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WealthWiseGuys Practical Mathematics

Chapter 8

Markups, Markdowns, Pricing, and Break-Even Analysis

Set a selling price and test whether it covers costs

A busy food stand can collect plenty of cash and still lose money. This chapter connects cost, selling price, markdowns, and sales volume so that you can explain where a business result comes from. All businesses and amounts are hypothetical; round money to cents, with a half cent rounded up, and round required whole-unit sales upward.

8.1

Markup on Cost and Margin on Selling Price

Learning objectives

After this section you will be able to:

  1. Find a selling price from cost and markup.
  2. Distinguish markup on cost from margin on selling price.
  3. Recover price from a target margin.

Real-life situation

A pop-up prices a tote

Use this hypothetical Philadelphia-area scenario to connect the calculation to a decision. All people, businesses, amounts, and rates in this chapter are invented; tax and payroll rules are teaching assumptions, not current law.

A pop-up prices a tote

  • Unit cost: $30.00
  • Selling price: $50.00
  • Markup dollars: $20.00
  • Markup on cost: 66.67%; margin: 40%
Figure 8.1 A pop-up prices a tote (hypothetical)

The concept

Cost is the amount assigned to acquiring or producing an item under the model being used. Selling price is what the customer pays before any separately stated tax. Markup dollars are selling price minus cost; this difference must help cover operating expenses as well as profit.

A markup rate on cost uses cost as its denominator. If cost is $30 and price is $50, the $20 markup is 20 ÷ 30, or approximately 66.67% of cost. Label the base whenever reporting a markup percentage.

Margin on selling price uses selling price instead. For the same tote, 20 ÷ 50 = 40%. These two rates describe the same dollars but are not interchangeable. Applying a 40% markup to cost would produce a different selling price.

To achieve a target margin m, the cost must be the remaining 1 − m share of selling price. Divide cost by that remaining share. A positive cost cannot be covered at a margin of 100% in this simple model. A pricing calculation is a plan, not a guarantee that customers will buy.

Cost + markup = selling price

  • 1. Cost: $30.00 (60.0% of total)
  • 2. Markup: $20.00 (40.0% of total)
  1. Start with $30 cost; add $20 markup to reach $50 selling price.
  2. Markup uses cost as its base: 20 ÷ 30 = 66.67%.
  3. Margin uses selling price as its base: 20 ÷ 50 = 40%. These are not interchangeable.

Original WealthWiseGuys diagram · Hypothetical teaching quantities.

Money Words

Plain-language words you will use in this lesson — learn them before we reach the formulas.

  • markup

    The amount added to a cost to set a selling price; the markup rate uses cost as its base.

  • profit margin

    Profit shown as a percent of revenue; a 10% margin means $10 of profit per $100 of sales.

  • gross profit

    Revenue minus the direct cost of the items sold, before other expenses.

Key terms

Cost
Amount assigned to acquiring or producing the item.
Selling price
Amount charged for the item before separately stated tax.
Markup dollars
Selling price minus cost.
Markup on cost
Markup dollars divided by cost.
Margin
Markup dollars divided by selling price in this model.

Formulas & rules

  • Markup dollars

    Markup = selling price − cost

  • Price from cost markup

    Selling price = cost × (1 + decimal markup rate)

  • Margin

    Decimal margin = markup ÷ selling price

  • Price from margin

    Selling price = cost ÷ (1 − decimal margin)

Worked examples

Example 1Apply markup on cost

An item costs $40. Set price using a 25% markup on cost.

Solution

  1. 1Markup: 40 × 0.25 = $10.00.
  2. 2Price: 40 + 10 = $50.00.
  3. 3Check markup rate: 10 ÷ 40 = 0.25 = 25%.

Answer: Price: $50.00.

Example 2Target a margin

A tote costs $30. Find the price for a 40% margin, then its markup rate on cost.

Solution

  1. 1Cost share: 1 − 0.40 = 0.60.
  2. 2Price: 30 ÷ 0.60 = $50.00.
  3. 3Markup: 50 − 30 = $20.00.
  4. 4Markup on cost: 20 ÷ 30 × 100% ≈ 66.67%.

Answer: Price $50.00; markup on cost approximately 66.67%.

Example 3Real-world friction: resale through an online marketplace

A student buys sneakers for $60 and resells them for $100 on a hypothetical marketplace that keeps a 12% seller fee. The student also pays $8 to ship. Compare the simple margin with the real net profit and net margin.

Solution

  1. 1Simple (no-fee) profit: 100 − 60 = $40.00, a 40% margin on the $100 price.
  2. 2Marketplace fee: 100 × 0.12 = $12.00.
  3. 3Total selling costs: 12 + 8 = $20.00.
  4. 4Net profit: 40 − 20 = $20.00.
  5. 5Net margin: 20 ÷ 100 × 100% = 20%.
  6. 6Judgment: fees and shipping cut the profit in half, so the price must clear all selling costs, not just the purchase cost.

Answer: Net profit $20.00; net margin 20%, half of the 40% simple margin.

Practice exercises 8.1

  1. Cost is $20 and price is $32. Find markup dollars.
  2. For that item, find markup on cost and margin.
  3. Price a $60 item with 30% markup on cost.
  4. Price a $45 item for a 25% margin.
  5. A seller says a 50% cost markup gives a 50% margin. Test with $10 cost.
  6. Decision: A hoodie costs $25 and sells for $40 on a platform charging a 10% fee plus $7 shipping paid by the seller. Find net profit and decide whether the flip is worth it if the student wants at least $10 profit.
  7. Decision: For that hoodie, what minimum price gives $10 net profit? (Price − 10% fee − $7 − $25 = $10.)

Recap quiz 8.1

1. $12 cost and $18 price give markup dollars of:
2. A 20% margin on a $50 price represents:
3. For a target 30% margin, divide cost by:

Section 8.1 recap

  • Markup and margin use different denominators.
  • Markup dollars must cover expenses before they can become profit.
Answer key with explanations: Section 8.1

Practice exercises

  1. 32 − 20 = $12.00.
  2. 12 ÷ 20 = 60% markup on cost; 12 ÷ 32 = 37.5% margin.
  3. Markup = 60 × 0.30 = $18.00; price = $78.00.
  4. 45 ÷ (1 − 0.25) = 45 ÷ 0.75 = $60.00.
  5. Price = 10 × 1.50 = $15.00; markup = $5.00; margin = 5 ÷ 15 ≈ 33.33%, not 50%.
  6. Fee = 40 × 0.10 = $4.00; selling costs = 4 + 7 = $11.00; net profit = 40 − 25 − 11 = $4.00. Below the $10 goal, so No-Go at this price.
  7. 0.90 × price = 10 + 7 + 25 = $42.00; price = 42 ÷ 0.90 ≈ $46.67.

Recap quiz

  1. $6: 18 − 12 = $6.
  2. $10: 50 × 0.20 = $10 markup dollars.
  3. 0.70: Cost is the remaining 70% of price.
8.2

Markdowns and Pricing Decisions

Learning objectives

After this section you will be able to:

  1. Calculate a markdown and reduced price.
  2. Measure markup remaining after a sale.
  3. Find an original price from a markdown price.

Real-life situation

A seasonal tote sale

Use this hypothetical Philadelphia-area scenario to connect the calculation to a decision. All people, businesses, amounts, and rates in this chapter are invented; tax and payroll rules are teaching assumptions, not current law.

A seasonal tote sale

  • Regular price: $80.00
  • Markdown: 25% = $20.00
  • Sale price: $60.00
  • Unit cost: $45.00; remaining markup: $15.00
Figure 8.2 A seasonal tote sale (hypothetical)

The concept

A markdown reduces a previously established selling price. Its rate is usually based on that regular price, not on the seller's cost. Calculate markdown dollars, round them to cents, and subtract them from the regular price.

A reduced selling price can still exceed cost. Compare the sale price with cost to find remaining markup dollars. That amount is not automatically profit because the seller may still need to cover rent, labor, payment fees, or other expenses.

A sale can also push price below cost. The difference is then a negative markup on that item under the stated cost model. A seller might accept this to clear storage, but the arithmetic should not hide the loss.

To recover a regular price from a sale price, divide by the remaining fraction after markdown. To evaluate several markdowns, apply them sequentially and round each monetary reduction. Compare actual final prices rather than simply adding advertised percentages.

Markdown and remaining markup

Regular price$ 80
Sale price$ 60
Unit cost$ 45
Remaining markup$ 15

Bars share a zero baseline.

Figure 8.2.V Markdown and remaining markup. Hypothetical teaching data.

Money Words

Plain-language words you will use in this lesson — learn them before we reach the formulas.

  • net profit

    The final profit after every cost is paid — sometimes called the bottom line.

  • fixed cost

    A cost that stays the same no matter how many items are sold, like rent.

  • variable cost

    A cost that rises or falls with the number of items made or sold.

Key terms

Markdown
Reduction from a regular selling price.
Regular price
Selling price before the stated markdown.
Markdown rate
Markdown dollars divided by regular price.
Remaining markup
Sale price minus cost.

Formulas & rules

  • Markdown

    Markdown dollars = regular price × decimal markdown rate

  • Reduced price

    Sale price = regular price − markdown dollars

  • Remaining markup

    Remaining markup = sale price − cost

  • Recover regular price

    Regular price = sale price ÷ (1 − decimal markdown rate)

Worked examples

Example 1Check a sale

A tote regularly sells for $80 and costs $45. Find sale price and remaining markup after 25% off.

Solution

  1. 1Markdown: 80 × 0.25 = $20.00.
  2. 2Sale price: 80 − 20 = $60.00.
  3. 3Remaining markup: 60 − 45 = $15.00.

Answer: Sale price $60.00; remaining markup $15.00 before operating expenses.

Example 2Recover a regular price

An item sells for $42 after a 30% markdown. Find its regular price.

Solution

  1. 1Remaining rate: 1 − 0.30 = 0.70.
  2. 2Regular price: 42 ÷ 0.70 = $60.00.
  3. 3Check: 60 × 0.30 = $18.00 markdown; 60 − 18 = $42.00.

Answer: Regular price: $60.00.

Practice exercises 8.2

  1. Reduce a $120 price by 15%.
  2. A $70 item is marked down to $56. Find the markdown rate.
  3. An item costing $35 sells on clearance for $30. Find remaining markup.
  4. A $72 sale price follows a 10% markdown. Find regular price.
  5. Apply successive 25% and 20% markdowns to $100.

Recap quiz 8.2

1. A 20% markdown on $90 is:
2. Cost is $40 and sale price is $52. Remaining markup is:
3. A $45 price after 25% off came from:

Section 8.2 recap

  • Markdowns use the regular selling price as their base.
  • A positive remaining markup is not necessarily net profit.
Answer key with explanations: Section 8.2

Practice exercises

  1. Markdown = 120 × 0.15 = $18.00; sale price = $102.00.
  2. Markdown = 70 − 56 = $14.00; 14 ÷ 70 = 20%.
  3. 30 − 35 = −$5.00; the sale is $5.00 below cost before other expenses.
  4. 72 ÷ 0.90 = $80.00; the markdown was $8.00.
  5. First reduction = $25.00, leaving $75.00; next reduction = $15.00, leaving $60.00. The combined markdown is 40%, not 45%.

Recap quiz

  1. $18: 90 × 0.20 = $18 markdown dollars.
  2. $12: 52 − 40 = $12.
  3. $60: 45 ÷ 0.75 = $60.
8.3

Contribution and Break-Even Analysis

Learning objectives

After this section you will be able to:

  1. Separate fixed and variable costs.
  2. Find contribution per unit and break-even volume.
  3. Calculate volume for a target operating profit.

Real-life situation

A neighborhood food stand plans a day

Use this hypothetical Philadelphia-area scenario to connect the calculation to a decision. All people, businesses, amounts, and rates in this chapter are invented; tax and payroll rules are teaching assumptions, not current law.

A neighborhood food stand plans a day

  • Fixed daily cost: $240.00
  • Price per meal: $10.00
  • Variable cost per meal: $4.00
  • Contribution: $6.00; break-even: 40 meals
Figure 8.3 A neighborhood food stand plans a day (hypothetical)

The concept

Fixed costs do not change with sales volume within the range being modeled. Variable costs change with each unit sold. A one-day permit may be fixed, while ingredients and a container may be variable. Classifications depend on the period and assumptions used.

Contribution per unit is selling price minus variable cost per unit. Each sale contributes this amount toward fixed costs. Only after all fixed costs are covered does additional contribution create operating profit in the model.

Break-even occurs when total revenue equals total fixed and variable costs. Divide fixed costs by positive contribution per unit. If the result is not a whole number, round up when deciding how many whole units must be sold to avoid a loss. If contribution is zero or negative and fixed costs are positive, no sales volume breaks even under these assumptions.

To plan for a target operating profit, add that target to fixed costs before dividing by contribution. This simple model assumes every unit made is sold, price and variable unit cost stay constant, and capacity is sufficient. Taxes, financing, and owner compensation are excluded unless explicitly included as costs.

Break-even: revenue meets total cost

00202004040060600Dollars ($)Meals sold (q)

Solid line: revenue = 10q. Dashed line: total cost = 240 + 4q.

  1. Each meal contributes $10 − $4 = $6 toward fixed cost.
  2. Break-even quantity = $240 ÷ $6 = 40 meals. At 40 meals, revenue and total cost both equal $400.
  3. Below the crossing, cost exceeds revenue; above it, revenue exceeds cost. This linear model assumes constant prices and costs.

Original WealthWiseGuys diagram · Hypothetical teaching quantities.

Money Words

Plain-language words you will use in this lesson — learn them before we reach the formulas.

  • break-even point

    The sales level where revenue exactly equals total cost, so there is no profit and no loss.

  • selling price

    The price a customer pays for an item or service.

  • cost

    The amount paid to make or buy an item before it is sold.

Key terms

Fixed cost
Cost constant over the modeled activity range.
Variable cost
Cost that changes with units sold.
Contribution per unit
Price minus variable cost per unit.
Break-even
Volume at which revenue equals modeled total cost.
Operating profit
Revenue less the modeled fixed and variable costs.

Formulas & rules

  • Contribution

    Contribution per unit = price − variable unit cost

  • Break-even

    Break-even units = fixed cost ÷ contribution per unit

  • Profit

    Operating profit = units × contribution per unit − fixed cost

  • Target volume

    Target units = (fixed cost + target profit) ÷ contribution per unit

Worked examples

Example 1Cover the stand's costs

Fixed daily cost is $240. Meals sell for $10 with $4 variable cost each. Find break-even units and profit at 55 sales.

Solution

  1. 1Contribution: 10 − 4 = $6.00 per meal.
  2. 2Break-even: 240 ÷ 6 = 40 meals.
  3. 3At 55 meals, contribution = 55 × 6 = $330.00.
  4. 4Operating profit: 330 − 240 = $90.00.

Answer: Break-even is 40 meals; 55 meals produce $90.00 operating profit.

Example 2Round required units upward

Fixed cost is $100, price is $9, and variable unit cost is $5. How many sales are needed for at least $50 operating profit?

Solution

  1. 1Contribution: 9 − 5 = $4.00.
  2. 2Required contribution: 100 + 50 = $150.00.
  3. 3Required volume: 150 ÷ 4 = 37.5; round up to 38 units.
  4. 4Check: 38 × 4 − 100 = $52.00 profit; 37 × 4 − 100 = $48.00.

Answer: Sell at least 38 units.

Practice exercises 8.3

  1. Find contribution when price is $15 and variable cost is $9.
  2. Fixed cost is $180 with $6 contribution. Find break-even.
  3. With those costs, find profit at 45 units.
  4. Fixed cost is $250 and contribution is $8. Find minimum whole-unit break-even volume.
  5. Price and variable cost are both $5 with $100 fixed cost. Can higher sales break even?

Recap quiz 8.3

1. Price $12 and variable cost $7 give contribution of:
2. $90 fixed cost and $4 contribution require at least:
3. At break-even, operating profit is:

Section 8.3 recap

  • Positive contribution first covers fixed costs, then produces operating profit.
  • Round required whole-unit volume upward and state the model assumptions.
Answer key with explanations: Section 8.3

Practice exercises

  1. 15 − 9 = $6.00 per unit.
  2. 180 ÷ 6 = 30 units.
  3. 45 × 6 − 180 = 270 − 180 = $90.00.
  4. 250 ÷ 8 = 31.25; round up to 32 units. Profit at 32 is 256 − 250 = $6.00.
  5. Contribution = 5 − 5 = $0.00; profit = units × 0 − 100 = −$100.00 at every volume, so no.

Recap quiz

  1. $5: 12 − 7 = $5.
  2. 23 units: 90 ÷ 4 = 22.5; 23 is the smallest whole-unit volume covering costs.
  3. $0: Revenue exactly covers the modeled fixed and variable costs.

Chapter 8 Organizer and Review

Chapter Organizer

Every section at a glance: key formulas, key terms, and the main idea.

  • 8.1 Markup on Cost and Margin on Selling Price

    Key formulas

    • Markup dollars: Markup = selling price − cost
    • Price from cost markup: Selling price = cost × (1 + decimal markup rate)
    • Margin: Decimal margin = markup ÷ selling price
    • Price from margin: Selling price = cost ÷ (1 − decimal margin)

    Key terms

    Cost, Selling price, Markup dollars, Markup on cost, Margin

    Remember

    Markup and margin use different denominators.

  • 8.2 Markdowns and Pricing Decisions

    Key formulas

    • Markdown: Markdown dollars = regular price × decimal markdown rate
    • Reduced price: Sale price = regular price − markdown dollars
    • Remaining markup: Remaining markup = sale price − cost
    • Recover regular price: Regular price = sale price ÷ (1 − decimal markdown rate)

    Key terms

    Markdown, Regular price, Markdown rate, Remaining markup

    Remember

    Markdowns use the regular selling price as their base.

  • 8.3 Contribution and Break-Even Analysis

    Key formulas

    • Contribution: Contribution per unit = price − variable unit cost
    • Break-even: Break-even units = fixed cost ÷ contribution per unit
    • Profit: Operating profit = units × contribution per unit − fixed cost
    • Target volume: Target units = (fixed cost + target profit) ÷ contribution per unit

    Key terms

    Fixed cost, Variable cost, Contribution per unit, Break-even, Operating profit

    Remember

    Positive contribution first covers fixed costs, then produces operating profit.

Cumulative Review

Mixed practice from this chapter and the two chapters before it.

  1. Section 8.1

    Decision: For that hoodie, what minimum price gives $10 net profit? (Price − 10% fee − $7 − $25 = $10.)

    Show answer

    0.90 × price = 10 + 7 + 25 = $42.00; price = 42 ÷ 0.90 ≈ $46.67.

  2. Section 8.2

    Apply successive 25% and 20% markdowns to $100.

    Show answer

    First reduction = $25.00, leaving $75.00; next reduction = $15.00, leaving $60.00. The combined markdown is 40%, not 45%.

  3. Section 8.3

    Price and variable cost are both $5 with $100 fixed cost. Can higher sales break even?

    Show answer

    Contribution = 5 − 5 = $0.00; profit = units × 0 − 100 = −$100.00 at every volume, so no.

  4. Review of Section 6.3

    Find the price of a $90 jacket after 30% off, before tax.

    Show answer

    Discount = 90 × 0.30 = $27.00; sale price = 90 − 27 = $63.00.

  5. Review of Section 7.3

    Adjust $700 merchandise for an $80 return and a $20 allowance.

    Show answer

    700 − 80 − 20 = $600.00 adjusted merchandise.

All prices, stores, people, and accounts in this chapter are hypothetical and for teaching only. Product references are generic categories; no brand endorses or is endorsed by this textbook.

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