Problem 30 - Algebra Form and Function

Question

One store sells 70 pounds of apples a week, and a second store sells 50 pounds of apples a week. Is the total number of pounds of apples sold, \(a\), proportional to the number of weeks, \(w\) ? If so, what is the constant of proportionality?

Step-by-Step Solution

Answer

If so, what is the constant of proportionality? Answer: Yes, the total number of pounds of apples sold is proportional to the number of weeks. The constant of proportionality is 120.

Step 1: Determine the Total Amount of Apples Sold per Week by Both Stores

Since the first store sells 70 pounds of apples each week and the second store sells 50 pounds of apples each week, we can add these two amounts to find the total amount of apples sold by both stores per week: \(70 + 50 = 120\) pounds per week.

Step 2: Determining if the Total Amount Sold is Proportional to the Number of Weeks

For a relationship to be proportional, it must take the form of a linear equation such as \(a = k\cdot w\), where \(a\) represents the total amount of apples sold, \(w\) represents the number of weeks, and \(k\) is the constant of proportionality. In this case, since we know the total amount of apples sold per week is 120 pounds, we can write the relationship as \(a = 120\cdot w\). This is a linear equation, so the total amount of apples sold is proportional to the number of weeks.

Step 3: Calculate the Constant of Proportionality

Now that we know the relationship between the total amount of apples sold and the number of weeks is proportional, we can find the constant of proportionality \(k\). From the previous step, we have the linear equation \(a = 120\cdot w\). Dividing both sides of the equation by \(w\), we get the constant of proportionality: \(k = \dfrac{a}{w} = 120\). Thus, the constant of proportionality is 120. This means that, for every week, the total amount of apples sold (\(a\)) by both stores is 120 times the number of weeks (\(w\)).

Key Concepts

Linear Equations

A linear equation is a foundational concept in mathematics. It represents a straight-line relationship between two variables. In the context of this problem, the linear equation is used to describe how the total pounds of apples sold by two stores are related to the number of weeks.
Here, a linear equation takes the form of \(a = k \cdot w\), where:

• \(a\) represents the total amount of apples sold,
• \(k\) is the constant of proportionality, and
• \(w\) denotes the number of weeks.

By understanding this structure, we can easily see how changes in the number of weeks affect the total apple sales. For instance, if the stores sell 120 pounds of apples each week, then over 2 weeks, they would sell \(a = 120 \cdot 2 = 240\) pounds. This linearity helps predict future sales based on the provided time frame.

Constant of Proportionality

The constant of proportionality is a crucial part of understanding proportional relationships. It serves as the multiplier that links the dependent variable to the independent variable. In our apple-selling scenario, this constant is 120 pounds per week.
This value, 120, implies that for every additional week, the stores sell an extra 120 pounds of apples.
Understanding the constant of proportionality allows one to predict outcomes quickly. For instance, if you were asked how many apples would be sold in 5 weeks, knowing the constant allows for an immediate calculation: \(a = 120 \cdot 5 = 600\) pounds.
In essence, the constant of proportionality maintains a consistent rate that defines the relationship between variables, letting us comprehend how one variable changes in response to the other.

Mathematical Problem Solving

Mathematical problem solving often involves recognizing whether relationships between variables are proportional. This task ensures that solutions can be efficiently applied or predicted for various scenarios.
The first step in problem solving is identifying the relevant information and determining the form of the relationship between variables, which in this case is linear and proportional. Such identification allows setting up equations like \(a = k \cdot w\).
Another step involves calculating solutions diligently. Here, by calculating the weekly total apples sold and embedding that in the equation, you readily determine proportions.
Overall, mathematical problem solving is about recognizing patterns, setting up equations, and calculating the correct outcome by applying constants and formulating logical steps to derive answers that satisfy the given conditions.