Question

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?

Answer 1

To determine whether the total profit is proportional to the number of t-shirts you sell, let's define the variables involved.

1. **Let \( n \)** be the number of t-shirts sold.
2. **Selling price per t-shirt:** $12
3. **Cost for supplies:** $40 (fixed cost)

**Revenue from selling \( n \) t-shirts:** This is calculated as the selling price multiplied by the number of t-shirts sold.

\[
\text{Revenue} = 12n
\]

**Total profit:** This is calculated by subtracting the total costs from the revenue. Since the fixed cost for supplies is $40, the total costs when selling \( n \) t-shirts is still $40 (ignoring variable costs).

\[
\text{Total Profit} = \text{Revenue} - \text{Total Costs}
\]
\[
\text{Total Profit} = 12n - 40
\]

Now let's analyze the profit formula:

\[
\text{Total Profit} = 12n - 40
\]

This equation can be rearranged to:

\[
\text{Total Profit} = 12n - 40 = 12(n - \frac{40}{12})
\]

The expression \( 12(n - \frac{40}{12}) \) shows that the total profit increases with \( n \), but it also has a fixed offset of -40.

In relation to whether it is proportional to \( n \), we can make the following observation:

- **Proportional relationship:** A situation is proportional if it can be expressed in the form \( y = kx \), where \( k \) is a constant and \( x \) is the independent variable. Here, the profit equation is not simply in the form of \( k \cdot n \) because of the -40 term (the fixed costs).

Thus, since the profit does not directly scale linearly with \( n \) alone (due to the -40 fixed cost term), we conclude that **the total profit is NOT proportional to the number of t-shirts sold**. The profit only becomes positive once you've sold enough t-shirts to cover the initial cost. Specifically, you need to sell at least \( \frac{40}{12} \approx 3.33 \) t-shirts, or at least 4, to start making a profit.