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Speed is a fundamental concept in algebra word problems, especially when dealing with moving objects. In this context, speed refers to how quickly something is moving from one place to another. It is determined by dividing the total distance traveled by the total time taken. This is represented in the equation: \[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} \] For example, in our ferry trip problem, the distance is 138 miles and the time is \(2 \frac{1}{2}\) hours. Using the speed formula, we can calculate the ferry's speed by substituting the given values into the equation.

• Distance: 138 miles
• Time: \(\frac{5}{2}\) hours (converted from mixed number)

This allows us to find that the ferry's speed is 55.2 miles per hour, demonstrating how speed connects distance and time in moving objects.

Fractions are crucial in algebra, particularly when dealing with problems that require division and multiplication. A fraction represents a part of a whole and is expressed as a division of two numbers, such as \(\frac{a}{b}\). In the context of our problem, the given time was a mixed number \(2 \frac{1}{2}\) hours. To simplify calculations, this was converted to an improper fraction. To convert a mixed number to an improper fraction:

• Multiply the whole number by the denominator: \(2 \times 2 = 4\).
• Add the numerator: \(4 + 1 = 5\).
• Place the result over the original denominator: \(\frac{5}{2}\).

This conversion allows us to work with the fraction more easily, aiding in calculations such as those required in speed determination. When plugging fractions into equations, always remember to consider the reciprocal if dividing by a fraction.

Distance-time problems often involve calculating one of these three elements—distance, time, or speed—when the other two are known. Understanding the relationship among these elements is key. In any distance-time scenario:

• Distance is the total path covered by the object.
• Time is the duration taken to cover the distance.
• Speed is how fast the object is moving through its path.

In our example, knowing the ferry's distance and time allows us to find its speed using the equation \(\text{Speed} = \frac{\text{Distance}}{\text{Time}}\). This formula serves as the foundation for solving many algebra word problems involving motion. Remember, when facing distance-time problems, always have a clear understanding of what you're solving for. Double-check your units to ensure consistent calculations, and use conversions like transforming mixed numbers to improper fractions to facilitate computation.