28. Break-even problems problems

28. Break-even problems problems - step by step solution procedure and detailed explanation for each problem.

Below are 10 break-even analysis problems with step-by-step solutions and detailed explanations. Break-even analysis determines the point at which total revenue equals total costs, resulting in zero profit or loss. The break-even point (BEP) can be calculated in units or dollars. The standard formulas are:
  • Break-even point in units:
    \text{BEP (units)} = \frac{\text{Fixed Costs}}{\text{Contribution Margin per Unit}}
    where Contribution Margin per Unit = Selling Price per Unit - Variable Cost per Unit.
  • Break-even point in dollars:
    \text{BEP (dollars)} = \frac{\text{Fixed Costs}}{\text{Contribution Margin Ratio}}
    where Contribution Margin Ratio =
    \frac{\text{Contribution Margin per Unit}}{\text{Selling Price per Unit}}
    .
Each problem will include a scenario, solution steps, and an explanation of the process and implications.
Problem 1: Basic Break-Even for a Coffee Shop
A coffee shop has fixed costs of $12,000 per month. Each coffee sells for $5, with a variable cost of $2 per coffee. How many coffees must the shop sell to break even?
Step-by-Step Solution:
  1. Identify given values:
    • Fixed Costs (FC) = $12,000
    • Selling Price per Unit (SP) = $5
    • Variable Cost per Unit (VC) = $2
  2. Calculate Contribution Margin per Unit:
    \text{CM per Unit} = \text{SP} - \text{VC} = 5 - 2 = 3
  3. Calculate Break-Even Point in Units:
    \text{BEP (units)} = \frac{\text{Fixed Costs}}{\text{CM per Unit}} = \frac{12,000}{3} = 4,000 \text{ coffees}
  4. Verify (optional):
    • Total Revenue at BEP = 4,000 × $5 = $20,000
    • Total Variable Costs = 4,000 × $2 = $8,000
    • Total Costs = Fixed Costs + Variable Costs = $12,000 + $8,000 = $20,000
    • Revenue ($20,000) = Total Costs ($20,000), confirming break-even.
Explanation: The shop must sell 4,000 coffees to cover its $12,000 fixed costs and $8,000 variable costs. Each coffee contributes $3 to fixed costs after covering its variable cost. This is a basic break-even problem, assuming linear costs and constant prices. The result helps the owner understand the minimum sales volume needed to avoid losses.
Problem 2: Break-Even in Dollars for a Bookstore
A bookstore has fixed costs of $50,000 per year. Books are sold for $20 each, with variable costs of $12 per book. What is the break-even point in sales dollars?
Step-by-Step Solution:
  1. Identify given values:
    • Fixed Costs (FC) = $50,000
    • Selling Price per Unit (SP) = $20
    • Variable Cost per Unit (VC) = $12
  2. Calculate Contribution Margin per Unit:
    \text{CM per Unit} = \text{SP} - \text{VC} = 20 - 12 = 8
  3. Calculate Contribution Margin Ratio:
    \text{CM Ratio} = \frac{\text{CM per Unit}}{\text{SP}} = \frac{8}{20} = 0.4 \text{ (or 40%)}
  4. Calculate Break-Even Point in Dollars:
    \text{BEP (dollars)} = \frac{\text{Fixed Costs}}{\text{CM Ratio}} = \frac{50,000}{0.4} = 125,000
  5. Verify (optional):
    • BEP in units =
      \frac{50,000}{8} = 6,250
      books
    • Sales Dollars = 6,250 × $20 = $125,000
    • Total Costs = $50,000 + (6,250 × $12) = $50,000 + $75,000 = $125,000
    • Revenue = Total Costs, confirming break-even.
Explanation: The bookstore needs $125,000 in sales to break even. The contribution margin ratio of 40% means 40 cents of each sales dollar contributes to fixed costs. This dollar-based break-even is useful for businesses tracking revenue rather than units. It shows the sales target needed to cover all costs.
Problem 3: Break-Even with Target Profit for a Bakery
A bakery has fixed costs of $30,000 per year. Each cake sells for $50, with variable costs of $20 per cake. How many cakes must be sold to earn a profit of $15,000?
Step-by-Step Solution:
  1. Identify given values:
    • Fixed Costs (FC) = $30,000
    • Selling Price per Unit (SP) = $50
    • Variable Cost per Unit (VC) = $20
    • Target Profit = $15,000
  2. Calculate Contribution Margin per Unit:
    \text{CM per Unit} = \text{SP} - \text{VC} = 50 - 20 = 30
  3. Calculate Units Needed for Target Profit:
    \text{Units} = \frac{\text{Fixed Costs} + \text{Target Profit}}{\text{CM per Unit}} = \frac{30,000 + 15,000}{30} = \frac{45,000}{30} = 1,500 \text{ cakes}
  4. Verify:
    • Revenue = 1,500 × $50 = $75,000
    • Variable Costs = 1,500 × $20 = $30,000
    • Total Costs = $30,000 + $30,000 = $60,000
    • Profit = Revenue - Total Costs = $75,000 - $60,000 = $15,000
Explanation: To achieve a $15,000 profit, the bakery must sell 1,500 cakes. The formula adjusts the break-even calculation by adding the target profit to fixed costs, as both must be covered by the contribution margin. This helps the bakery set sales goals beyond just breaking even.
Problem 4: Multi-Product Break-Even for a Restaurant
A restaurant sells burgers ($10 each, $4 variable cost) and fries ($5 each, $2 variable cost). Fixed costs are $20,000 per month. Sales are split 60% burgers, 40% fries (by units). How many of each must be sold to break even?
Step-by-Step Solution:
  1. Identify given values:
    • Fixed Costs (FC) = $20,000
    • Burgers: SP = $10, VC = $4, Weight = 0.6
    • Fries: SP = $5, VC = $2, Weight = 0.4
  2. Calculate Contribution Margin per Unit:
    • Burgers:
      \text{CM} = 10 - 4 = 6
    • Fries:
      \text{CM} = 5 - 2 = 3
  3. Calculate Weighted Average Contribution Margin:
    \text{Weighted CM} = (0.6 \times 6) + (0.4 \times 3) = 3.6 + 1.2 = 4.8
  4. Calculate Total Break-Even Units:
    \text{Total Units} = \frac{\text{Fixed Costs}}{\text{Weighted CM}} = \frac{20,000}{4.8} \approx 4,167 \text{ units}
  5. Allocate Units:
    • Burgers = 0.6 × 4,167 ≈ 2,500
    • Fries = 0.4 × 4,167 ≈ 1,667
  6. Verify:
    • Revenue = (2,500 × $10) + (1,667 × $5) = $25,000 + $8,335 = $33,335
    • Variable Costs = (2,500 × $4) + (1,667 × $2) = $10,000 + $3,334 = $13,334
    • Total Costs = $20,000 + $13,334 = $33,334
    • Revenue ≈ Total Costs (minor rounding).
Explanation: The restaurant must sell approximately 2,500 burgers and 1,667 fries to break even. The weighted contribution margin accounts for the sales mix, ensuring the calculation reflects the proportion of each product sold. This is critical for multi-product businesses, as product mix affects profitability.
Problem 5: Break-Even with Increased Fixed Costs
A manufacturer has fixed costs of $100,000, which will increase by $20,000 if production exceeds 5,000 units. Each unit sells for $50, with variable costs of $30. What is the break-even point?
Step-by-Step Solution:
  1. Identify given values:
    • Initial Fixed Costs = $100,000
    • Additional Fixed Costs = $20,000 if >5,000 units
    • Selling Price per Unit = $50
    • Variable Cost per Unit = $30
  2. Calculate Contribution Margin per Unit:
    \text{CM per Unit} = 50 - 30 = 20
  3. Calculate Break-Even with Initial Fixed Costs:
    \text{BEP} = \frac{100,000}{20} = 5,000 \text{ units}
  4. Check if BEP exceeds 5,000 units:
    • At 5,000 units, no additional costs are incurred, so BEP = 5,000 units.
  5. Verify:
    • Revenue = 5,000 × $50 = $250,000
    • Variable Costs = 5,000 × $30 = $150,000
    • Total Costs = $100,000 + $150,000 = $250,000
    • Revenue = Total Costs.
Explanation: The break-even point is 5,000 units, exactly at the threshold for additional fixed costs. If the BEP had exceeded 5,000 units, we would recalculate with total fixed costs of $120,000, yielding
\frac{120,000}{20} = 6,000
units. Since 5,000 units is valid, no adjustment is needed. This scenario highlights how step-fixed costs affect break-even analysis.
Problem 6: Break-Even with Sales Commission
A company sells gadgets for $100 each, with variable costs of $60. Salespeople earn a 10% commission on each sale. Fixed costs are $80,000. How many gadgets must be sold to break even?
Step-by-Step Solution:
  1. Identify given values:
    • Fixed Costs = $80,000
    • Selling Price per Unit = $100
    • Variable Cost per Unit = $60
    • Commission = 10% of $100 = $10 per unit
  2. Calculate Total Variable Cost per Unit:
    \text{Total VC} = 60 + 10 = 70
  3. Calculate Contribution Margin per Unit:
    \text{CM per Unit} = 100 - 70 = 30
  4. Calculate Break-Even Point:
    \text{BEP} = \frac{80,000}{30} \approx 2,667 \text{ gadgets}
  5. Verify:
    • Revenue = 2,667 × $100 = $266,700
    • Variable Costs = 2,667 × $70 = $186,690
    • Total Costs = $80,000 + $186,690 = $266,690
    • Revenue ≈ Total Costs (minor rounding).
Explanation: The commission increases the variable cost per unit, reducing the contribution margin to $30. The company needs to sell 2,667 gadgets to break even. This problem shows how sales commissions, treated as variable costs, impact the break-even point.
Problem 7: Break-Even with Limited Capacity
A gym has fixed costs of $24,000 per year and can serve up to 500 members. Membership fees are $600 per year, with variable costs of $100 per member. What is the break-even point, and is it feasible?
Step-by-Step Solution:
  1. Identify given values:
    • Fixed Costs = $24,000
    • Selling Price per Unit = $600
    • Variable Cost per Unit = $100
    • Capacity = 500 members
  2. Calculate Contribution Margin per Unit:
    \text{CM per Unit} = 600 - 100 = 500
  3. Calculate Break-Even Point:
    \text{BEP} = \frac{24,000}{500} = 48 \text{ members}
  4. Check Feasibility:
    • 48 members < 500, so the BEP is within capacity.
  5. Verify:
    • Revenue = 48 × $600 = $28,800
    • Variable Costs = 48 × $100 = $4,800
    • Total Costs = $24,000 + $4,800 = $28,800
    • Revenue = Total Costs.
Explanation: The gym needs only 48 members to break even, well within its 500-member capacity. This low break-even point suggests the business model is viable, as it requires only 9.6% of capacity to cover costs. Capacity constraints are critical in such analyses.
Problem 8: Break-Even with Price Discount
A retailer sells shirts for $40 each, with variable costs of $15. Fixed costs are $60,000. If the retailer offers a 20% discount, how many shirts must be sold to break even?
Step-by-Step Solution:
  1. Identify given values:
    • Fixed Costs = $60,000
    • Original Selling Price = $40
    • Variable Cost per Unit = $15
    • Discount = 20%, so New SP = $40 × (1 - 0.2) = $32
  2. Calculate Contribution Margin per Unit:
    \text{CM per Unit} = 32 - 15 = 17
  3. Calculate Break-Even Point:
    \text{BEP} = \frac{60,000}{17} \approx 3,529 \text{ shirts}
  4. Compare with No Discount:
    • Original CM = $40 - $15 = $25
    • Original BEP =
      \frac{60,000}{25} = 2,400
      shirts
  5. Verify:
    • Revenue = 3,529 × $32 = $112,928
    • Variable Costs = 3,529 × $15 = $52,935
    • Total Costs = $60,000 + $52,935 = $112,935
    • Revenue ≈ Total Costs.
Explanation: The 20% discount reduces the contribution margin, increasing the break-even point from 2,400 to 3,529 shirts. This shows the impact of pricing decisions on break-even volume. The retailer must assess if the discount drives enough additional sales to justify the higher BEP.
Problem 9: Break-Even with Mixed Costs
A delivery service has fixed costs of $10,000 per month and semi-variable costs: $2 per delivery plus $500 per month. Each delivery is charged at $10. How many deliveries are needed to break even?
Step-by-Step Solution:
  1. Identify given values:
    • Fixed Costs = $10,000
    • Semi-Variable Costs = $500 (fixed) + $2 per delivery (variable)
    • Total Fixed Costs = $10,000 + $500 = $10,500
    • Variable Cost per Unit = $2
    • Selling Price per Unit = $10
  2. Calculate Contribution Margin per Unit:
    \text{CM per Unit} = 10 - 2 = 8
  3. Calculate Break-Even Point:
    \text{BEP} = \frac{10,500}{8} = 1,312.5 \text{ deliveries}
    • Since deliveries are whole units, round up to 1,313.
  4. Verify:
    • Revenue = 1,313 × $10 = $13,130
    • Variable Costs = 1,313 × $2 = $2,626
    • Total Costs = $10,500 + $2,626 = $13,126
    • Profit = $13,130 - $13,126 = $4 (small profit, as rounded up).
Explanation: The semi-variable cost’s fixed portion is added to fixed costs, while the variable portion is treated as a per-unit cost. The service needs 1,313 deliveries to break even. This problem illustrates how to handle mixed costs in break-even analysis.
Problem 10: Break-Even with Tax Consideration
A consultant charges $200 per hour, with variable costs of $50 per hour. Fixed costs are $40,000 per year. If profits are taxed at 25%, how many hours must be billed to earn an after-tax profit of $15,000?
Step-by-Step Solution:
  1. Identify given values:
    • Fixed Costs = $40,000
    • Selling Price per Unit = $200
    • Variable Cost per Unit = $50
    • After-Tax Profit = $15,000
    • Tax Rate = 25%
  2. Calculate Before-Tax Profit:
    \text{After-Tax Profit} = \text{Before-Tax Profit} \times (1 - \text{Tax Rate})
    15,000 = \text{Before-Tax Profit} \times (1 - 0.25)
    \text{Before-Tax Profit} = \frac{15,000}{0.75} = 20,000
  3. Calculate Contribution Margin per Unit:
    \text{CM per Unit} = 200 - 50 = 150
  4. Calculate Units Needed:
    \text{Units} = \frac{\text{Fixed Costs} + \text{Before-Tax Profit}}{\text{CM per Unit}} = \frac{40,000 + 20,000}{150} = \frac{60,000}{150} = 400 \text{ hours}
  5. Verify:
    • Revenue = 400 × $200 = $80,000
    • Variable Costs = 400 × $50 = $20,000
    • Total Costs = $40,000 + $20,000 = $60,000
    • Before-Tax Profit = $80,000 - $60,000 = $20,000
    • Tax = $20,000 × 0.25 = $5,000
    • After-Tax Profit = $20,000 - $5,000 = $15,000
Explanation: The consultant must bill 400 hours to achieve a $15,000 after-tax profit. The tax adjustment increases the required profit before applying the break-even formula. This problem shows how taxes complicate target profit calculations.
General Notes
  • Assumptions: These problems assume linear cost and revenue functions, constant prices, and no external factors (e.g., demand changes).
  • Applications: Break-even analysis helps businesses set sales targets, evaluate pricing strategies, and assess cost structures.
  • Limitations: It doesn’t account for market dynamics, non-linear costs, or multiple break-even points in complex scenarios.
Each problem demonstrates a unique aspect of break-even analysis, from basic calculations to handling commissions, taxes, and multi-product scenarios. 

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