Problem 7-12 (Algo)
A firm is selling two products—chairs and bar stools—each at $55 per unit. Chairs have a variable cost of $30, and bar stools $20. The fixed cost for the firm is $20,000.

If the sales mix is 1:1 (one chair sold for every bar stool sold), what is the break-even point in dollars of sales? In units of chairs and bar stools?
Note: Round the number of chairs to the nearest whole number BEFORE computing the number of bar stools. Use these rounded breakeven quantities in your calculation for the breakeven point in dollar sales.



If the sales mix changes to 1:4 (one chair sold for every four bar stools sold), what is the break-even point in dollars of sales? In units of chairs and bar stools?
Note: Round the number of chairs to the nearest whole number BEFORE computing the number of bar stools. Use these rounded breakeven quantities in your calculation for the breakeven point in dollar sales.



To find the break-even point for a firm selling chairs and bar stools, we need to consider both the sales mix and the cost structure. Here’s how we can solve it:

### Given Data
- **Selling Price per unit**:
  - Chairs: $55
  - Bar Stools: $55
- **Variable Cost per unit**:
  - Chairs: $30
  - Bar Stools: $20
- **Fixed Costs**: $20,000

### Contribution Margin
Contribution Margin (CM) per unit = Selling Price - Variable Cost

- CM for Chairs: $55 - 30 = 25$
- CM for Bar Stools: $55 - 20 = 35$

### Sales Mix 1:1

1. **Average Contribution Margin per unit**:
   Since the sales mix is 1:1, the average CM per unit is:
   \[
   \text{Average CM} = \frac{25 + 35}{2} = 30
   \]

2. **Break-even Point in Units**:
   Total break-even units = $\frac{\text{Fixed Costs}}{\text{Average CM}}$
   \[
   \text{Total break-even units} = \frac{20000}{30} \approx 667 \text{ units}
   \]
   Since the sales mix is 1:1, the break-even units for chairs and bar stools will both be:
   \[
   \text{Chairs} = 334 \text{ units (rounded)}
   \]
   \[
   \text{Bar Stools} = 334 \text{ units (rounded)}
   \]

3. **Break-even Point in Dollars**:
   Total break-even sales = Total units * Selling Price per unit
   \[
   \text{Total break-even sales} = (334 + 334) \times 55 = 668 \times 55 = 36,740 \text{ dollars}
   \]

### Sales Mix 1:4

1. **Average Contribution Margin per unit**:
   With a sales mix of 1:4 (one chair for every four bar stools), the weighted average CM is:
   \[
   \text{Average CM} = \frac{1 \times 25 + 4 \times 35}{1 + 4} = \frac{25 + 140}{5} = 33
   \]

2. **Break-even Point in Units**:
   Total break-even units = $\frac{\text{Fixed Costs}}{\text{Average CM}}$
   \[
   \text{Total break-even units} = \frac{20000}{33} \approx 606 \text{ units}
   \]
   For the 1:4 mix, break-even units for chairs:
   \[
   \text{Chairs} = \frac{606}{5} \approx 121 \text{ units (rounded)}
   \]
   Break-even units for bar stools:
   \[
   \text{Bar Stools} = 121 \times 4 = 484 \text{ units}
   \]

3. **Break-even Point in Dollars**:
   Total break-even sales = (Units of Chairs + Units of Bar Stools) * Selling Price per unit
   \[
   \text{Total break-even sales} = (121 + 484) \times 55 = 605 \times 55 = 33,275 \text{ dollars}
   \]

### Summary
- **1:1 Sales Mix**:
  - Break-even Units: 334 Chairs, 334 Bar Stools
  - Break-even Sales: $36,740

- **1:4 Sales Mix**:
  - Break-even Units: 121 Chairs, 484 Bar Stools
  - Break-even Sales: $33,275

Would you like more details or have any questions? Here are some related questions to expand on:

1. How does the change in sales mix affect the overall profitability of the firm?
2. What would be the impact on the break-even point if the fixed costs were reduced by 10%?
3. How would an increase in the selling price of bar stools to $60 affect the break-even point in a 1:1 sales mix?
4. What is the margin of safety in both sales mixes if the firm sells 800 units in total?
5. How would the break-even analysis change if the firm introduced a third product with different costs and sales mix?

**Tip:** Always consider the impact of changes in sales mix and cost structures on the break-even point to make informed business decisions.

A firm is selling two products—chairs and bar stools—each at $55 per unit. Chairs have a variable cost of $30, and bar stools $20. The fixed cost for the firm is $20,000.

If the sales mix is 1:1 (one chair sold for every bar stool sold), what is the break-even point in dollars of sales? In units of chairs and bar stools?

If the sales mix changes to 1:4 (one chair sold for every four bar stools sold), what is the break-even point in dollars of sales? In units of chairs and bar stools?

Discover how to calculate the break-even point for a firm selling chairs and bar stools with different sales mixes. Learn how to determine break-even units and sales using contribution margins and fixed costs in this detailed business problem analysis.

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