Problem 46 - Algebra 1

Question

The charge for a cab ride is S11.50, and you give a 20\% tip. Using the model, find the total cost of the cab ride. Describe what the variable a represents. Model \(1: a\) is \(20 \%\) of \(\$ 11.50\)

Step-by-Step Solution

Answer

The total cost of the taxi ride is $13.80. The variable a represents the tip given, which is 20% of the fare.

Step 1: Understand the variable a

According to the provided model, the variable a represents 20% of the taxi fare. This translates into the task of finding out 20% of $11.50.

Step 2: Calculate 20% of $11.50

To calculate 20% of $11.50, multiply 11.50 by 0.20. \(a = 11.50 * 0.20\)

Step 3: Calculate the total cost

To find the total cost of the taxi ride, add the calculated tip (a) to the original cost of the taxi ride ($11.50). \(total cost = 11.50 + a\)

Key Concepts

Taxi Fare Calculation

A taxi fare calculation starts with understanding the base cost of using the taxi service, which is essentially the charge that everyone has to pay to cover the service the taxi company provides.
For our exercise, this fare is given as $11.50. This base cost does not include any additional charges such as tips or taxes.
To calculate the total fare, we start by adding any additional costs to this base fare. In taxi fare discussions, additional costs might include:

• Tips: A customary extra amount given to the driver to show appreciation for the service.
• Taxes: Some regions include sales tax or other applicable taxes in the fare.

In our exercise, we have a tip, which requires us to calculate an additional percentage on the base cost, making our calculations slightly more complex than just dealing with a static fare.

Total Cost Calculation

Once you have your base taxi fare and the additional costs, such as tips, you can easily compute the total cost of the taxi ride.
To find this, you simply add the calculated additional expenses (like tips) to the base fare.
The formula for total cost here is:

• If base fare is $11.50 and the tip is 20%, first find the tip amount by multiplying $11.50 by 0.20.
• Then add this tip to the base fare; thus, if the tip is found to be $2.30, the total cost becomes $11.50 + $2.30.
• The end result in our exercise is a total cost of $13.80.

This process is a universal method for finding any total cost involving a percentage increase, ensuring you know exactly what you will pay after considering all charges.

Mathematical Modeling

Mathematical modeling helps us represent real-world scenarios with equations, making complex calculations more streamlined and understandable.
In our exercise, we use a mathematical model to represent the taxi fare and tip problem. The model simplifies the calculation through a series of steps:

• Firstly, identify the variable (a), which here represents 20% of the taxi fare—a portion calculated by multiplying the base fare by 0.20.
• The second step involves creating an equation that adds this calculated portion (the tip) to the base fare to find the total cost.
• This approach makes adjusting to different variables, like a change in fare or tip percentage, simple and quick.

By formalizing the process through mathematical models, you not only make everyday problems like calculating taxi fares easier to solve but also lay the groundwork for tackling much larger and complex mathematical problems.