Problem 66 - Algebra 1

Question

You start a daily flower club and charge \(\$ 10\) to join and \(\$ .50\) per day. Every day each member of the club gets a fresh flower. Let \(n\) represent the number of club members and let \(I\) represent your income for four weeks. A model for the situation is \(I=[10+4 \cdot 7(0.5)] n .\) Write an input-output table that shows your income for \(2,4,6,8,\) and 10 club members.

Step-by-Step Solution

Answer

The income for 2, 4, 6, 8, and 10 members would be 48, 96, 144, 192, and 240 respectively

Step 1: Understand and simplify the income model

The model of the given situation is \(I = [10 + 4 \cdot 7(0.5)]n\). This formula can be simplified as \(I = [10 + 14]n\), which is \(I = 24n\). So, for any given number of \(n\) members, the income \(I\) would be \(24n\).

Step 2: From the simplified model, compute the income for 2 members

To find the income for 2 members, substitute \(n=2\) into the simplified model equation: \(I = 24(2) = 48\)

Step 3: Compute the income for 4 members

To find the income for 4 members, substitute \(n=4\) into the equation: \(I = 24(4) = 96\)

Step 4: Compute the income for 6 members

To find the income for 6 members, substitute \(n=6\) into the equation: \(I = 24(6) = 144\)

Step 5: Compute the income for 8 members

To find the income for 8 members, substitute \(n=8\) into the equation: \(I = 24(8) = 192\)

Step 6: Compute the income for 10 members

To find the income for 10 members, substitute \(n=10\) into the equation: \(I = 24(10) = 240\)

Key Concepts

Income Calculation

Understanding income calculation is vital in deriving meaningful insights into your business's financial status. In our flower club example, we need to calculate the total income based on several club members. This involves looking at two components: the initial membership fee and daily fees over four weeks. To break this down:

• Each member pays a fixed fee of \(10 to join the club.
• The club incurs a \)0.50 daily charge per member for fresh flowers, calculated over four weeks. Remember, one week has seven days, so that's 28 days in total.

Thus, the member pays $14 in total for daily services over four weeks. Therefore, the total income from each member is: \[ I = (10 + 14)n = 24n \] where \( n \) is the number of members. By understanding these crucial ingredients of income calculation, you can predict financial outcomes effectively.

Input-Output Table

An input-output table is a simple and powerful tool used to represent relationships between inputs and outputs systematically. In our scenario, it helps us visualize how changes in the number of club members affect total income. By creating an input-output table, you assign different values to the input (the number of members \( n \)) and calculate the resulting output (income \( I \)). Here’s how this looks with the table filled in for our example:

• For \( n = 2 \), the income \( I = 24 \times 2 = 48 \)
• For \( n = 4 \), the income \( I = 24 \times 4 = 96 \)
• For \( n = 6 \), the income \( I = 24 \times 6 = 144 \)
• For \( n = 8 \), the income \( I = 24 \times 8 = 192 \)
• For \( n = 10 \), the income \( I = 24 \times 10 = 240 \)

Using this table simplifies the process of checking various scenarios and makes it easier to spot trends and patterns in data distribution.

Linear Equation

Linear equations form the backbone of many algebraic processes and are instrumental in modeling real-world scenarios. In essence, a linear equation expresses a relationship where the change in one variable is proportional to a change in another.For our flower club problem, the linear equation is \( I = 24n \), where \( I \) is the income and \( n \) is the number of members. This equation is linear because each additional club member contributes a constant \(24 to the total income, keeping the relationship direct and proportional. The characteristic features of linear equations include:

• They graph as a straight line on a coordinate plane.
• The slope of the line represents the rate of change — here, the \)24 per member.
• The intercept is the starting point on the axis, which in this case, would be at zero, meaning there's no income with zero members.

This makes a linear equation straightforward to understand and apply, especially when predicting outcomes from variable inputs. Embracing these concepts provides clarity in algebraic predictions.