Given the information in the table below, how many units will be required to be sold to hit the break-even point?
| Fixed Cost | $456,000 |
|---------------|----------|
| Variable Cost | $0.20 |
| Selling Price | $6.20 |
[?] units
Answer (2)
Define x as the number of units to be sold.
Express total cost as the sum of fixed costs and variable costs: Total Cost = 456 , 000 + 0.20 x .
Express total revenue as the product of selling price and number of units: Total Revenue = 6.20 x .
Set total cost equal to total revenue and solve for x : 456 , 000 + 0.20 x = 6.20 x , which gives x = 76 , 000 .
The break-even point is 76 , 000 units.
Explanation
Understanding the Problem Let's analyze the problem. We need to find the number of units that must be sold to cover all costs, both fixed and variable. This is the break-even point. We are given the fixed costs, variable cost per unit, and selling price per unit.
Calculating Total Cost Let x be the number of units to be sold. The total cost is the sum of the fixed cost and the variable cost, which is the variable cost per unit times the number of units:
Total Cost = Fixed Cost + (Variable Cost per unit × Number of Units)
Total Cost = $456,000 + 0.20 x
Calculating Total Revenue The total revenue is the selling price per unit times the number of units:
Total Revenue = Selling Price per unit × Number of Units
Total Revenue = $6.20x
Setting Up the Equation At the break-even point, the total cost equals the total revenue:
Total Cost = Total Revenue
456 , 000 + 0.20 x = 6.20 x
Solving for x Now, we solve for x :
456 , 000 = 6.20 x − 0.20 x
456 , 000 = 6.00 x
x = 6.00 456 , 000
x = 76 , 000
Final Answer Therefore, the number of units that need to be sold to reach the break-even point is 76,000 units.
Examples
Understanding break-even points is crucial in business. For example, if you're starting a small business selling handmade jewelry, you need to know how many pieces you need to sell to cover your initial investment in materials, tools, and studio space (fixed costs) as well as the cost of materials for each piece (variable costs). Calculating the break-even point helps you set realistic sales goals and pricing strategies. If your fixed costs are $1000, your variable cost per piece is $5, and you sell each piece for $25, you need to sell 50 pieces to break even, which is calculated as $\frac{1000}{(25-5)} = 50$.
To break even, the student needs to sell 76,000 units. This is calculated by setting total costs equal to total revenues and solving for the number of units. The derived equation shows that at this point, costs are fully covered by revenue from sales.
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