Problem 63 - Understanding Elementary Algebra with Geometry

Question

Mary is working as a clerk in a bank, where she inspects documents. It takes her 12 minutes to check a loan application and 8 minutes to check a credit report. On a certain day she is scheduled to spend 360 minutes inspecting documents. (a) Let \(a\) represent the number of loan applications and \(c\) represent the number of credit reports. Write an equation that reflects the given situation. (b) Sketch the graph of this relationship. Be sure to label the coordinate axes clearly. (c) If Mary inspected 25 loan applications, use the equation you obtained in part (a) to find the number of credit reports she checked.

Step-by-Step Solution

Answer

The equation is \(12a + 8c = 360\). For 25 loan applications, Mary cannot inspect a feasible number of credit reports (7.5).

Step 1: Identify and Define Variables

Let the number of loan applications be represented by \(a\) and the number of credit reports be represented by \(c\). We are given that inspecting a loan application takes 12 minutes and inspecting a credit report takes 8 minutes.

Step 2: Formulate the Equation

Mary spends a total of 360 minutes inspecting documents. The equation is formed by summing the time spent on loan applications and credit reports: \(12a + 8c = 360\).

Step 3: Solve for Maximum Values

To define the graph, determine the maximum values of \(a\) and \(c\) by setting the other variable to 0. When \(a = 0\): \(8c = 360\) \(c = 45\) When \(c = 0\): \(12a = 360\) \(a = 30\)

Step 4: Graph the Equation

Sketch the graph on a coordinate plane with \(a\) on the x-axis and \(c\) on the y-axis. Plot the points (30, 0) and (0, 45). Draw a straight line connecting these two points, representing the equation \(12a + 8c = 360\).

Step 5: Substitute and Solve for Part (c)

If Mary inspected 25 loan applications (\(a = 25\)), substitute this value into the equation:\(12(25) + 8c = 360\)\(300 + 8c = 360\)Subtract 300 from both sides:\(8c = 60\)Divide by 8:\(c = 7.5\)Since the number of credit reports must be an integer, there seems to be an inconsistency, suggesting that inspecting 25 loan applications may not be feasible within the 360 minutes, based on this linear model.

Key Concepts

Variable Identification

In algebra, understanding how to identify variables is crucial for solving equations. Variables are symbols used to represent unknown values. In the problem, we need to identify the variables that represent different document types Mary inspects.
We see that Mary spends 12 minutes per loan application and 8 minutes per credit report. We can define:
- Let \(a\) be the number of loan applications.
- Let \(c\) be the number of credit reports.
By assigning these variables, we simplify the process of forming and solving the equation. Think of variables as placeholders that allow you to translate word problems into mathematical language.

Graphing Linear Equations

Graphing linear equations can help visualize the relationship between two variables. In our problem, after forming the equation \(12a + 8c = 360\), we can plot this on a graph.
First, find the intercepts where one variable is zero:
- When \(a = 0\): \(8c = 360\) so \(c = 45\)
- When \(c = 0\): \(12a = 360\) so \(a = 30\)
These points (0, 45) and (30, 0) are plotted on the coordinate plane. Connect these points with a straight line.
The x-axis (horizontal) represents the number of loan applications \(a\). The y-axis (vertical) represents the number of credit reports \(c\). By graphing, you can easily see which combinations of \(a\) and \(c\) fit within the given timeframe.

Solving Linear Equations

Solving linear equations involves isolating the variable to find its value. Let's say Mary inspected 25 loan applications. We substitute \(a = 25\) into our equation \(12a + 8c = 360\).
- Plug in the value: \(12(25) + 8c = 360\)
- Simplify: \(300 + 8c = 360\)
- Solve for \(c\):\(8c = 60\) then \(c = 7.5\)
Since \(c = 7.5\) is not an integer, it implies that 25 loan applications aren't feasible given the constraints.
This example highlights the importance of checking whether solutions make practical sense. If the number of reports or applications doesn't align with whole numbers, it suggests revisiting the problem setup or considering alternative solutions.