Question

A store buys 15 sweaters for $90 and sells them for $405. If p represents the total profit in dollars and cents when the store buys any number of sweaters, s, write a proportional equation for p in terms of s that matches the context.

Answer 1

To create a proportional equation for \( p \) (the total profit in dollars and cents) in terms of \( s \) (the number of sweaters), we first need to determine both the cost and the selling price per sweater.

1. **Calculate the cost per sweater**:
\[
\text{Cost per sweater} = \frac{\text{Total cost}}{\text{Number of sweaters}} = \frac{90}{15} = 6 \text{ dollars}
\]

2. **Calculate the selling price per sweater**:
\[
\text{Selling price per sweater} = \frac{\text{Total selling price}}{\text{Number of sweaters}} = \frac{405}{15} = 27 \text{ dollars}
\]

3. **Determine the profit per sweater**:
\[
\text{Profit per sweater} = \text{Selling price} - \text{Cost price} = 27 - 6 = 21 \text{ dollars}
\]

Now, we can express the total profit \( p \) in terms of the number of sweaters bought \( s \):

\[
p = (\text{Profit per sweater}) \times s = 21s
\]

Thus, the proportional equation for \( p \) in terms of \( s \) is:

\[
p = 21s
\]

This equation indicates that for every sweater bought, the store earns a profit of $21, and \( p \) will represent the total profit from selling \( s \) sweaters.