Problem 85 - 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 smallest profit?

Step-by-Step Solution

Answer

Tacos had the smallest profit.

Step 1: Calculate Profits for each food type for Day 1

For Day 1, subtract the expenses from the income for each food type: Hamburgers: \(72 - 14 = 58\), Hot dogs: \(85 - 18 = 67\), Tacos: \(46 - 19 =27\)

Step 2: Calculate Profits for each food type for Day 2

For Day 2, subtract the expenses from the income for each food type: Hamburgers: \(62 - 10 = 52\), Hot dogs: \(52 - 11 = 41\), Tacos: \(72 -26 = 46\)

Step 3: Calculate total profits

Add the profits of Day1 and Day2 for each food type: Hamburgers: \(58 + 52 = 110\), Hotdogs: \(67 + 41 = 108\), Tacos: \(27 + 46 = 73\)

Step 4: Identify the food with the smallest profit

Compare the total profits for all food items. The smallest profit is made by tacos, with a total of $73.

Key Concepts

Matrices in Algebra

Matrices are incredibly powerful tools in algebra that simplify the representation and manipulation of data. They are collections of numbers, symbols, or expressions arranged in rows and columns, forming a rectangular array. Matrices enable us to perform multiple calculations at once, often used to solve systems of linear equations, to represent and work with transformations in geometry, and, as in our snack stand example, to organize and compute financial information efficiently.

Each position in a matrix is called an 'element,' and the size of a matrix is given by the number of rows and columns it contains, typically noted as 'm x n' where 'm' is the number of rows and 'n' the number of columns. In the snack stand scenario, matrix notation aids in visualizing the income and expenses per food item over two days in a concise way, which then allows for quicker analysis when calculating profits.

Profit Calculation

Profit calculation is a fundamental business concept that involves determining the financial gain after all expenses have been subtracted from the income. It is a clear indicator of financial performance. To compute profit for each item, we simply calculate the difference between the income and expenses that each food item generates. In algebra, this calculation can also be represented and simplified using matrix operations.

Understanding how to calculate profit is crucial, not just in a business context, but as a skill in algebra problems where financial literacy is applied to mathematical concepts. The snack stand's income and expenses can be swiftly calculated using matrix operations to determine the profitability of hamburgers, hot dogs, and tacos.

Subtraction of Matrices

Subtraction of matrices is another matrix operation commonly used in algebra. This operation can only be performed on two matrices of the same size, and it involves subtracting the corresponding elements from each matrix. For the snack stand example, we subtracted the expense matrices from the income matrices for each food item to calculate the profits.

When we say 'corresponding elements,' we mean the elements that occupy the same row and column position in each matrix. For instance, if we have two 2x2 matrices, A and B, the subtraction C=A-B would result in a third 2x2 matrix, C, where each element, 'c_ij', is the result of 'a_ij' minus 'b_ij' (e.g., c₁₁ = a₁₁ - b₁₁). This operation is directly applicable to calculating daily profits by comparing income and expenses across the same types of food and days.