the formula is: \[ \text{Rate} = \frac{\text{Distance}}{\text{Time}} \]

This means:

• Distance = 174.6 miles
• Time = 4.25 hours (as a decimal)

Thus, substituting the known values, we compute:\[ \text{Rate} = \frac{174.6}{4.25} \approx 41.1 \text{ mph} \]

This result is rounded to the nearest tenth, giving a clear and precise measurement of how fast the car was moving.

Understanding Decimal Conversion

Decimal conversion involves changing fractions and whole numbers into a decimal form. This action makes calculations easier in most cases, especially when computing speed or rate.

For instance, in our problem, the time given was 4 \( \frac{1}{4} \) hours. To convert this, we changed the fraction to a decimal:

\( \frac{1}{4} \) can be expressed as 0.25 when converted to a decimal.

With this simple conversion, we could then express the total time as 4.25 hours.

Using decimals is advantageous because:

• They simplify complex calculations.
• They allow for precision in scientific and mathematical contexts.

Comprehending decimal conversion is crucial for calculations, particularly when dealing with mixed numbers in real-world problems.