To determine whether the total profit is proportional to the number of t-shirts sold, let's analyze the current scenario.
Current Scenario:
- Selling price per t-shirt: $12
- Cost for supplies: $40
- Let \( n \) be the number of t-shirts sold.
The revenue from selling \( n \) t-shirts is: \[ \text{Revenue} = 12n \]
The total cost for supplies is a fixed amount of $40, regardless of how many t-shirts are sold. Therefore, the profit \( P \) is given by: \[ P = \text{Revenue} - \text{Cost} = 12n - 40 \]
This profit equation shows that the profit is not proportional to the number of t-shirts sold. This is because of the fixed cost of $40, which creates a constant term in the profit equation.
To make this scenario proportional, you need to eliminate the fixed cost. One way to do this is to ensure that there are no fixed costs in the calculation. For example:
Assume we have no upfront costs. So, you can change the scenario as follows:
- Suppose you decide to buy supplies in bulk for each t-shirt as needed. Let’s assume the cost for each t-shirt is $8 (which is the variable cost).
- Now, you sell each t-shirt for $12 with no upfront cost.
In this adjusted scenario, the profit \( P \) when selling \( n \) t-shirts would be:
\[ \text{Revenue} = 12n \] \[ \text{Cost} = 8n \] \[ P = \text{Revenue} - \text{Cost} = 12n - 8n = 4n \]
In this case, the profit is clearly proportional to the number of t-shirts sold because it can be expressed as \( P = 4n \), where 4 is the constant of proportionality. Thus, the profit increases linearly with the number of t-shirts sold.
In summary, by changing the fixed cost to a variable cost per item, we create a proportional scenario where total profit is directly related to the quantity sold.