Problem 22 - Algebra 2

Question

Multiple Choice At a restaurant, Elena wants to leave a tip that is 15\(\%\) of the bill \(B\) plus \(\$ 1 .\) Which linear model best describes the total \(T ?\) $$ \begin{array}{ll}{\text { A } T=15 B+1} & {\text { B } T=B+15} \\ {\text { C } T=1.15 B} & {\text { D } T=1.15 B+1}\end{array} $$

Step-by-Step Solution

Answer

The best linear model that describes the total T Elena will pay is D) T = 1.15B + 1.

Step 1: Understand the concept of tipping

A tip is typically a percentage of the total bill amount. Elena wants to leave a 15% tip on the bill amount B. This means she will tip 0.15 times the bill amount, since 15% as a decimal is 0.15.

Step 2: Set up the expression for the tip

To calculate the tip amount, multiply the bill amount B by 0.15. This gives an expression for the tip as 0.15B.

Step 3: Add the additional dollar to the tip

Elena wants to add an additional $1 to the tip calculated in step 2. The total tip she will leave is therefore 0.15B + 1.

Step 4: Identify the total cost

The total T that Elena will pay is the sum of the original bill amount B and the total tip. Therefore, the total cost T is the bill B plus the tip 0.15B plus 1, which gives T = B + (0.15B + 1).

Step 5: Identify the correct linear model

The correct linear model must reflect the total cost, including the original bill and the tip with the added dollar. The expression from step 4 can be simplified as T = 1B + 0.15B + 1, which further simplifies to T = 1.15B + 1. Hence, the linear model that best describes the total T is T = 1.15B + 1.

Key Concepts

Percentages in Algebra

In algebra, percentages are used to describe a ratio as a fraction of 100. This allows us to easily compare proportions and make calculations on various quantities. Understanding percentages is fundamental in real-world scenarios like calculating discounts, interest rates, and tips in restaurants, as seen in the given exercise.

To convert a percentage to a decimal, which is necessary for algebraic operations, you simply divide by 100. For instance, a 15% tip becomes 0.15 when used in an equation because 15% is the same as 0.15 of the total bill. This conversion is crucial because it allows the percentage to be multiplied by the quantity in question within an algebraic expression.

In the context of our exercise, by understanding that the tip is 15% of the bill represented by the variable 'B,' we are able to formulate the tip as 0.15B. This conversion from percentage to a decimal factor is the first step in creating a linear model to represent a real-world financial scenario.

Writing Linear Equations

A linear equation is the mathematical representation of a straight line and is at the heart of many algebraic problems. When writing linear equations, we are often looking to model a relationship between two variables. The standard form for a linear equation in two variables is \(y = mx + b\), where \(m\) is the slope, \(x\) is the independent variable, and \(b\) is the y-intercept.

In the context of the tipping problem, we sought to find a linear equation that represented the total cost \(T\) based on the bill amount \(B\). The 'slope' in this context is the percentage converted to a decimal (the rate at which the tip increases per dollar on the bill), and the 'y-intercept' is the fixed additional amount — the extra $1 tip.

By recognizing that the equation needs to reflect both the original bill and the percentage-based tip, it's clear that the equation \(T = 1.15B + 1\) encapsulates the linear relationship between the bill and the total cost after tipping. This demonstrates the process of writing a linear equation based on given conditions and converting real-world details into algebraic form.

Solving Real-World Math Problems

Applying algebra to solve real-world problems involves translating a scenario into mathematical language. The first step is to understand the context of the problem, what is being asked, and what relationships exist between the quantities involved. This comprehension allows us to define variables and create an equation or a set of equations that model the situation.

In our example, we translated Elena's tipping scenario into an algebraic model. Recognizing the practical application of a percentage tip and a fixed dollar amount, we defined a linear equation to represent the total cost. The ability to create and work with such models is powerful, allowing us to predict, analyze, and make informed decisions based on a variety of factors. This particular problem also shows the importance of being able to simplify expressions to identify the most appropriate mathematical model from multiple choices.

Overall, the skill of solving these kinds of real-world math problems can be applied to various fields ranging from personal finance to engineering, making algebra a critical tool in many aspects of life.