Problem 66 - 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 that the distance from a hotel to the airport is 8.2 miles, and the ride takes 20 minutes. Is it more expensive to take a taxi to the airport or to just sit in the taxi?

Step-by-Step Solution

Answer

Taking the taxi ride is more expensive than just sitting in the taxi.

Step 1: Determine the Cost of the Taxi Ride to the Airport

First, calculate the cost for the first \( \frac{1}{5} \) of a mile, which is \\( 1.25. Then, find how many \( \frac{1}{5} \) mile segments are in 8.2 miles. Since \( \frac{8.2}{0.2} = 41 \), there are 41 segments. The cost for the remaining 40 segments (after the first) is \( 40 \times 0.25 = \\) 10.00 \). So, the total cost for the ride is \\( 1.25 (first segment) + \\) 10.00 (additional segments) = \$ 11.25.\

Step 2: Determine the Cost of Waiting in the Taxi

Calculate the cost for the taxi to wait for 20 minutes. Since the charge is \\( 12.00 per hour, convert 20 minutes to hours: \( \frac{20}{60} = \frac{1}{3} \) hours. Thus, the cost for waiting is \( \frac{1}{3} \times 12.00 = \\) 4.00. \)

Step 3: Compare Both Costs

Compare the cost of the taxi ride, which is \\( 11.25, with the cost of waiting, which is \\) 4.00. Since \\( 11.25 is greater than \\) 4.00, it is more expensive to take the taxi ride to the airport than to just sit in the taxi.

Key Concepts

Taxi Fare Calculation

Understanding how taxi fares are calculated is crucial. This involves knowing the basic cost structure set by the taxi service. In this case, the taxi fare in Austin for a short distance was \( \\(1.25 \) for the first \( \frac{1}{5} \) of a mile. This base fare covers the initial work of picking up a passenger. For each additional \( \frac{1}{5} \) of a mile, the cost is \( \\)0.25 \).

To calculate the total cost of a taxi ride over a certain distance, start by determining the number of segments the ride covers. Each segment is \( \frac{1}{5} \) of a mile, so for a distance like 8.2 miles, the process involves dividing 8.2 by 0.2, resulting in 41 segments.

• The first segment costs \( \\(1.25 \).
• Each subsequent segment costs \( \\)0.25 \), totaling \( 40 \times 0.25 = \\(10.00 \) for the remaining segments.

The overall fare then adds up as: base fare plus cost for additional segments, making it \( \\)11.25 \) for the full distance.

Distance and Rate Problems

Distance and rate problems are a key part of prealgebra, involving calculations to determine time or costs based on distance traveled and rates of speed or cost per unit distance. For taxi rides, handling these calculations involves a few straightforward steps.

Firstly, determine how far and at what rate the taxi is traveling. In the scenario of traveling 8.2 miles with a rate of \( \$0.25 \) per \( \frac{1}{5} \) mile beyond the first segment, calculating the full cost requires understanding how distance and rate impact the fare.

• The initial portion of the journey is readily calculated with a fixed cost.
• Subsequent parts involve multiplying the additional segments by their rate, neatly illustrating the connection between distance, rate, and total cost.

This approach fosters an easier grasp of word problems covering distance and rate, where understanding the multiplication of segments with fixed rates is critical.

Cost Comparison in Math

Cost comparison involves reviewing different expenses to figure out which option is more financially sensible. In this taxi fare problem, compare the direct cost of a taxi ride to the airport with the alternative cost of simply waiting in the taxi without traveling.

First, calculate each cost separately:

• The taxi ride to the airport costs \( \\(11.25 \).
• Waiting for 20 minutes incurs a \( \\)4.00 \) cost based on a rate of \( \$12.00 \) per hour or \( \frac{1}{3} \) of that rate.

Comparing these values illustrates that it is significantly cheaper to wait in the taxi. Such comparisons can help with broader financial decisions, showcasing the importance of understanding different calculations and factors such as time, distance, and rates.