Problem 64 - Algebra 1

Question

Your aunt lends you \(\$ 175\) to buy a guitar. She will decrease the amount you owe by \(\$ 25\) for each day you help her by doing odd jobs. Write a verbal model that you can use to find the decrease in the amount you owe your aunt depending on the number of days you help her out.

Step-by-Step Solution

Answer

The verbal model is: 'The amount owed is equal to $175 minus the product of the number of days worked and $25.'

Step 1: Define the Variables

Let's denote the total amount owed by the letter 'A' and the number of days the student works with the letter 'D'.

Step 2: Establish the initial condition

Initially, the student owes $175 to the aunt, so we can write this as \( A = 175 \) when \( D = 0 \).

Step 3: Establish the rate of decrease

Next, it is stated that for each day the student works, the amount owed decreases by $25. In other words, every increase in 'D' by 1 results in a decrease of 'A' by $25.

Step 4: Write down the verbal model

The verbal model can thus be expressed as follows: The amount owed (A) is equal to $175 minus the product of the number of days worked (D) and 25.

Key Concepts

Variables

In every algebraic expression, variables play a crucial role. They act as placeholders or symbols that represent quantities which can change. In this exercise, we have two main variables: the amount owed to your aunt, represented by 'A', and the number of days you work for her, represented by 'D'. This dynamic lets us express mathematical relationships.
Using variables makes it easier to formulate and solve equations, as they provide a way to generalize problems and manipulate expressions. So, when you see a variable like 'D', think about all the possible different values it could take and how it directly affects the solution.

Initial Condition

An initial condition in a problem helps us set the stage for how everything starts. It's like the starting point of our math journey. In the exercise, you initially owe your aunt $175. Mathematically, we express this as:

• At the beginning (when you haven’t worked any days, meaning \( D = 0 \) ), you owe: \[ A = 175 \]

This initial setup allows us to see exactly where you start before any changes (like working days) come into play. It is crucial because it ties all other components of the expression together and impacts how we calculate changes over time.

Rate of Change

The rate of change describes how a quantity alters with respect to another. Here, it's all about how the amount you owe changes with each day you work. Specifically, every day you help your aunt is worth a $25 reduction in what you owe her. Mathematically, when 'D' increases by 1, 'A' decreases by 25, giving us the equation:

• The rate of change is established as: \\[ A = 175 - 25D \]

Understanding the rate of change helps us predict future outcomes, such as how much longer you need to work to pay off your debt entirely. It creates a straightforward method to calculate ongoing adjustments between two related quantities.