Problem 65 - 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. Suppose the distance from one of the hotels to one of the western dance clubs in Austin is 12.4 miles. If the fare meter in the taxi gives the charge for that trip as \(\$ 16.50,\) is the meter working correctly?

Step-by-Step Solution

Answer

Yes, the meter is working correctly as the calculated fare matches \$16.50.

Step 1: Calculate the Fare for the First 1/5 Mile

The fare for the first \(\frac{1}{5}\) of a mile is \\(1.25. Since this is a fixed initial charge, the cost of the first \(\frac{1}{5}\) mile is \\)1.25.

Step 2: Determine Additional Miles After First 1/5 Mile

The total distance is 12.4 miles. After the first \(\frac{1}{5}\) mile, the remaining distance to cover is \(12.4 - 0.2 = 12.2\) miles.

Step 3: Calculate the Number of Additional 1/5 Miles

Each additional distance segment is \(\frac{1}{5}\) of a mile. Therefore, there are \(\frac{12.2}{0.2} = 61\) additional segments of \(\frac{1}{5}\) of a mile after the first segment.

Step 4: Calculate the Cost of Additional 1/5 Miles

The cost for each additional \(\frac{1}{5}\) mile is \\(0.25. Therefore, the cost for these 61 additional segments is \(61 \times 0.25 = \\)15.25\).

Step 5: Calculate Total Fare

Add the initial cost to the cost of the additional miles: \(1.25 + 15.25 = \$16.50\). This is the total fare for the trip.

Key Concepts

Distance and rate calculations

Understanding distance and rate calculations is crucial when it comes to solving taxi fare problems. Essentially, we need to figure out how distances and costs work together. Let's start by looking at a short cab ride. If the cab fare starts with a fixed charge for the first fraction of a mile, that's the base rate. Knowing this base rate is essential because it sets the stage for calculating the additional miles that come after that starting distance.

In this case, the fare for the first \(\frac{1}{5}\) of a mile is \(\\(1.25\). This is your starting point. Any distance traveled beyond this initial segment is calculated at a different rate. In our exercise, this is \(\\)0.25\) for each additional \(\frac{1}{5}\) mile.

Learning to calculate the remaining miles correctly and knowing the rate for those additional miles helps you know if the taxi meter is calculating fares correctly. It's all about knowing rates and distances and applying them correctly.

Cost analysis in word problems

Cost analysis in word problems requires carefully interpreting the details given in a question to calculate expenses accurately. In taxi fare problems, one common pitfall is misunderstanding how costs add up over distances. Your job is to break down each segment of the trip and apply the correct costs.

For taxi rides, the initial charge often covers a small distance, after which a different charge applies. In our taxi fare scenario, once the first \(\frac{1}{5}\) mile is covered, we need to calculate the cost for the remaining distance. The problem gives us all we need to handle this: the first mile segment cost of \(\\(1.25\) and the \(\\)0.25\) for any extra \(\frac{1}{5}\) mile.

We identified from the math that after subtracting the initial \(\frac{1}{5}\) mile, there were 61 segments remaining, leading to an additional cost of \(61 \times \\(0.25 = \\)15.25\). Adding this to the initial \(\$1.25\) gives the correct total taxi fare. Analyzing the cost in such parts helps students better understand where each dollar comes from.

Step-by-step problem solving

Step-by-step problem solving in mathematics is like having a roadmap for a journey. Each step guides you progressively until you reach the final solution. Understanding each step ensures that you do not get lost along the way.

When tackling a problem, start by identifying what you are given and what you need to find out. Here, our problem was to verify if the taxi fare matches a certain distance rate. We did this by following each logical step:

• Calculate the cost of the initial \(\frac{1}{5}\) mile.
• Determine how much distance is left after the initial charge.
• Break down the remaining distance into its cost-effective segments.
• Solve for the total, ensuring your figures are consistent with the given problem.

Each step builds on the previous one, making sure that each calculation is grounded in fact. Being linear in your approach allows you to pinpoint errors and correct them early. It's about being systematic and ensuring each phase of the problem contributes to the solution.