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WealthWiseGuys Practical Mathematics

Chapter 17

Depreciation and Business Assets

Allocate asset cost, compare methods, and read book value

Equipment can help a business for years even when the purchase is paid for today. Depreciation allocates that cost across the periods or output receiving the benefit; it does not predict resale prices. All assets, businesses, estimates, and amounts are hypothetical, and the methods here are accounting examples rather than current tax rules; round money to cents.

17.1

Asset Cost and Straight-Line Depreciation

Learning objectives

After this section you will be able to:

  1. Determine an asset's depreciable cost basis.
  2. Calculate straight-line depreciation and book value.
  3. Distinguish accounting allocation from market value.

Real-life situation

A shared workshop buys equipment

A hypothetical Philadelphia workshop buys a machine for $9,000, pays $600 delivery and $400 installation, and expects five years of service with a $1,000 residual value. It is ready for use at the start of the first year.

Machine record

  • Purchase price: $9,000
  • Delivery: $600; installation: $400
  • Recorded cost: $10,000
  • Estimated residual value: $1,000
  • Estimated useful life: 5 years
  • Straight-line depreciation: $1,800 per full year
Figure 17.1 Hypothetical asset cost and depreciation estimates

The concept

A long-lived business asset helps operations beyond the current period. Its recorded cost generally includes the purchase price and necessary costs to bring it to the location and condition required for use. This illustration capitalizes delivery and installation; routine later maintenance is treated separately.

Depreciable amount is recorded cost minus estimated residual value. Residual value is the expected amount remaining at the end of the useful life under the estimates used. Neither the life nor the residual is a promise; actual outcomes may differ.

Straight-line depreciation allocates the same depreciable amount to each full year. Accumulated depreciation adds all depreciation recorded so far, while book value equals cost minus that accumulated amount. Under unchanged estimates, the scheduled ending book value is the residual value, not necessarily zero.

Depreciation is an allocation of cost, not an appraisal and not a cash savings account. A machine's market value may rise or fall differently from its book value. For partial years, this section uses an explicitly stated months-in-service fraction; real policies and tax rules may use different conventions.

Spread depreciable cost over useful life

  1. 1

    Asset cost

    $9,000 + $600 + $400 = $10,000

  2. 2

    Depreciable base

    $10,000 − $1,000 residual = $9,000

  3. 3

    Annual expense

    $9,000 ÷ 5 years = $1,800

Figure 17.1.V Spread depreciable cost over useful life. Hypothetical teaching data.

Money Words

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

  • depreciation

    The way a costly item, such as a machine, loses value and its cost is spread over its useful life.

  • asset

    Something valuable that a person or business owns, like cash, equipment, or a building.

  • liability

    Money or obligations a person or business owes to others.

Key terms

Capitalized cost
An amount recorded as part of an asset rather than expensed immediately.
Useful life
The estimated period over which an asset benefits the business.
Residual value
The estimated value remaining at the end of the useful life.
Depreciable amount
Recorded cost minus estimated residual value.
Accumulated depreciation
Total depreciation recorded on an asset to date.
Book value
Recorded asset cost minus accumulated depreciation in this simplified model.

Formulas & rules

  • Recorded cost

    Cost = purchase price + necessary delivery + necessary installation

  • Straight-line expense

    Annual depreciation = (cost − residual value) ÷ useful life

  • Book value

    Book value = cost − accumulated depreciation

  • Partial-year expense

    Partial-year depreciation = annual depreciation × months in service ÷ 12

    The convention used in these examples.

Worked examples

Example 1Build the annual expense

Find the workshop machine's recorded cost and annual straight-line depreciation.

Solution

  1. 1Recorded cost: $9,000 + $600 + $400 = $10,000.
  2. 2Depreciable amount: $10,000 − $1,000 = $9,000.
  3. 3Annual expense: $9,000 ÷ 5 = $1,800.

Answer: Record the machine at $10,000 and allocate $1,800 per full year.

Example 2Read book value after three years

Find accumulated depreciation and book value after three full years.

Solution

  1. 1Annual straight-line depreciation is $1,800.
  2. 2Accumulated depreciation: 3 × $1,800 = $5,400.
  3. 3Book value: $10,000 − $5,400 = $4,600.
  4. 4Remaining depreciable amount: $4,600 − $1,000 = $3,600.

Answer: Accumulated depreciation is $5,400 and book value is $4,600, leaving two $1,800 full-year allocations.

Practice exercises 17.1

  1. Find book value after two full years.
  2. The machine is first used halfway through a year. Under the stated six-month convention, find first-year depreciation.
  3. Find accumulated depreciation and book value after five full years.
  4. What percent of recorded cost is the $1,000 residual estimate?
  5. A routine $200 maintenance bill occurs in year 2. Under the stated policy, does it increase the machine's recorded cost?

Recap quiz 17.1

1. The machine's depreciable amount is:
2. Straight-line depreciation on $9,000 over five years is:
3. Book value necessarily equals:

Section 17.1 recap

  • Depreciate cost minus residual, not automatically the entire cost.
  • Book value, market value, and cash on hand are different quantities.
Answer key with explanations: Section 17.1

Practice exercises

  1. Accumulated depreciation = 2 × $1,800 = $3,600. Book value = $10,000 − $3,600 = $6,400.
  2. $1,800 × 6 ÷ 12 = $900. Book value at that year-end is $10,000 − $900 = $9,100.
  3. Accumulated depreciation = 5 × $1,800 = $9,000. Book value = $10,000 − $9,000 = $1,000, the estimated residual.
  4. $1,000 ÷ $10,000 × 100 = 10%. The remaining 90%, or $9,000, is depreciable.
  5. No. Record the $200 as maintenance expense under this simplified policy; cost stays $10,000 and scheduled annual depreciation stays $1,800.

Recap quiz

  1. $9,000: $10,000 cost − $1,000 residual = $9,000.
  2. $1,800: $9,000 ÷ 5 = $1,800.
  3. Cost minus accumulated depreciation in this model: Book value is an accounting amount, not a market-price estimate.
17.2

Units-of-Production Depreciation

Learning objectives

After this section you will be able to:

  1. Calculate depreciation per unit of expected output.
  2. Allocate expense according to actual production.
  3. Track cumulative depreciation without crossing residual value.

Real-life situation

A machine works harder in some years

Use the same $10,000 machine and $1,000 residual value, but assume expected lifetime output is 45,000 units. A units-of-production method connects expense to actual use rather than equal calendar years.

PeriodUnitsDepreciationEnding book value
Year 112000$2,400$7,600
Year 215000$3,000$4,600
Remaining expected output18000$3,600 remaining$1,000 after full allocation
Figure 17.2 Hypothetical output-based depreciation record

The concept

Units-of-production depreciation is useful when wear or benefit is better represented by output than by elapsed time. The activity measure might be units produced, machine hours, or another consistent measure. Estimated lifetime activity must use the same unit as actual activity.

Divide depreciable amount by total expected activity to get a per-unit allocation. Multiply that rate by the period's actual activity. Keep enough rate precision for the schedule; premature rounding can distort total expense when expected output is large.

The method can produce unequal annual expenses without changing the original total depreciable amount. A busier year records more depreciation under this method, while straight-line records equal full-year amounts under unchanged estimates. Neither method is automatically a prediction of selling price.

Track cumulative activity and accumulated depreciation. Under unchanged estimates, total depreciation must not exceed cost minus residual. Unexpected additional use does not justify pushing book value below the residual in this simplified schedule; revised estimates require a separate, clearly explained calculation.

Units-of-Production Depreciation: solution map

  1. 1

    Identify the question

    Find the depreciation rate per unit and the first year's expense for 12,000 units.

  2. 2

    Calculation 1

    Depreciable amount: $10,000 − $1,000 = $9,000.

  3. 3

    Calculation 2

    Per-unit amount: $9,000 ÷ 45,000 = $0.20.

  4. 4

    Calculation 3

    First-year depreciation: 12,000 × $0.20 = $2,400.

  5. 5

    Calculation 4

    Ending book value: $10,000 − $2,400 = $7,600.

  6. 6

    Interpret and check

    Allocate $0.20 per unit; year 1 expense is $2,400 and book value is $7,600.

Figure 17.2.V Units-of-Production Depreciation: solution map. Hypothetical teaching data.

Money Words

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

  • equity

    What is left of an asset after debts are subtracted; in a home, value minus what you still owe.

  • balance sheet

    A snapshot listing what a business owns (assets) and owes (liabilities) on one date.

  • cash flow

    The movement of money into and out of a business or account.

Key terms

Units-of-production method
Depreciation based on actual activity relative to estimated lifetime activity.
Activity base
The measurable use, such as units produced or machine hours.
Per-unit depreciation
Depreciable amount allocated to one unit of activity.
Cumulative output
Total units produced from the start through the current period.
Remaining depreciable amount
Cost minus residual minus depreciation already recorded.

Formulas & rules

  • Per-unit allocation

    Depreciation per unit = (cost − residual value) ÷ estimated total units

  • Period expense

    Period depreciation = actual units × depreciation per unit

    Cap expense at the remaining depreciable amount.

  • Remaining allocation

    Remaining depreciable amount = cost − residual value − accumulated depreciation

Worked examples

Example 1Allocate the first year's use

Find the depreciation rate per unit and the first year's expense for 12,000 units.

Solution

  1. 1Depreciable amount: $10,000 − $1,000 = $9,000.
  2. 2Per-unit amount: $9,000 ÷ 45,000 = $0.20.
  3. 3First-year depreciation: 12,000 × $0.20 = $2,400.
  4. 4Ending book value: $10,000 − $2,400 = $7,600.

Answer: Allocate $0.20 per unit; year 1 expense is $2,400 and book value is $7,600.

Example 2Continue the output record

The machine produces 15,000 units in year 2. Find expense, book value, and remaining expected units.

Solution

  1. 1Year 2 depreciation: 15,000 × $0.20 = $3,000.
  2. 2Accumulated depreciation: $2,400 + $3,000 = $5,400.
  3. 3Book value: $10,000 − $5,400 = $4,600.
  4. 4Cumulative output: 12,000 + 15,000 = 27,000 units.
  5. 5Remaining expected output: 45,000 − 27,000 = 18,000 units.

Answer: Year 2 expense is $3,000, book value is $4,600, and 18,000 expected units remain.

Practice exercises 17.2

  1. Find year 1's remaining expected units after 12,000 units.
  2. A separate year has 8,000 units of output with sufficient depreciable amount remaining. Find expense.
  3. After the first two years, find the remaining depreciable amount.
  4. Compare first-year units-of-production expense with the $1,800 straight-line expense.
  5. After the first two years, actual additional output is 20,000 units. Under unchanged estimates, apply the depreciation cap.

Recap quiz 17.2

1. $9,000 depreciable amount over 45,000 units gives:
2. At $0.20 per unit, 15,000 units produce expense of:
3. With unchanged estimates, total depreciation is limited to:

Section 17.2 recap

  • Match the activity measure in the rate with the activity actually recorded.
  • Track accumulated depreciation so the allocation does not exceed its cap.
Answer key with explanations: Section 17.2

Practice exercises

  1. 45,000 − 12,000 = 33,000 units remain under the original lifetime estimate.
  2. 8,000 × $0.20 = $1,600.
  3. $10,000 − $1,000 − $5,400 = $3,600. This also equals 18,000 expected units × $0.20.
  4. $2,400 − $1,800 = $600 more under units of production. The difference changes timing, not the original $9,000 total depreciable amount.
  5. Uncapped expense = 20,000 × $0.20 = $4,000. Only $3,600 remains depreciable, so record $3,600 and leave book value at $1,000.

Recap quiz

  1. $0.20 per unit: $9,000 ÷ 45,000 = $0.20 per unit.
  2. $3,000: 15,000 × $0.20 = $3,000.
  3. Cost minus residual: The method allocates the depreciable amount, leaving the estimated residual.
17.3

Declining Balance and Asset Disposal

Learning objectives

After this section you will be able to:

  1. Construct a capped double-declining-balance schedule.
  2. Compare accelerated and straight-line expense timing.
  3. Calculate a gain or loss when an asset is sold.

Real-life situation

More expense early, less later

For a separate method comparison, depreciate the $10,000 machine using double-declining balance over five years. Residual remains $1,000; this illustration uses no switch to straight-line and adjusts the last year's expense to reach residual.

YearBeginning book valueExpenseEnding book value
1$10,000$4,000$6,000
2$6,000$2,400$3,600
3$3,600$1,440$2,160
4$2,160$864$1,296
5$1,296$296 adjusted$1,000
Figure 17.3 Hypothetical double-declining-balance schedule

The concept

An accelerated depreciation method records more expense early than later under typical conditions. Double-declining balance starts with twice the straight-line rate, then applies that rate to beginning book value rather than repeatedly to original cost.

For a five-year life, the straight-line rate is 1 ÷ 5 = 20%, so the double-declining rate is 40%. Do not subtract residual before multiplying book value by this rate. Instead, residual acts as a floor that limits the expense allowed.

A pure declining-balance calculation may not land exactly on residual at the end of the estimated life. Figure 17.3 therefore states a specific final-year adjustment to allocate the remaining amount. Other schedules may switch to straight-line earlier; identifying the convention is part of solving the problem.

When an asset is sold, compare net proceeds with its book value on the sale date after recording depreciation through that date. Proceeds above book value create an accounting gain; proceeds below it create a loss. This gain or loss does not equal the total lifetime cash profit from owning the asset, and any tax consequences depend on rules outside this illustration.

Declining Balance and Asset Disposal: solution map

  1. 1

    Identify the question

    Find the rate and the first two years of double-declining depreciation.

  2. 2

    Calculation 1

    Rate: 2 ÷ 5 = 0.40, or 40%.

  3. 3

    Calculation 2

    Year 1 expense: $10,000 × 0.40 = $4,000.

  4. 4

    Calculation 3

    Year 1 ending book value: $10,000 − $4,000 = $6,000.

  5. 5

    Calculation 4

    Year 2 expense: $6,000 × 0.40 = $2,400.

  6. 6

    Calculation 5

    Year 2 ending book value: $6,000 − $2,400 = $3,600.

  7. 7

    Interpret and check

    The rate is 40%; expenses are $4,000 and $2,400, leaving $3,600 after year 2.

Figure 17.3.V Declining Balance and Asset Disposal: solution map. Hypothetical teaching data.

Money Words

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

  • cost

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

  • profit

    What is left when expenses are subtracted from revenue.

  • overhead

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

Key terms

Accelerated depreciation
A method that generally allocates more depreciation in earlier periods.
Double-declining balance
A method applying twice the straight-line rate to beginning book value.
Residual floor
The minimum scheduled book value under the stated unchanged estimates.
Net sale proceeds
Sale receipts minus direct selling costs.
Gain on disposal
Net proceeds exceeding book value at disposal.
Loss on disposal
Book value exceeding net proceeds at disposal.

Formulas & rules

  • Double-declining rate

    Double-declining rate = 2 ÷ useful life

  • Ordinary period calculation

    Depreciation = beginning book value × double-declining rate

    Apply the residual floor and the stated final-year convention.

  • Final-year adjustment

    Final-year depreciation = beginning book value − residual value

    Used in this chapter's no-switch schedule.

  • Disposal result

    Gain or loss = net sale proceeds − book value

    Positive is a gain; negative is a loss.

Worked examples

Example 1Begin an accelerated schedule

Find the rate and the first two years of double-declining depreciation.

Solution

  1. 1Rate: 2 ÷ 5 = 0.40, or 40%.
  2. 2Year 1 expense: $10,000 × 0.40 = $4,000.
  3. 3Year 1 ending book value: $10,000 − $4,000 = $6,000.
  4. 4Year 2 expense: $6,000 × 0.40 = $2,400.
  5. 5Year 2 ending book value: $6,000 − $2,400 = $3,600.

Answer: The rate is 40%; expenses are $4,000 and $2,400, leaving $3,600 after year 2.

Example 2Sell an asset at a gain or loss

For a separate disposal example, a machine has $10,000 cost and $5,400 accumulated depreciation at sale. Compare net proceeds of $5,000 and $4,000.

Solution

  1. 1Book value: $10,000 − $5,400 = $4,600.
  2. 2At $5,000 net proceeds: $5,000 − $4,600 = $400.
  3. 3The positive $400 result is an accounting gain.
  4. 4At $4,000 net proceeds: $4,000 − $4,600 = −$600, an accounting loss.

Answer: $5,000 net proceeds create a $400 gain; $4,000 net proceeds create a $600 loss.

Practice exercises 17.3

  1. Calculate year 3 of the double-declining schedule.
  2. Calculate year 4 of the schedule.
  3. Find the final-year adjusted expense and explain why ordinary multiplication is not used.
  4. Compare first-year double-declining expense with first-year straight-line expense.
  5. An asset with $4,600 book value sells for $5,000, but selling costs are $200. Find the gain or loss.

Recap quiz 17.3

1. The double-declining rate for a five-year life is:
2. Ordinary double-declining expense uses:
3. $4,000 net proceeds minus $4,600 book value gives:

Section 17.3 recap

  • Accelerated depreciation changes expense timing, not the original depreciable total.
  • Compare net proceeds with updated book value to determine a disposal gain or loss.
Answer key with explanations: Section 17.3

Practice exercises

  1. Expense = $3,600 × 0.40 = $1,440. Ending book value = $3,600 − $1,440 = $2,160.
  2. Expense = $2,160 × 0.40 = $864. Ending book value = $2,160 − $864 = $1,296.
  3. Final expense = $1,296 − $1,000 = $296. Ordinary 40% expense would be $518.40 and push book value to $777.60, below the residual floor.
  4. $4,000 − $1,800 = $2,200 more expense under double-declining balance. Both schedules still allocate $9,000 in total under the stated full-life conventions.
  5. Net proceeds = $5,000 − $200 = $4,800. Gain = $4,800 − $4,600 = $200.

Recap quiz

  1. 40%: 2 ÷ 5 = 0.40 = 40%.
  2. Beginning book value: The rate is applied to the declining beginning book value.
  3. A $600 loss: $4,000 − $4,600 = −$600, so the disposal produces a $600 loss.

Chapter 17 Organizer and Review

Chapter Organizer

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

  • 17.1 Asset Cost and Straight-Line Depreciation

    Key formulas

    • Recorded cost: Cost = purchase price + necessary delivery + necessary installation
    • Straight-line expense: Annual depreciation = (cost − residual value) ÷ useful life
    • Book value: Book value = cost − accumulated depreciation
    • Partial-year expense: Partial-year depreciation = annual depreciation × months in service ÷ 12

    Key terms

    Capitalized cost, Useful life, Residual value, Depreciable amount, Accumulated depreciation, Book value

    Remember

    Depreciate cost minus residual, not automatically the entire cost.

  • 17.2 Units-of-Production Depreciation

    Key formulas

    • Per-unit allocation: Depreciation per unit = (cost − residual value) ÷ estimated total units
    • Period expense: Period depreciation = actual units × depreciation per unit
    • Remaining allocation: Remaining depreciable amount = cost − residual value − accumulated depreciation

    Key terms

    Units-of-production method, Activity base, Per-unit depreciation, Cumulative output, Remaining depreciable amount

    Remember

    Match the activity measure in the rate with the activity actually recorded.

  • 17.3 Declining Balance and Asset Disposal

    Key formulas

    • Double-declining rate: Double-declining rate = 2 ÷ useful life
    • Ordinary period calculation: Depreciation = beginning book value × double-declining rate
    • Final-year adjustment: Final-year depreciation = beginning book value − residual value
    • Disposal result: Gain or loss = net sale proceeds − book value

    Key terms

    Accelerated depreciation, Double-declining balance, Residual floor, Net sale proceeds, Gain on disposal, Loss on disposal

    Remember

    Accelerated depreciation changes expense timing, not the original depreciable total.

Cumulative Review

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

  1. Section 17.1

    A routine $200 maintenance bill occurs in year 2. Under the stated policy, does it increase the machine's recorded cost?

    Show answer

    No. Record the $200 as maintenance expense under this simplified policy; cost stays $10,000 and scheduled annual depreciation stays $1,800.

  2. Section 17.2

    After the first two years, actual additional output is 20,000 units. Under unchanged estimates, apply the depreciation cap.

    Show answer

    Uncapped expense = 20,000 × $0.20 = $4,000. Only $3,600 remains depreciable, so record $3,600 and leave book value at $1,000.

  3. Section 17.3

    An asset with $4,600 book value sells for $5,000, but selling costs are $200. Find the gain or loss.

    Show answer

    Net proceeds = $5,000 − $200 = $4,800. Gain = $4,800 − $4,600 = $200.

  4. Review of Section 15.3

    Reproduce month 2 in Figure 15.3.

    Show answer

    Interest = $179,820.81 × 0.005 = $899.10405 → $899.10. Principal = $1,079.19 − $899.10 = $180.09. Balance = $179,820.81 − $180.09 = $179,640.72.

  5. 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.

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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