Problem 63 - Prealgebra

Question

Recently, the Texas Junior College Teachers Association annual conference was held in Austin. At that time a taxi ride in Austin was \(\$ 1.25\) for the first \(\frac{1}{5}\) of a mile and \(\$ 0.25\) for each additional \(\frac{1}{5}\) of a mile. The charge for a taxi to wait is \(\$ 12.00\) per hour. Use this information for Problems 63 through 66. If the distance from one of the convention hotels to the airport is 7.5 miles, how much will it cost to take a taxi from that hotel to the airport?

Step-by-Step Solution

Answer

The cost to take the taxi from the hotel to the airport is \( \$ 10.38 \).

Step 1: Determine Initial Fare

The initial fare for the taxi includes the first \( \frac{1}{5} \) of a mile. According to the information, this initial section costs \( \$ 1.25 \).

Step 2: Calculate Distance Beyond Initial Fare

The total distance to the airport is 7.5 miles. Since the initial fare covers \( \frac{1}{5} = 0.2 \) miles, we need to calculate the remaining distance: \( 7.5 - 0.2 = 7.3 \) miles.

Step 3: Determine Number of Additional Increments

Each additional \( \frac{1}{5} \) mile beyond the initial 0.2 miles costs \( \$ 0.25 \). To find the number of additional \( \frac{1}{5} \) mile increments in 7.3 miles, divide by 0.2: \( \frac{7.3}{0.2} = 36.5 \). This means there are 36.5 additional \( \frac{1}{5} \) mile increments.

Step 4: Calculate Cost of Additional Distance

The cost for each additional increment is \( \\( 0.25 \). Therefore, the cost for 36.5 increments is \( 36.5 \times 0.25 = \\) 9.125 \).

Step 5: Calculate Total Cost

Add the initial fare to the cost of the additional increments: \( \\( 1.25 + \\) 9.125 = \\( 10.375 \). The total cost is rounded to the nearest cent, \( \\) 10.38 \).

Key Concepts

Unit Rate

Understanding unit rate can help break down costs into manageable, per-measurement increments. In a taxi fare scenario, the unit rate is the cost per unit of distance. Here, for each additional \( \frac{1}{5} \) of a mile, the unit rate is \( \\(0.25 \). For the first \( \frac{1}{5} \) mile, it's \( \\)1.25 \). Knowing this, you can easily calculate costs over longer distances.

• Identify the unit you're charged for. In this case, \( \frac{1}{5} \) of a mile.
• Understand the cost associated with that unit. Initial \( \frac{1}{5} \) mile costs \( \\(1.25 \), subsequent ones cost \( \\)0.25 \) each.
• Utilize the unit rate to calculate the total cost for given increments. Multiply the unit rate by the number of units to find this.

Once you grasp unit rates, calculating costs for various distances becomes straightforward. You can apply this concept to other contexts like hourly wages or costs per item at a supermarket.

Linear Equation

A linear equation is essential in connecting different concepts, such as unit rates, to real-world scenarios. It's used to calculate total costs based on distance, among other applications. For the taxi problem, the fare model is a piecewise linear equation:

• Initial Charge: \( \\(1.25 \) for the first \( \frac{1}{5} \) of a mile.
• Additional Charge: \( \\)0.25 \) per additional \( \frac{1}{5} \) mile.

You build the equation from these parts to represent the total cost. Given the distance exceeds the first \( \frac{1}{5} \) of a mile, use the equation:\[ \text{Total cost} = 1.25 + 0.25(x) \]Here, \( x \) is the number of additional \( \frac{1}{5} \) mile increments. Solving this results in our total trip cost. Linear equations aren't just for taxi fares — they help solve countless problems involving constant rates of change.

Distance Calculations

When figuring out costs or travel time, knowing how to work with distances is key. In our taxi example, the goal is to determine the total fare based on how far you're traveling. Start by breaking down the journey and calculating expenses for each part.
Calculating distance involves:

• Understanding the total distance covered. Here, the trip is 7.5 miles.
• Deducting covered distances from the total mileage to find remaining figures. The first \( \frac{1}{5} \) mile is already included, leaving 7.3 miles.
• Converting this remaining distance into the number of billable units — increments of \( \frac{1}{5} \) mile, specifically.

When calculating, always verify if your measurements and units are consistent. Gain confidence in distance calculations by practicing with other variables like speed or time, which work similarly in providing vital information for everyday decisions.