Alright, let's go through the problem step-by-step to find the break-even point and the number of units for each product that will be sold at that point.
### Step 1: Calculate the Contribution Margin for Each Product
The contribution margin is calculated by subtracting the variable cost from the selling price for each product.
- Contribution margin for windows:
[tex]\[ \text{Selling price of windows} - \text{Variable cost of windows} = \$340 - \$195 = \$145 \][/tex]
- Contribution margin for doors:
[tex]\[ \text{Selling price of doors} - \text{Variable cost of doors} = \$730 - \$490 = \$240 \][/tex]
### Step 2: Calculate the Weighted Average Contribution Margin
The weighted average contribution margin is calculated by multiplying the contribution margin of each product by its sales ratio and then summing the results.
- Weighted average contribution margin:
[tex]\[ \text{Weighted average contribution margin} = (\text{Contribution margin of windows} \times \text{Sales ratio of windows}) + (\text{Contribution margin of doors} \times \text{Sales ratio of doors}) \][/tex]
[tex]\[ = (\$145 \times 0.60) + (\$240 \times 0.40) \][/tex]
[tex]\[ = \$87 + \$96 = \$183 \][/tex]
### Step 3: Calculate the Break-Even Point in Units
To find the break-even point in units, we divide the total fixed costs by the weighted average contribution margin.
- Break-even point in units:
[tex]\[ \text{Break-even point} = \frac{\text{Total fixed costs}}{\text{Weighted average contribution margin}} \][/tex]
[tex]\[ = \frac{\$446,950}{\$183} \][/tex]
[tex]\[ \approx 2442.35 \text{ units} \][/tex]
### Step 4: Calculate the Number of Units for Each Product at the Break-Even Point
Now, to find the number of units for each product at the break-even point, we multiply the total break-even units by the sales ratio of each product.
- Number of windows:
[tex]\[ \text{Number of windows} = \text{Break-even units} \times \text{Sales ratio of windows} \][/tex]
[tex]\[ = 2442.35 \times 0.60 \][/tex]
[tex]\[ = 1465.41 \text{ units} \][/tex]
- Number of doors:
[tex]\[ \text{Number of doors} = \text{Break-even units} \times \text{Sales ratio of doors} \][/tex]
[tex]\[ = 2442.35 \times 0.40 \][/tex]
[tex]\[ = 976.94 \text{ units} \][/tex]
### Summary
- Contribution margin for windows: \[tex]$145
- Contribution margin for doors: \$[/tex]240
- Weighted average contribution margin: \$183
- Break-even point in units: 2442.35 units
- Number of windows at break-even point: 1465.41 units
- Number of doors at break-even point: 976.94 units
This is the step-by-step solution to determine the break-even point in units and the number of units for each product that will be sold at that point.
