A yo-yo factory has fixed operating costs of $475,000 per year. In addition to the fixed costs, the cost to produce one yo-yo is $1.10. The yo-yo company sells the yo-yos to a distributor for $4.50 each. How many yo-yos must be sold in a year to earn a profit of at least $180,000?

To solve this problem, we need to determine the number of yo-yos that must be sold to cover the fixed costs, variable costs, and earn the desired profit.

Let's break this down:

### Known Information:
- Fixed operating costs: \$475,000
- Cost to produce one yo-yo (variable cost per yo-yo): \$1.10
- Selling price per yo-yo: \$4.50
- Desired profit: \$180,000

### Step 1: Define the variables
Let $ x $ be the number of yo-yos to be sold.

### Step 2: Total Revenue
The revenue from selling $ x $ yo-yos at \$4.50 each is:
\[
\text{Revenue} = 4.50x
\]

### Step 3: Total Cost
The total cost includes both the fixed costs and the variable costs. The variable cost to produce $ x $ yo-yos is $ 1.10x $, so the total cost is:
\[
\text{Total Cost} = 475,000 + 1.10x
\]

### Step 4: Profit Equation
The profit is the revenue minus the total costs:
\[
\text{Profit} = \text{Revenue} - \text{Total Cost}
\]
Substitute the expressions for revenue and total cost:
\[
\text{Profit} = 4.50x - (475,000 + 1.10x)
\]
Simplify the equation:
\[
\text{Profit} = 4.50x - 475,000 - 1.10x
\]
\[
\text{Profit} = 3.40x - 475,000
\]

### Step 5: Solve for $ x $
We want the profit to be at least \$180,000, so set the equation equal to 180,000 and solve for $ x $:
\[
180,000 = 3.40x - 475,000
\]
Add 475,000 to both sides:
\[
180,000 + 475,000 = 3.40x
\]
\[
655,000 = 3.40x
\]
Now, divide both sides by 3.40:
\[
x = \frac{655,000}{3.40}
\]
\[
x = 192,647.06
\]

Since the number of yo-yos must be a whole number, round up to the nearest whole number:
\[
x = 192,648
\]

### Conclusion:
The company must sell at least **192,648 yo-yos** in a year to earn a profit of at least \$180,000.

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Would you like more details on any step? Here are five related questions to expand on this topic:

1. How would the number of yo-yos change if the variable cost per yo-yo increased?
2. What is the breakeven point where the company makes zero profit?
3. How does the profit change if the selling price per yo-yo decreases to \$4.00?
4. What if the company aims to make a profit of \$250,000 instead?
5. How would a change in fixed operating costs affect the number of yo-yos needed?

**Tip:** To determine profit, always compare your revenue with both fixed and variable costs to get the most accurate picture of your earnings.

Learn how to calculate the number of yo-yos a factory must sell to cover fixed costs of $475,000 and earn a profit of at least $180,000. This problem involves analyzing linear equations and cost-revenue relationships. Suitable for high school students in Grades 9-12.

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