Problem 45 - Algebra 1

Question

MULTIPLE CHOICE You drove 200 miles in 3 hours 20 minutes. Which expression represents your average speed if \(d\) represents distance and \(t\) represents time? A \(d t\) B \(\frac{d}{t}\) C \(\frac{t}{d}\) D \(d+t\)

Step-by-Step Solution

Answer

The correct answer is option B: \(\frac{d}{t}\). This is the correct mathematical expression for average speed.

Step 1: Identify the Correct Formula for Average Speed

Average speed is calculated as total distance traveled divided by total time taken. This can be represented as \(\frac{d}{t}\).

Step 2: Match the Formula to the Given Options

The only option that matches our formula \(\frac{d}{t}\) is option B.

Step 3: Validation Check

Replace \(d\) with 200 miles and \(t\) with 3.33 hours in the formula. The result (approximately 60 miles/hour) stands valid for the concept of average speed. Therefore, confirmation is obtained that the identified option B is correct.

Key Concepts

Distance and Time

To solve problems related to movement, it is essential to understand the concepts of distance and time. Distance is the total length of the path covered by a moving object, while time refers to the period over which the movement occurs. The units of measurement for distance are often in miles or kilometers, and time is measured in seconds, minutes, or hours.

Such problems typically involve determining how far an object has traveled (distance) over a certain period (time), or inversely, how much time it takes to cover a specific distance. Remember that to obtain accurate results, the units for distance and time must be consistent when performing calculations. For instance, if the distance is in miles, then the speed should be calculated in miles per hour.

Average Speed Formula

Understanding the formula for average speed is crucial for solving motion related problems. The average speed is defined as the total distance travelled divided by the total time taken to travel that distance. Mathematically, we represent it with the formula: \[ \text{Average Speed} = \frac{\text{Total Distance Traveled}}{\text{Total Time Taken}} \]

For example, if a car traveled 200 miles in 3 hours and 20 minutes, to find the average speed we first convert the time to a single unit, preferably hours. In this case, 3 hours and 20 minutes equates to 3.33 hours. The average speed would then be \( \frac{200 \text{ miles}}{3.33 \text{ hours}} \) which simplifies to approximately 60 miles per hour. This is a fundamental concept that often serves as the foundation for more complex problems in physics and real-world applications such as travel planning.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain numbers, variables, and operation symbols. Variables, represented by letters such as \(d\) for distance and \(t\) for time, stand in for unknown or changing quantities. Operations include addition, subtraction, multiplication, and division.

The beauty of algebraic expressions lies in their ability to model real-world situations in a simplified manner. For example, the expression \( \frac{d}{t} \) represents the average speed, where \(d\) is the distance travelled and \(t\) is the time taken. Such expressions allow us to manipulate the represented quantities algebraically to solve for unknowns, make predictions, and understand relationships between variables.