Question

Write and solve an equation to find your average speed on a trip from St. Louis to Dallas. You drove 630 miles in \(10 \frac{1}{2}\) hours.

Short answer

The average speed for the trip from St. Louis to Dallas was 60 miles per hour.

Step-by-step solution

Identification of Key Details

In the given problem, the distance covered is 630 miles and the time taken is \(10 \frac{1}{2}\) hours, which is equivalent to 10.5 hours in decimal form. We need to find the average speed.

Set up the Formula for Average Speed

The formula for average speed is calculated by dividing the total distance travelled by the total time taken. Thus, the equation becomes: Average speed = Total Distance / Total Time. Plugging in the known values into the equation gives: Average speed = 630 miles/10.5 hours.

Solve for Average Speed

Dividing 630 by 10.5, the average speed for the journey equals to 60 miles per hour.

Key concepts

Understanding the Average Speed Formula

The concept of average speed is commonly misunderstood, yet it is an integral part of travel and movement. It is calculated by taking the total distance travelled and dividing it by the total time taken to cover that distance. The formula is written as:
\[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} \]
In practical terms, if you're driving from one city to another, as in the textbook exercise, you would note the miles covered, measured by your car's odometer, and the time you took, which might be measured by a simple wristwatch or your car's clock. It's crucial to convert all your measurements to compatible units to use this formula properly. For instance, if the distance is in miles and the time is in hours, the average speed will come out in miles per hour, as seen in the textbook problem where a journey of 630 miles took 10.5 hours, resulting in an average speed of 60 mph.

Decoding Distance-Time Ratio

The relationship between distance and time is fundamental in understanding motion. Distance-time ratio gives us an insight into how fast an object is moving. In the context of our textbook problem, the total distance is a fixed figure – 630 miles. However, the time taken can vary depending on the speed. If you're slower, the time will be longer; if you're faster, the time will be shorter.
To grasp this concept better, consider this: If someone else made the same St. Louis to Dallas trip in half the time, their average speed would be double yours. And if they took twice as long, their average speed would be half. The ratio provides insights into travel planning and can help you estimate arrival times based on different speeds. It's a simple yet powerful tool to compare speeds without actually calculating the average speed for different scenarios.

Solving Problems Using Algebraic Equations

Algebraic equations are a cornerstone of mathematics, used to solve problems involving unknown variables. In our textbook example, we set up an equation to find the average speed, which is our unknown variable. Algebraic manipulation allows us to isolate this variable and solve for it.
Here's a simple step-by-step rundown:

• Identify the variables: in this case, distance (D) and time (T).
• Write down the equation: Average speed (S) equals distance divided by time (S = D/T).
• Plug in the known values: D = 630 miles, T = 10.5 hours.
• Perform the division: S = 630 / 10.5, which simplifies to 60 mph.

Algebra can seem intimidating, but with practice, it becomes a powerful skill that allows for the solution of various practical problems, such as finding average speeds, calculating budgets, or predicting outcomes.