Problem 93 - Algebra for College Students

Question

Distance traveled. Bonita drove 100 miles in \(x\) hours. Assuming she continued to drive at the same speed, write a rational expression for the distance that she traveled in the next \(\frac{3}{4}\) of an hour.

Step-by-Step Solution

Answer

In the next \(\frac{3}{4}\) of an hour, Bonita will travel \(\frac{75}{x}\) miles.

Step 1 - Find the speed

To find the speed, divide the total distance traveled by the time taken. Given that Bonita drove 100 miles in \(x\) hours, the speed is \(\frac{100}{x}\) miles per hour.

Step 2 - Calculate the distance for the next time period

To find the distance traveled in the next \(\frac{3}{4}\) of an hour, multiply the speed by this time period. Thus, the distance traveled in the next \(\frac{3}{4}\) of an hour is \(\frac{100}{x} \times \frac{3}{4} = \frac{300}{4x} = \frac{75}{x}\).

Key Concepts

Distance and Speed

To start, let’s understand the relationship between distance, speed, and time. Distance, speed, and time are connected by the formula: \[ \text{Distance} = \text{Speed} \times \text{Time} \] This means if you know any two of these elements, you can find the third. In Bonita’s situation, she traveled 100 miles in x hours. To find her speed, we use the formula: \[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} = \frac{100 \text{ miles}}{x \text{ hours}} \] This expresses her speed as a fraction, \(\frac{100}{x}\) miles per hour. By understanding this formula, you can solve many problems related to distance, speed, and time.

Multiplication of Fractions

Next, let's look into multiplying fractions. This is critical as we need to multiply Bonita's speed by the new time period to find the distance she traveled. Multiplication of fractions follows the rule: \[ \frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d} \] Using Bonita’s speed, \(\frac{100}{x}\), we multiply it by \(\frac{3}{4}\) hours: \[ \frac{100}{x} \times \frac{3}{4} = \frac{100 \times 3}{x \times 4} = \frac{300}{4x} \] Simplifying the fraction, we get: \[ \frac{300}{4x} = \frac{75}{x} \] Thus, the distance traveled in the next \(\frac{3}{4}\) of an hour is \(\frac{75}{x}\) miles. Understanding how to multiply fractions properly is essential to solving this part of the problem.

Algebraic Manipulation

Finally, algebraic manipulation is a key skill in handling rational expressions. Let's break down the steps we used: 1. First, we found the speed as a rational expression \(\frac{100}{x}\). This is called forming a rational expression, where both the numerator (100 miles) and the denominator (x hours) are polynomials. 2. Next, we multiplied this rational expression by another fraction, \(\frac{3}{4}\), to find the distance for a new time period. Remember: When multiplying fractions, multiply the numerators together and the denominators together. 3. Simplifying the resulting fraction \(\frac{300}{4x}\) to \(\frac{75}{x}\) involved dividing both the numerator and the denominator by a common factor. This is an essential algebraic skill to simplify complex fractions. Mastering these skills allows you to manipulate expressions and solve complex problems efficiently.