Problem 61 - Prealgebra

Question

Martin drives 6 hours at an average rate of 65 miles per hour. What is the distance Martin travels? Use \(d=r t\).

Step-by-Step Solution

Answer

Martin travels 390 miles.

Step 1: Identify the Known Values

First, identify the values given in the problem. Martin's driving time is 6 hours, and his average speed is 65 miles per hour.

Step 2: Use the Distance Formula

Apply the formula for distance, which is given by \[ d = r \, t \] where \( d \) is the distance, \( r \) is the rate of speed (65 miles per hour), and \( t \) is the time (6 hours).

Step 3: Substitute the Known Values into the Formula

Replace \( r \) with 65 and \( t \) with 6 in the distance formula:\[ d = 65 \, \text{miles/hour} \times 6 \, \text{hours} \]

Step 4: Perform the Calculation

Multiply 65 by 6 to find the distance traveled:\[ d = 390 \, \text{miles} \]

Key Concepts

Rate of Speed

Rate of speed is a fundamental concept when calculating how far you travel over a period of time. It is typically expressed in terms such as miles per hour (mph) or kilometers per hour (km/h). Here, it tells us how many miles Martin covers per hour on average. To break it down:

• The rate of speed helps to measure movement over time.
• In Martin's case, this is 65 miles per hour.
• This means, every hour, Martin travels 65 miles if the speed remains constant.

Understanding rate of speed is crucial. It is part of the formula used to calculate travel distance, which combines speed with the time spent moving. It tells us that both speed and time are directly proportional to distance. If speed increases, more distance is covered in less time.

Time Calculation

Time plays a key role in determining how far an object will travel. When you know how long someone or something has traveled, you can use this information in tandem with speed to find the distance. In Martin's situation, he drove for 6 hours. Consider these aspects of time calculation:

• Time is measured in units such as hours, minutes, or seconds.
• For distance problems, ensure the time units match the speed units for accurate calculations.
• In the given problem, time is expressed in hours, so it pairs smoothly with Martin's speed in miles per hour.

Accurate time tracking is essential. It ensures your calculations remain precise. Mixing up units or miscounting time can lead to significant discrepancies in your distance calculations.

Multiplication

Multiplication is the mathematical operation we use to combine the rate of speed and the time to find the distance traveled. In Martin's example, this operation is crucial in the final step to solve the problem.Why multiplication matters here:

• It allows us to consider the impact of both speed and time together.
• In the formula \(d = r \, t\), distance (\(d\)) equals speed (\(r\)) multiplied by time (\(t\)).
• In Martin's case, multiplying 65 miles per hour by 6 hours results in 390 miles.

Multiplication ensures the results are scaled properly. It provides a straightforward way to relate speed and time to a measurable outcome, such as distance. Mastering multiplication aligns seamlessly with using formulas efficiently.