Question

The Baynard High School Boosters are selling hotdogs and pretzels at the concession stand. They have 300 hot dogs and pretzels in stock and want to earn an income of $800. They plan to sell the hotdogs for $4 each and the pretzels for $2 each. How many hotdogs and how many pretzels do the boosters need to sell reach their goal of $800?

Answer 1

Let \( x \) be the number of hot dogs sold and \( y \) be the number of pretzels sold.

We have the following two equations based on the problem description:

1. **Total items sold**:
\[
x + y = 300
\]

2. **Total income generated**:
\[
4x + 2y = 800
\]

Now, we can solve this system of equations.

First, we can simplify the second equation by dividing everything by 2:
\[
2x + y = 400
\]

Now we have the simplified system:
1. \( x + y = 300 \)
2. \( 2x + y = 400 \)

Next, we can subtract the first equation from the second:
\[
(2x + y) - (x + y) = 400 - 300
\]
\[
2x + y - x - y = 100
\]
\[
x = 100
\]

Now that we found \( x \), we can substitute it back into the first equation to find \( y \):
\[
100 + y = 300
\]
\[
y = 300 - 100
\]
\[
y = 200
\]

Thus, the boosters need to sell:
- **Hot dogs**: \( \mathbf{100} \)
- **Pretzels**: \( \mathbf{200} \)

To verify, we can check the income:
- Income from hot dogs: \( 100 \times 4 = 400 \)
- Income from pretzels: \( 200 \times 2 = 400 \)
- Total income: \( 400 + 400 = 800 \), which matches the goal.

Therefore, the final answer is:
**They need to sell 100 hot dogs and 200 pretzels.**