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The problem states that the total amount collected was \$2,400 from the sale of hamburgers and hot dogs, and that the amounts earned from both hamburgers and hot dogs were equal. A hamburger sold for \$3 and a hot dog sold for \$2. We are asked to determine how many of each were sold.

Let's denote:
- $ x $ = number of hamburgers sold
- $ y $ = number of hot dogs sold

From the information given:
1. The total amount collected was \$2,400.
2. The earnings from hamburgers and hot dogs were equal.

### Step 1: Set up the equations

The total amount earned from hamburgers is $ 3x $ (since each hamburger costs \$3), and the total amount earned from hot dogs is $ 2y $ (since each hot dog costs \$2). The amounts earned from hamburgers and hot dogs are equal, so:

\[
3x = 2y
\]

Also, the total earnings from both hamburgers and hot dogs is \$2,400:

\[
3x + 2y = 2400
\]

### Step 2: Solve the system of equations

From the first equation $ 3x = 2y $, solve for $ y $:

\[
y = \frac{3x}{2}
\]

Substitute this expression for $ y $ into the second equation:

\[
3x + 2\left(\frac{3x}{2}\right) = 2400
\]

Simplifying:

\[
3x + 3x = 2400
\]

\[
6x = 2400
\]

\[
x = 400
\]

Now substitute $ x = 400 $ back into $ y = \frac{3x}{2} $:

\[
y = \frac{3(400)}{2} = 600
\]

### Final Answer:
- 400 hamburgers were sold.
- 600 hot dogs were sold.

Let me know if you want further details or have any questions.

Here are some related questions:
1. What if hamburgers sold for \$4 each instead of \$3?
2. What if the total collected was \$3,000 with the same conditions?
3. How many hamburgers would need to be sold if only 500 hot dogs were sold?
4. How would the equations change if the amounts from hamburgers and hot dogs were not equal?
5. What if both items sold for the same price?

**Tip:** When setting up equations, always check if there is any relationship or proportion between the variables to simplify the solution process.

The band boosters collected $2,400 from the sale of hamburgers and hot dogs. The amounts earned from hamburgers and hot dogs were equal. A hamburger sold for $3, and a hot dog sold for $2. How many of each were sold?

Learn how to solve the system of equations involving the sale of hamburgers and hot dogs where the total collected is $2,400, and the earnings from both are equal. This algebra problem uses key concepts like setting up and solving equations. Suitable for Grades 7-9.

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