Question

You want to sell t-shirts for $12 a piece. It cost you $40 for supplies. Is the total profit proportional to the number of t-shirts you sell? Change this scenario in to be a proportional scenario.

Answer 1

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:

1. 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).
2. 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.