Problem 114 - College Algebra

Question

At noon, a barista notices that she has \(\$ 20\) in her tip jar. If she makes an average of \(\$ 0.50\) from each customer, how much will she have in her tip jar if she serves \(n\) more customers during her shift.

Step-by-Step Solution

Answer

The tip jar will have \( 20 + 0.50n \) dollars after serving \( n \) more customers.

Step 1: Understand the Problem

The problem provides the initial amount in the tip jar, which is $20. The barista makes $0.50 per additional customer served. We need to find out how much money will be in the tip jar if she serves 'n' more customers.

Step 2: Setup the Equation

Let \( T \) represent the total amount of money in the tip jar after serving \( n \) more customers. We start with \(20 and add \)0.50 for each of the \( n \) customers. The equation can be set up as: \[ T = 20 + 0.50n \]

Step 3: Solve the Equation

To find the total amount in the tip jar, substitute the value of \( n \) into the equation. This general formula \( T = 20 + 0.50n \) will give the accumulated total tip amount for any given \( n \).

Key Concepts

Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating these symbols. It allows us to express real-world situations in the form of equations or expressions. The primary goal of algebra is to find the unknown values that satisfy the equations.

In the context of our problem, algebra helps us create an equation to find out how much money the barista will have after serving additional customers. We use numbers and variables to set up a formula that describes the situation. This is powerful because it allows us to quickly calculate outcomes for different scenarios by simply changing the variable values. For example, if we know the initial amount in a jar and the average tip per customer, we can use algebra to predict the total tips based on the number of customers served.

Understanding algebraic equations like the one in our exercise—\( T = 20 + 0.50n \)—is crucial because it sets the foundation for more complex problem-solving skills. Algebra can represent a wide variety of situations, demonstrating its versatility in mathematics and real-world applications.

Variables in Equations

Variables are like placeholders in mathematics. They represent unknown quantities that can change or vary. In equations, we use variables to generalize problems so we can solve them for different conditions.

In our exercise, the variable used is \( n \), which stands for "the number of additional customers" the barista serves. Using \( n \), we can write the equation as \( T = 20 + 0.50n \). In this expression, \( T \) stands for the total amount in the tip jar. We can solve for \( T \) once \( n \) is known, which shows how variables allow us to make computations without specific numbers until needed.

The use of variables makes equations flexible and powerful tools in problem solving. They enable us to change one aspect (like the number of customers) while keeping the relationship between all parts of the equation intact. Learning how to work with variables is a key component in mastering algebra and developing mathematical literacy.

Problem Solving with Equations

When it comes to problem-solving with equations, the main task is to translate a real-world situation into a mathematical statement. This usually involves setting up an equation using known values and variables to find unknown quantities.

In our example, the problem-solving process began by recognizing important details: an initial tip amount and a per-customer tip. We translated these details into the equation \( T = 20 + 0.50n \). This shows how each element of the scenario (initial amount, additional earnings per customer, and the variable number of customers) comes together to form an equation.

The power of equations lies in their ability to give us a simple method for solving complex problems. By substituting different values of \( n \), we can quickly find how much money the barista will have for any number of additional customers served. This technique allows us to efficiently and accurately compute outcomes in a structured manner.Using equations involves careful reading of problems, identifying the necessary numerical relationships, and applying basic algebraic principles. Learning this process is essential for those looking to develop strong analytical and mathematical skills.