Math Problem Statement

The break-even point for the company as a whole, given data on selling prices, variable costs, fixed costs, and expected sales volumes for two products.

Solution

To calculate the break-even point for the company as a whole, we first need to use the data provided in the table. Let's break down the process step by step:

Given Data:

For product F1:

  • Selling price (LE): 50
  • Variable cost per unit (LE): 30
  • Fixed cost for product (LE): 2000
  • Expected sales volume: 500

For product F2:

  • Selling price (LE): 36
  • Variable cost per unit (LE): 27
  • Fixed cost for product (LE): 6000
  • Expected sales volume: 2000

Step 1: Contribution margin per unit for each product

The contribution margin is calculated as: Contribution Margin=Selling PriceVariable Cost\text{Contribution Margin} = \text{Selling Price} - \text{Variable Cost}

For F1: Contribution Margin (F1)=5030=20\text{Contribution Margin (F1)} = 50 - 30 = 20

For F2: Contribution Margin (F2)=3627=9\text{Contribution Margin (F2)} = 36 - 27 = 9

Step 2: Total fixed costs

Total fixed costs are the sum of the fixed costs for both products: Total Fixed Costs=2000+6000=8000LE\text{Total Fixed Costs} = 2000 + 6000 = 8000 \, \text{LE}

Step 3: Sales mix (proportion of each product's sales)

Sales mix is the proportion of sales volume of each product to the total sales volume.

Sales Volume (F1)=500,Sales Volume (F2)=2000\text{Sales Volume (F1)} = 500, \quad \text{Sales Volume (F2)} = 2000 Total Sales Volume=500+2000=2500\text{Total Sales Volume} = 500 + 2000 = 2500 Sales Mix (F1)=5002500=0.2,Sales Mix (F2)=20002500=0.8\text{Sales Mix (F1)} = \frac{500}{2500} = 0.2, \quad \text{Sales Mix (F2)} = \frac{2000}{2500} = 0.8

Step 4: Weighted average contribution margin

The weighted average contribution margin (WACM) is calculated as the sum of the contribution margins of both products, weighted by their sales mix: WACM=(0.2×20)+(0.8×9)\text{WACM} = (0.2 \times 20) + (0.8 \times 9) WACM=4+7.2=11.2LE\text{WACM} = 4 + 7.2 = 11.2 \, \text{LE}

Step 5: Break-even point (in units)

The break-even point in units can be calculated using the formula: Break-even point (units)=Total Fixed CostsWeighted Average Contribution Margin\text{Break-even point (units)} = \frac{\text{Total Fixed Costs}}{\text{Weighted Average Contribution Margin}} Break-even point (units)=800011.2714.29units\text{Break-even point (units)} = \frac{8000}{11.2} \approx 714.29 \, \text{units}

Step 6: Break-even point (in sales volume)

Now, to find the break-even point in terms of sales for each product, we distribute the 714.29 units based on the sales mix:

  • For F1: 714.29×0.2142.86units714.29 \times 0.2 \approx 142.86 \, \text{units}
  • For F2: 714.29×0.8571.43units714.29 \times 0.8 \approx 571.43 \, \text{units}

Thus, the company must sell approximately 143 units of F1 and 571 units of F2 to break even.

Would you like a further explanation of any step or additional details?

Here are 5 related questions you might find useful:

  1. How does changing the fixed cost impact the break-even point?
  2. How would changes in the sales mix affect the weighted average contribution margin?
  3. What is the break-even point in terms of revenue instead of units?
  4. How can this analysis be adapted for more than two products?
  5. What role does the contribution margin play in determining profitability?

Tip: Always consider the sales mix when calculating the break-even point for multiple products—it can significantly impact the overall result.

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Math Problem Analysis

Mathematical Concepts

Break-even analysis
Cost accounting
Profit planning

Formulas

Contribution Margin = Selling Price - Variable Cost
Total Fixed Costs = Fixed Costs for F1 + Fixed Costs for F2
Weighted Average Contribution Margin = (Sales Mix F1 * Contribution Margin F1) + (Sales Mix F2 * Contribution Margin F2)
Break-even Point (Units) = Total Fixed Costs / Weighted Average Contribution Margin

Theorems

Break-even point theory

Suitable Grade Level

Grades 10-12 (Business and Economics)