Max is starting a tropical soda stand. Empty bottles cost $0.32 each and ingredients cost $0.10 per bottle. Additionally, Max spent $54 on a bottle-cap sealer. He will sell his soda for $1.50 per bottle.
How many bottles of soda must Max sell for his sales to equal his expenses?

We need to find how many bottles of soda Max must sell for his sales to equal his total expenses. To do this, we can break the problem down into the following components:

### 1. **Expenses**
- **Fixed expense**: The bottle-cap sealer costs $54 (this is a one-time cost).
- **Variable expenses** per bottle:
  - Each empty bottle costs $0.32.
  - Ingredients for each bottle cost $0.10.
  
Thus, the total cost per bottle is:
\[
\text{Total cost per bottle} = 0.32 + 0.10 = 0.42 \, \text{dollars}
\]

### 2. **Revenue per bottle**
Max will sell each bottle of soda for $1.50.

### 3. **Set up the equation**
Let $ x $ be the number of bottles Max must sell. The total cost includes both the fixed cost and the variable cost per bottle. So, the total cost is:
\[
\text{Total cost} = 54 + 0.42x
\]
The revenue from selling $ x $ bottles is:
\[
\text{Revenue} = 1.50x
\]
We want to find $ x $ such that total revenue equals total cost:
\[
1.50x = 54 + 0.42x
\]

### 4. **Solve for $ x $**
First, subtract $ 0.42x $ from both sides of the equation:
\[
1.50x - 0.42x = 54
\]
Simplifying:
\[
1.08x = 54
\]
Now, divide both sides by 1.08:
\[
x = \frac{54}{1.08} = 50
\]

Thus, Max must sell **50 bottles** for his sales to equal his expenses.

Would you like further details or explanations? Here are some related questions for additional practice:

1. How much profit will Max make if he sells 60 bottles of soda?
2. What will Max’s total expenses be after selling 100 bottles of soda?
3. How would the break-even number change if the bottle cost increased to $0.40?
4. If Max decides to sell the soda for $2.00 per bottle, what would be his new break-even point?
5. What is the total revenue generated if Max sells 80 bottles?

**Tip:** In any break-even problem, it's important to separate fixed costs (one-time) from variable costs (which scale with each unit sold).

Max is starting a tropical soda stand. Empty bottles cost $0.32 each and ingredients cost $0.10 per bottle. Additionally, Max spent $54 on a bottle-cap sealer. He will sell his soda for $1.50 per bottle. How many bottles of soda must Max sell for his sales to equal his expenses?

Learn how to calculate the break-even point for a soda stand by solving linear equations. This problem involves algebraic concepts such as fixed costs, variable costs, and revenue calculations. Perfect for students in Grades 6-8 learning how to set up and solve equations in real-world scenarios.

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