Problem 84 - Algebra 1

Question

Use the following information. You open a snack stand at a fair. The income and expenses (in dollars) for selling each type of food are shown in the matrices. \(\begin{array}{ccccccc}\text { Day 1 } & \text { Income } & \text { Expenses } & \text { Day 2 } & \text { Income } & \text { Expenses } \\ \text { Hamburgers } & 72 & 14 & \text { Hamburgers } & 62 & 10 \\ \text { Hot dogs } & 85 & 18 & \text { Hot dogs } & 52 & 11 \\ \text { Tacos } & 46 & 19 & \text { Tacos } & 72 & 26\end{array}\) Which type of food had the largest profit?

Step-by-Step Solution

Answer

The food item with the largest profit is Hamburgers with a profit of 110 dollars.

Step 1: Compute Profit for Each Food on Day 1

The profit for each food can be calculated by subtracting its expenses from its income. For Hamburgers, profit on Day 1 is \(72 - 14 = 58\) dollars. For Hot dogs, the profit on Day 1 is \(85 - 18 = 67\) dollars. For Tacos, the profit amount on Day 1 is \(46 - 19 = 27\) dollars.

Step 2: Compute Profit for Each Food on Day 2

Similar to Day 1, the profit for each food item can be calculated. For Hamburgers, the profit on Day 2 is \(62 - 10 = 52\) dollars. For Hot dogs, the profit on Day 2 is \(52 - 11 = 41\) dollars. For Tacos, the profit on Day 2 is \(72 - 26 = 46\) dollars.

Step 3: Calculate Total Two-Day Profit for Each Food

The total profit for each food over two days is found by adding up the profits from Day 1 and Day 2. For Hamburgers, the total profit is \(58 + 52 = 110\) dollars. For Hot dogs, the total profit is \(67 + 41 = 108\) dollars. For Tacos, the total profit is \(27 + 46 = 73\) dollars.

Step 4: Identify the Food with the Highest Total Profit

Since Hamburgers' total profit (110 dollars) is greater than both Hot dogs' and Tacos' total profits, they are the most profitable food item.

Key Concepts

Profit Computation

Profit computation is the process of determining the financial gain made from business transactions after all expenses are deducted from income. In a snack stand scenario like the one presented in the exercise, profit is calculated separately for each product. To calculate the profit for a particular food item, you subtract the total expenses from the total income generated by selling that food item.

For example, the profit calculation for hamburgers on Day 1 is illustrated through the equation: \(72 - 14 = 58\) dollars. Here, \(72\) represents the income from hamburgers, and \(14\) denotes the cost associated with selling them. By subtracting the expenses from the income, you get the net profit. This is an essential calculation in business as it helps determine the viability of products and can influence future business decisions such as pricing, promotions, and inventory management.

Matrix Operations

Matrix operations offer a structured way to handle multiple data points, which is particularly useful in business scenarios where transactions involve several variables. In the case of the snack stand, the use of matrices can streamline the process of organizing and calculating profits. A matrix is composed of rows and columns that hold numerical values, and basic operations include addition, subtraction, multiplication, and division.

In the exercise, the matrices are used to list incomes and expenses side by side for easy comparison and calculation. You would perform element-wise subtraction (expenses from income) to find the profit for each item on each day, which simplifies to performing a series of arithmetic operations. While the step-by-step solution separates these calculations, matrix operations could allow you to organize and carry out these calculations in a single step if extended to matrix form.

Algebraic Application in Business

Algebra is not just a subject for classroom study; its principles are regularly applied in the business world. Algebraic application in business involves the use of variables and equations to model situations and solve problems related to finance, economics, and operational efficiency.

In our snack stand profit analysis, algebraic equations are used to represent the relationship between income, expenses, and profit. These equations help to understand and communicate the financial performance of each product. Algebra helps simplify complex problems, so business owners can make informed decisions about managing resources to maximize profits. In modern business practices, these calculations are often handled by software, but understanding the underlying principles of algebra is critical in ensuring accurate financial planning and analysis.