WealthWiseGuys Practical Mathematics

Chapter 18

Inventory, Overhead, and Operating Costs

From the stockroom count to the break-even point

A busy shop can sell plenty and still lose money if stock disappears or costs are overlooked. Learn to track inventory, share overhead, and test an operating plan. All businesses, people, prices, and cost assumptions in this chapter are hypothetical; round monetary answers to cents.

18.1

Inventory Counts, Cost, and Turnover

Learning objectives

After this section you will be able to:

  1. Reconcile goods available with ending inventory.
  2. Compute cost of goods sold and shrinkage.
  3. Use average inventory to calculate turnover.

Real-life situation

Counting a neighborhood pop-up

A hypothetical Philadelphia student cooperative sells plain tote bags. Its stock sheet separates goods sold from bags still on the shelf; every bag has the same $8 cost.

ItemAmount or quantity
Beginning inventory100 bags at $8
Purchases150 bags at $8
Recorded sales180 bags
Physical ending count65 bags
Figure 18.1 Tote-bag stock sheet (hypothetical)

The concept

Inventory is merchandise held for sale. A physical count tells a business what it actually has; a stock record tells it what it expects to have. Compare both in units before assigning dollar values. In the figure, 100 + 150 − 180 = 70 bags should remain, but only 65 are counted.

For this simplified model, goods available at cost equals beginning inventory plus net purchases. Net purchases means purchases after returns and discounts, including any assigned freight-in. Cost of goods sold, or COGS, is the cost assigned to merchandise sold, not the selling price charged to customers.

A shortage between expected and counted inventory is shrinkage. Here 70 − 65 = 5 bags are missing, a $40 cost. Businesses investigate counting errors, damage, and theft rather than assuming every shortage is theft. Keep shrinkage separate from recorded sales cost in this lesson; subtracting physical ending inventory from goods available would combine both.

Turnover compares a period's COGS with average inventory at cost. Using the simple mean of beginning and ending inventory gives a useful estimate, but seasonal businesses may need monthly averages. Higher turnover can mean efficient selling or dangerously low stock, so compare similar businesses over the same period. Round money to cents and turnover to two decimal places.

Inventory Counts, Cost, and Turnover: solution map

  1. 1

    Identify the question

    Use the figure to find recorded COGS, physical ending value, and shrinkage.

  2. 2

    Calculation 1

    Available units = 100 + 150 = 250; available cost = 250 × $8 = $2,000.00.

  3. 3

    Calculation 2

    Recorded COGS = 180 × $8 = $1,440.00.

  4. 4

    Calculation 3

    Expected ending units = 250 − 180 = 70; physical ending cost = 65 × $8 = $520.00.

  5. 5

    Calculation 4

    Shrinkage = (70 − 65) × $8 = $40.00.

  6. 6

    Calculation 5

    Check: $1,440 + $520 + $40 = $2,000.

  7. 7

    Interpret and check

    Recorded COGS is $1,440.00, ending inventory is $520.00, and shrinkage is $40.00.

Figure 18.1.V Inventory Counts, Cost, and Turnover: solution map. Hypothetical teaching data.

Money Words

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

  • inventory

    The stock of goods a business holds ready to sell.

  • overhead

    Ongoing business costs such as rent, utilities, and insurance that do not change with each sale.

  • fixed cost

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

Key terms

Inventory
Goods held for sale, measured in units or at cost.
Net purchases
Purchase cost after returns and discounts, plus assigned freight-in.
Cost of goods sold
Inventory cost assigned to goods actually sold.
Shrinkage
Expected inventory minus actual inventory, when positive.
Inventory turnover
Period COGS divided by average inventory at cost.

Formulas & rules

  • Goods available

    Available cost = beginning inventory cost + net purchases

  • Recorded sales cost

    COGS = units sold × cost per unit (equal-cost goods)

  • Shrinkage

    Shrinkage cost = (expected units − counted units) × unit cost

  • Turnover

    Turnover = COGS ÷ [(beginning inventory + ending inventory) ÷ 2]

Worked examples

Example 1Reconcile the tote bags

Use the figure to find recorded COGS, physical ending value, and shrinkage.

Solution

  1. 1Available units = 100 + 150 = 250; available cost = 250 × $8 = $2,000.00.
  2. 2Recorded COGS = 180 × $8 = $1,440.00.
  3. 3Expected ending units = 250 − 180 = 70; physical ending cost = 65 × $8 = $520.00.
  4. 4Shrinkage = (70 − 65) × $8 = $40.00.
  5. 5Check: $1,440 + $520 + $40 = $2,000.

Answer: Recorded COGS is $1,440.00, ending inventory is $520.00, and shrinkage is $40.00.

Example 2Measure turnover

A shop has $3,000 beginning inventory, $5,000 ending inventory, and $24,000 annual COGS. Find annual turnover.

Solution

  1. 1Average inventory = ($3,000 + $5,000) ÷ 2 = $4,000.00.
  2. 2Turnover = $24,000 ÷ $4,000 = 6.
  3. 3The annual COGS is six times the average inventory investment.

Answer: Annual inventory turnover is 6.00 times.

Practice exercises 18.1

  1. A booth begins with 40 hats, buys 75, and sells 90. How many should remain?
  2. The booth counts 22 hats instead of 25. Each cost $6. Find shrinkage.
  3. Beginning inventory costs $700; net purchases cost $1,100. Ending inventory costs $500, with no shrinkage. Find COGS.
  4. Beginning and ending inventory are $2,000 and $4,000. Annual COGS is $15,000. Find turnover.
  5. Ten equal-cost cases cost $240 total; three remain with no loss. Find unit cost and COGS.

Recap quiz 18.1

1. Which amount values goods at cost rather than selling price?
2. Expected stock is 48 units; the count is 45. Each costs $9. Shrinkage is:
3. COGS is $18,000 and average inventory is $3,000. Turnover is:

Section 18.1 recap

  • Reconcile units first, then multiply by cost; keep shrinkage separate from sales.
  • Turnover uses COGS and average inventory at cost for the same period.
Answer key with explanations: Section 18.1

Practice exercises

  1. Available = 40 + 75 = 115 hats; expected ending = 115 − 90 = 25 hats.
  2. Missing units = 25 − 22 = 3; shrinkage cost = 3 × $6 = $18.00.
  3. Available cost = $700 + $1,100 = $1,800; COGS = $1,800 − $500 = $1,300.00.
  4. Average = ($2,000 + $4,000) ÷ 2 = $3,000; turnover = $15,000 ÷ $3,000 = 5.00 times per year.
  5. Unit cost = $240 ÷ 10 = $24; units sold = 10 − 3 = 7; COGS = 7 × $24 = $168.00.

Recap quiz

  1. COGS: COGS assigns acquisition cost to the merchandise sold.
  2. $27: (48 − 45) × $9 = $27.00.
  3. 6 times: $18,000 ÷ $3,000 = 6.00 times for the stated period.
18.2

Overhead and Cost Allocation

Learning objectives

After this section you will be able to:

  1. Distinguish direct costs from overhead.
  2. Allocate shared overhead using a stated activity base.
  3. Calculate a full unit cost without double-counting.

Real-life situation

Sharing a kitchen

Two hypothetical food stalls share a Philadelphia preparation kitchen. They allocate $1,200 of monthly kitchen overhead by preparation hours, not by the number of meals sold.

ItemAmount or quantity
Stall A preparation hours80
Stall B preparation hours40
Shared kitchen overhead$1,200.00
Stall A meals400
Stall A direct cost per meal$4.00
Figure 18.2 Monthly kitchen allocation (hypothetical)

The concept

Direct costs can be traced to a chosen product or job, such as ingredients used for a meal. Overhead supports operations but cannot conveniently be traced to a single unit, such as shared kitchen rent. The same cost can be direct for one purpose and indirect for another; always state the cost object.

An allocation base is a measurable activity used to divide a shared cost. Preparation hours may be reasonable for a kitchen; floor area might fit rent, and machine hours might fit equipment costs. A fair-looking base is not automatically a perfect measure of cost caused.

Divide the overhead pool by total activity to get an overhead rate. Then multiply each job's activity by that rate. Allocations must add back to the full overhead pool, aside from any disclosed rounding difference. Use a single consistent base within an allocation.

Full unit cost combines direct unit cost with allocated overhead per unit. This is useful for planning prices, but allocating fixed rent to a meal does not mean the rent disappears if one fewer meal is made. Do not add allocated overhead again after already including it in full cost. Round final money to cents; retain the unrounded allocation rate when necessary.

Overhead and Cost Allocation: solution map

  1. 1

    Identify the question

    Use the figure to allocate the $1,200 pool.

  2. 2

    Calculation 1

    Total preparation time = 80 + 40 = 120 hours.

  3. 3

    Calculation 2

    Overhead rate = $1,200 ÷ 120 = $10.00 per hour.

  4. 4

    Calculation 3

    Stall A receives 80 × $10 = $800.00.

  5. 5

    Calculation 4

    Stall B receives 40 × $10 = $400.00; check $800 + $400 = $1,200.

  6. 6

    Interpret and check

    Allocate $800.00 to A and $400.00 to B.

Figure 18.2.V Overhead and Cost Allocation: solution map. Hypothetical teaching data.

Money Words

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

  • variable cost

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

  • cost

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

  • revenue

    The total money a business brings in from sales, before any costs are paid.

Key terms

Direct cost
A cost traceable to the product or job being measured.
Overhead
Indirect operating costs shared by products or jobs.
Cost pool
The total shared cost to be allocated.
Allocation base
The activity measure used to divide a cost pool.
Full unit cost
Direct unit cost plus allocated overhead per unit.

Formulas & rules

  • Overhead rate

    Rate = overhead pool ÷ total allocation-base units

  • Allocated overhead

    Job overhead = rate × job activity

  • Full unit cost

    Full unit cost = direct unit cost + (allocated overhead ÷ units made)

Worked examples

Example 1Split the shared kitchen bill

Use the figure to allocate the $1,200 pool.

Solution

  1. 1Total preparation time = 80 + 40 = 120 hours.
  2. 2Overhead rate = $1,200 ÷ 120 = $10.00 per hour.
  3. 3Stall A receives 80 × $10 = $800.00.
  4. 4Stall B receives 40 × $10 = $400.00; check $800 + $400 = $1,200.

Answer: Allocate $800.00 to A and $400.00 to B.

Example 2Price information, not a price rule

Find Stall A's full cost for each of 400 meals, using $4 direct cost per meal and $800 allocated overhead.

Solution

  1. 1Total direct cost = 400 × $4 = $1,600.00.
  2. 2Overhead per meal = $800 ÷ 400 = $2.00.
  3. 3Full unit cost = $4 + $2 = $6.00.
  4. 4Check: 400 × $6 = $2,400 = $1,600 + $800.

Answer: The full unit cost is $6.00; this alone does not determine an appropriate selling price.

Practice exercises 18.2

  1. Allocate $900 of overhead over 300 machine hours. What is the rate?
  2. At $3 per machine hour, how much overhead goes to a 45-hour job?
  3. A batch has $250 direct cost and $150 allocated overhead for 50 units. Find full unit cost.
  4. Allocate $600 rent between rooms of 200 and 400 square feet.
  5. A seller counts $5 direct cost and $2 allocated overhead per item, then adds the same $2 overhead again. Correct the cost.

Recap quiz 18.2

1. A $2,400 pool covers 600 labor hours. The overhead rate is:
2. Which is a plausible base for allocating shared floor rent?
3. Direct unit cost is $7 and overhead per unit is $3. Full unit cost is:

Section 18.2 recap

  • Allocate the whole pool with an explicit, reasonable base.
  • Full cost informs planning; allocated fixed costs are not necessarily avoidable.
Answer key with explanations: Section 18.2

Practice exercises

  1. Rate = $900 ÷ 300 = $3.00 per machine hour.
  2. Allocated overhead = 45 × $3 = $135.00.
  3. Full batch cost = $250 + $150 = $400; unit cost = $400 ÷ 50 = $8.00.
  4. Total area = 200 + 400 = 600 square feet; rate = $600 ÷ 600 = $1 per square foot. Allocations are $200.00 and $400.00, totaling $600.00.
  5. Correct full cost = $5 + $2 = $7.00, not $9.00. The extra $2 duplicates the allocation.

Recap quiz

  1. $4 per hour: $2,400 ÷ 600 = $4.00 per hour.
  2. Floor area occupied: Floor area measures use of the shared space.
  3. $10: $7 + $3 = $10.00, counting overhead once.
18.3

Operating Profit and Break-Even Planning

Learning objectives

After this section you will be able to:

  1. Separate fixed and variable operating costs.
  2. Compute contribution margin and operating profit.
  3. Find whole-unit break-even and target-profit sales.

Real-life situation

A weekend print booth

A hypothetical booth near a Philadelphia community event sells posters. The simplified monthly model assumes constant selling price and variable cost, enough capacity, and no sales tax in revenue.

ItemAmount or quantity
Price per poster$20.00
Variable cost per poster$8.00
Monthly fixed costs$600.00
Planned sales80
Figure 18.3 Poster booth monthly plan (hypothetical)

The concept

Operating profit is revenue minus the costs of running the business. In a simplified cost-volume-profit model, variable costs change in proportion to units sold, while total fixed costs remain unchanged within the relevant activity range. A monthly booth fee can stay fixed even when no posters sell.

Contribution margin per unit is price minus variable cost per unit. It first covers fixed costs; only after those costs are covered does additional contribution become operating profit. It is not the same as net income, which may also reflect interest, income taxes, and other items excluded here.

Break-even occurs when total contribution equals fixed costs. Divide fixed costs by contribution margin per unit. If products can only be sold whole, round the required unit count up, never to the nearest integer. A nonpositive contribution margin cannot cover positive fixed costs in this model.

To plan for a target operating profit, add that target to fixed costs before dividing. A plan is not a guarantee: price cuts, overtime, waste, or capacity limits can change the inputs. Use consistent time periods and round money to cents; do not round contribution margin before calculating required units.

Operating Profit and Break-Even Planning: solution map

  1. 1

    Identify the question

    Use the figure to find break-even units and operating profit at 80 posters.

  2. 2

    Calculation 1

    Contribution per poster = $20 − $8 = $12.00.

  3. 3

    Calculation 2

    Break-even = $600 ÷ $12 = 50 posters.

  4. 4

    Calculation 3

    At 80 posters, contribution = 80 × $12 = $960.00.

  5. 5

    Calculation 4

    Operating profit = $960 − $600 = $360.00.

  6. 6

    Interpret and check

    Break-even is 50 posters; operating profit at 80 is $360.00.

Figure 18.3.V Operating Profit and Break-Even Planning: solution map. Hypothetical teaching data.

Money Words

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

  • profit margin

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

  • break-even point

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

  • net profit

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

Key terms

Fixed cost
A total cost unchanged within the stated activity range and period.
Variable cost
A cost whose total changes with the number of units.
Contribution margin
Revenue minus variable costs, available to cover fixed costs and profit.
Break-even point
Sales volume at which operating profit is zero.
Operating profit
Revenue less operating costs in the defined model.

Formulas & rules

  • Unit contribution

    Contribution per unit = price − variable cost per unit

  • Operating profit

    Profit = units × contribution per unit − fixed costs

  • Break-even units

    Break-even units = fixed costs ÷ contribution per unit; round up

  • Target-profit units

    Required units = (fixed costs + target profit) ÷ contribution per unit; round up

Worked examples

Example 1Can the booth cover its costs?

Use the figure to find break-even units and operating profit at 80 posters.

Solution

  1. 1Contribution per poster = $20 − $8 = $12.00.
  2. 2Break-even = $600 ÷ $12 = 50 posters.
  3. 3At 80 posters, contribution = 80 × $12 = $960.00.
  4. 4Operating profit = $960 − $600 = $360.00.

Answer: Break-even is 50 posters; operating profit at 80 is $360.00.

Example 2Round up to meet a goal

A seller earns $7 contribution per item and has $500 fixed costs. Find sales needed for at least $200 operating profit.

Solution

  1. 1Required contribution = $500 + $200 = $700.00.
  2. 2Required units = $700 ÷ $7 = 100.
  3. 3Check: 100 × $7 − $500 = $200.00.

Answer: Sell at least 100 items.

Practice exercises 18.3

  1. A snack sells for $5 and costs $2 in variable costs. Find contribution per snack.
  2. With $250 fixed costs and $3 contribution per snack, find whole-unit break-even sales.
  3. A booth sells 120 items at $10 each, variable cost $6 each, with $300 fixed cost. Find profit.
  4. Find units needed for $450 profit with $350 fixed costs and $8 contribution per unit.
  5. Price is $9, variable cost is $10, and fixed cost is $100. Can greater sales produce break-even?

Recap quiz 18.3

1. A $15 item has $9 variable cost. Unit contribution is:
2. Fixed cost is $101 and contribution is $10 per unit. Minimum whole-unit sales to avoid a loss?
3. At 70 units, $5 contribution each and $200 fixed costs, profit is:

Section 18.3 recap

  • Contribution covers fixed costs before it produces profit.
  • Round required whole-unit sales upward and check the model assumptions.
Answer key with explanations: Section 18.3

Practice exercises

  1. Contribution = $5 − $2 = $3.00 per snack.
  2. $250 ÷ $3 = 83.333...; round up to 84 snacks. Profit at 84 = 84 × $3 − $250 = $2.00, the first nonnegative whole-unit result.
  3. Revenue = 120 × $10 = $1,200; variable costs = 120 × $6 = $720; profit = $1,200 − $720 − $300 = $180.00.
  4. Required contribution = $350 + $450 = $800; units = $800 ÷ $8 = 100.
  5. Unit contribution = $9 − $10 = −$1. Profit = units × (−$1) − $100, which remains negative for all nonnegative sales. No break-even is possible without changing the inputs.

Recap quiz

  1. $6: $15 − $9 = $6.00.
  2. 11: $101 ÷ $10 = 10.1; round up to 11. Ten units leave a $1 loss.
  3. $150: 70 × $5 − $200 = $150.00.

Chapter 18 Organizer and Review

Chapter Organizer

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

  • 18.1 Inventory Counts, Cost, and Turnover

    Key formulas

    • Goods available: Available cost = beginning inventory cost + net purchases
    • Recorded sales cost: COGS = units sold × cost per unit (equal-cost goods)
    • Shrinkage: Shrinkage cost = (expected units − counted units) × unit cost
    • Turnover: Turnover = COGS ÷ [(beginning inventory + ending inventory) ÷ 2]

    Key terms

    Inventory, Net purchases, Cost of goods sold, Shrinkage, Inventory turnover

    Remember

    Reconcile units first, then multiply by cost; keep shrinkage separate from sales.

  • 18.2 Overhead and Cost Allocation

    Key formulas

    • Overhead rate: Rate = overhead pool ÷ total allocation-base units
    • Allocated overhead: Job overhead = rate × job activity
    • Full unit cost: Full unit cost = direct unit cost + (allocated overhead ÷ units made)

    Key terms

    Direct cost, Overhead, Cost pool, Allocation base, Full unit cost

    Remember

    Allocate the whole pool with an explicit, reasonable base.

  • 18.3 Operating Profit and Break-Even Planning

    Key formulas

    • Unit contribution: Contribution per unit = price − variable cost per unit
    • Operating profit: Profit = units × contribution per unit − fixed costs
    • Break-even units: Break-even units = fixed costs ÷ contribution per unit; round up
    • Target-profit units: Required units = (fixed costs + target profit) ÷ contribution per unit; round up

    Key terms

    Fixed cost, Variable cost, Contribution margin, Break-even point, Operating profit

    Remember

    Contribution covers fixed costs before it produces profit.

Cumulative Review

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

  1. Section 18.1

    Ten equal-cost cases cost $240 total; three remain with no loss. Find unit cost and COGS.

    Show answer

    Unit cost = $240 ÷ 10 = $24; units sold = 10 − 3 = 7; COGS = 7 × $24 = $168.00.

  2. Section 18.2

    A seller counts $5 direct cost and $2 allocated overhead per item, then adds the same $2 overhead again. Correct the cost.

    Show answer

    Correct full cost = $5 + $2 = $7.00, not $9.00. The extra $2 duplicates the allocation.

  3. Section 18.3

    Price is $9, variable cost is $10, and fixed cost is $100. Can greater sales produce break-even?

    Show answer

    Unit contribution = $9 − $10 = −$1. Profit = units × (−$1) − $100, which remains negative for all nonnegative sales. No break-even is possible without changing the inputs.

  4. Review of Section 16.3

    Receivables increase $700 instead of $500. Recompute operating cash flow.

    Show answer

    $800 + $200 − $700 + $300 = $600. The additional $200 uncollected revenue lowers operating cash by $200.

  5. Review of Section 17.3

    Calculate year 3 of the double-declining schedule.

    Show answer

    Expense = $3,600 × 0.40 = $1,440. Ending book value = $3,600 − $1,440 = $2,160.

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.