Question
A proposed project has fixed costs of $39,480, depreciation expense of $8,724, and a sales quantity of 1,330 units. The total variable costs are $5,607. What is the contribution margin per unit if the projected level of sales is the accounting break-even point?
Answers
GPT-4o mini
To find the contribution margin per unit at the accounting break-even point, we first need to understand a few key concepts.
1. **Fixed Costs (FC)**: These are the costs that do not change with the level of output, which in this case are $39,480.
2. **Variable Costs (VC)**: These change with the level of output; in this case, the total variable costs are $5,607 for 1,330 units.
3. **Sales Quantity (Q)**: The number of units sold, which is 1,330 in this situation.
The accounting break-even point is the level of sales at which total revenues equal total costs (both fixed and variable). At this point, the contribution margin must cover all fixed costs.
### Step 1: Calculate Total Variable Cost per Unit
To find the variable cost per unit (VC per unit):
\[
\text{Variable cost per unit} = \frac{\text{Total Variable Costs}}{\text{Sales Quantity}} = \frac{5,607}{1,330}
\]
Calculating this gives:
\[
\text{Variable cost per unit} = 4.22
\]
### Step 2: Determine Total Revenue at Break-even Point
At the break-even point, total revenue (TR) must equal total costs (TC), which includes both fixed and variable costs. The formula can be represented as:
\[
\text{Total Revenue} = \text{Total Fixed Costs} + \text{Total Variable Costs}
\]
### Step 3: Express Total Revenue and Solve for Contribution Margin
The total revenue can also be expressed in terms of selling price per unit (P) and sales quantity (Q):
\[
TR = P \times Q
\]
From the break-even condition:
\[
P \times Q = \text{Fixed Costs} + \text{Variable Costs}
\]
Substituting known values:
\[
P \times 1,330 = 39,480 + 5,607
\]
\[
P \times 1,330 = 45,087
\]
### Step 4: Solve for Selling Price per Unit
\[
P = \frac{45,087}{1,330} \approx 33.88
\]
### Step 5: Find Contribution Margin per Unit
The contribution margin per unit (CM) is calculated as the difference between the selling price per unit (P) and the variable cost per unit:
\[
CM = P - \text{Variable cost per unit}
\]
Substituting the values we've calculated:
\[
CM = 33.88 - 4.22
\]
\[
CM \approx 29.66
\]
### Conclusion
The contribution margin per unit at the projected level of sales (accounting break-even point) is approximately **$29.66**.
1. **Fixed Costs (FC)**: These are the costs that do not change with the level of output, which in this case are $39,480.
2. **Variable Costs (VC)**: These change with the level of output; in this case, the total variable costs are $5,607 for 1,330 units.
3. **Sales Quantity (Q)**: The number of units sold, which is 1,330 in this situation.
The accounting break-even point is the level of sales at which total revenues equal total costs (both fixed and variable). At this point, the contribution margin must cover all fixed costs.
### Step 1: Calculate Total Variable Cost per Unit
To find the variable cost per unit (VC per unit):
\[
\text{Variable cost per unit} = \frac{\text{Total Variable Costs}}{\text{Sales Quantity}} = \frac{5,607}{1,330}
\]
Calculating this gives:
\[
\text{Variable cost per unit} = 4.22
\]
### Step 2: Determine Total Revenue at Break-even Point
At the break-even point, total revenue (TR) must equal total costs (TC), which includes both fixed and variable costs. The formula can be represented as:
\[
\text{Total Revenue} = \text{Total Fixed Costs} + \text{Total Variable Costs}
\]
### Step 3: Express Total Revenue and Solve for Contribution Margin
The total revenue can also be expressed in terms of selling price per unit (P) and sales quantity (Q):
\[
TR = P \times Q
\]
From the break-even condition:
\[
P \times Q = \text{Fixed Costs} + \text{Variable Costs}
\]
Substituting known values:
\[
P \times 1,330 = 39,480 + 5,607
\]
\[
P \times 1,330 = 45,087
\]
### Step 4: Solve for Selling Price per Unit
\[
P = \frac{45,087}{1,330} \approx 33.88
\]
### Step 5: Find Contribution Margin per Unit
The contribution margin per unit (CM) is calculated as the difference between the selling price per unit (P) and the variable cost per unit:
\[
CM = P - \text{Variable cost per unit}
\]
Substituting the values we've calculated:
\[
CM = 33.88 - 4.22
\]
\[
CM \approx 29.66
\]
### Conclusion
The contribution margin per unit at the projected level of sales (accounting break-even point) is approximately **$29.66**.