Depreciation: practice problems with step-by-step solutions

Free Depreciation practice problems with diagrams, hints and full worked solutions. New values every time, at three levels: straight-line, double-declining balance, sum of the years’ digits.

Free practice problemEasy
0€20 000123BV = ?45yearBook value at the end of each year

A delivery van costs €20 000, has a useful life of 5 years and a salvage value of €0. Using straight-line depreciation, find the book value after 3 years.

cost = €20 000life = 5salvage = €0
€
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What you practise

  1. Easystraight-line
  2. Mediumdouble-declining balance
  3. Hardsum of the years’ digits

Key formulas

Straight-line
(cost − salvage)/life
Double-declining balance
2/life × book value at the start of the year
Sum of the years’ digits
(cost − salvage) × remaining life/SYD
SYD
n(n + 1)/2
Book value
cost − accumulated depreciation

Learn Depreciation from scratch

1Why assets are depreciated

A company buys a delivery van for €20 000. It will use the van for 5 years and then sell it for about €2000.

Charging the full €20 000 as a cost in the year of purchase would make that year look terrible and the next four look too good. The van helps earn revenue in all five years.

Depreciation spreads the cost of an asset over its useful life. Each year a part of the cost appears as an expense, the depreciation charge D.

Four terms come back in every question:

  • Cost: the price paid for the asset.
  • Salvage value (or residual value): what the asset is expected to be worth at the end of its life.
  • Useful life n: the number of years the asset will be used.
  • Book value BV: cost minus all depreciation so far (the accumulated depreciation). It is the value shown on the balance sheet.

Only cost − salvage is used up, so that is the total amount to depreciate. Here that is €18 000.

Self-check A machine cost €10 000. So far €3000 of depreciation has been charged. What is its book value?
  1. €10 000
  2. €13 000
  3. €3000
  4. €7000

Answer: €7000. Book value = cost − accumulated depreciation = 10 000 − 3000 = €7000. Adding the depreciation (€13 000) goes the wrong way: an asset loses value as it is used.

2Straight-line depreciation

The simplest method charges the same amount every year:

D = (cost − salvage) / n

For the van: D = (20 000 − 2000) / 5 = €3600 per year.

After k years the book value is:

BVk = cost − k · D

After 3 years: BV = 20 000 − 3 · 3600 = €9200. After 5 years the book value equals the salvage value, €2000.

Self-check An asset costs €10 000, has a salvage value of €2000 and a life of 4 years. What is the straight-line depreciation per year?
  1. €2000
  2. €2500
  3. €3000
  4. €8000

Answer: €2000. (10 000 − 2000) / 4 = 8000 / 4 = €2000. €2500 forgets to subtract the salvage value, €3000 adds it, and €8000 is the total over all four years.

3Accelerated methods

Many assets lose most of their value early. Accelerated methods charge more in the first years and less later.

Double-declining balance (DDB) uses a fixed rate of 2/n on the book value at the start of each year:

D = (2/n) · BVstart of year

The salvage value is not subtracted first. It only acts as a floor: the book value never goes below it.

Sum of the years’ digits (SYD) depreciates cost − salvage with a fraction that counts down. For a life of n years, add the digits 1 + 2 + … + n:

SYD = n(n + 1)/2
Dk = (cost − salvage) · (n − k + 1) / SYD

For the van, SYD = 5 · 6 / 2 = 15. Year 1 uses 5/15, year 2 uses 4/15, and so on: €6000, €4800, €3600, €2400, €1200.

tBV012345€20000€2000

The figure shows the van’s book value. The dashed line is straight-line: an even fall. The solid line is DDB: a steep fall at first that flattens out. Both end at the salvage value.

Self-check An asset has a useful life of 5 years. What is the yearly rate for double-declining balance?
  1. 10%
  2. 20%
  3. 40%
  4. 50%

Answer: 40%. The straight-line rate is 1/5 = 20%, and DDB doubles it: 2/5 = 40%. 20% is the straight-line rate, not the doubled one.

4The method step by step

  1. Write down cost, salvage value, life n and the year k that is asked.
  2. Straight-line: D = (cost − salvage)/n, the same every year. Book value: cost − k · D.
  3. Double-declining balance: rate 2/n. Build a table year by year: charge = rate × book value at the start of the year, then subtract it.
  4. Sum of the years’ digits: SYD = n(n + 1)/2. The charge in year k is (cost − salvage) · (n − k + 1)/SYD.
  5. Check: the charges never take the book value below the salvage value.

5A worked example

Follow every step. Try to predict the next one before you open it.

Worked example

A delivery van costs €20 000, has a useful life of 5 years and a salvage value of €2000. Using the double-declining-balance method, find the depreciation charge D in year 3.

1

Find the rate

rate = 2/n = 2/5 = 0.4

The salvage value is not subtracted: DDB works on the full book value.

2

Year 1

The book value at the start is the cost, €20 000.

D1 = 0.4 · 20 000 = 8000
BV = 20 000 − 8000 = 12 000
3

Year 2

D2 = 0.4 · 12 000 = 4800
BV = 12 000 − 4800 = 7200
4

Year 3

D3 = 0.4 · 7200 = 2880

The book value becomes 7200 − 2880 = €4320.

5

Check the answer

€4320 is still above the salvage value of €2000, so the full charge is allowed. The charges fall each year (8000, 4800, 2880), as DDB should.

Answer

D = €2880 in year 3.

6Common mistakes

  • Forgetting to subtract the salvage value in straight-line and SYD.
  • Subtracting the salvage value in DDB. There the rate applies to the full book value.
  • In DDB, applying the rate to the original cost every year. It applies to the book value at the start of that year.
  • In SYD, counting the fraction up instead of down: year 1 uses n/SYD, the largest fraction.
  • Confusing the charge for one year with the book value or the accumulated depreciation.
Self-check A machine costs €10 000, has a life of 5 years and a salvage value of €1000. With double-declining balance, what is the charge in year 2?
  1. €1800
  2. €2160
  3. €2400
  4. €4000

Answer: €2400. Year 1: 0.4 · 10 000 = 4000, book value 6000. Year 2: 0.4 · 6000 = €2400. €4000 is the year-1 charge, €2160 subtracts the salvage value first, and €1800 is straight-line.

7How exam questions look

  • Easy: straight-line. Find the yearly depreciation, or the book value after a number of years.
  • Medium: double-declining balance. Find the charge in a given year, working through the book value year by year.
  • Hard: sum of the years’ digits. Find SYD, then the fraction for the year asked.

Write a small table with the year, the book value at the start, the charge and the book value at the end. It prevents most mistakes.

Learn it step by step in the appLearn it step by step in the app

Worked examples

Example 1Easy
0€50 000123BV = ?45678yearBook value at the end of each year

A machine costs €50 000, has a useful life of 8 years and a salvage value of €0. Using straight-line depreciation, find the book value after 3 years.

cost = €50 000life = 8salvage = €0

Find BV (€)

Show hints
  1. Straight-line: the same amount every year.
  2. Annual depreciation = (cost − salvage)/life.
  3. Book value = cost − accumulated depreciation.
Show the answer and the step-by-step solution

Depreciation spreads the cost of an asset over its useful life. Straight-line: (cost − salvage value)/life every year. Double-declining balance: 2/life times the book value at the start of each year, so the charge falls each year (salvage is not subtracted, but the book value never goes below it). Sum-of-the-years’-digits: (cost − salvage) × remaining life/(1 + 2 + … + n).

Annual depreciation

D = (50 000 − 0)/8 = €6 250.00

Book value

BV = 50 000 − 3·6 250.00 = €31 250.00

Answer

BV = €31 250.00
Example 2Medium
0€120 00012345678salvage €12 000D = ?yearBook value at the end of each year

A computer system costs €120 000, has a useful life of 8 years and a salvage value of €12 000. Using the double-declining-balance method (without switching to straight-line), find the depreciation charge D in year 6.

cost = €120 000life = 8salvage = €12 000year = 6

Find D (€)

Show hints
  1. Rate = 2/life = 25% per year.
  2. Each year: depreciation = rate × book value at the start of the year.
  3. book value at the start of year 6: €28 476.56.
Show the answer and the step-by-step solution

Depreciation spreads the cost of an asset over its useful life. Straight-line: (cost − salvage value)/life every year. Double-declining balance: 2/life times the book value at the start of each year, so the charge falls each year (salvage is not subtracted, but the book value never goes below it). Sum-of-the-years’-digits: (cost − salvage) × remaining life/(1 + 2 + … + n).

Rate

2/8 = 0.25

Year by year

year 1: 0.25·120 000.00 = €30 000.00
year 2: 0.25·90 000.00 = €22 500.00
year 3: 0.25·67 500.00 = €16 875.00
year 4: 0.25·50 625.00 = €12 656.25
year 5: 0.25·37 968.75 = €9 492.19
year 6: 0.25·28 476.56 = €7 119.14

Answer

D = €7 119.14
Example 3Hard
0€45 0001234salvage €4 500D = ?yearBook value at the end of each year

A delivery van costs €45 000, has a useful life of 4 years and a salvage value of €4 500. Using the sum-of-the-years’-digits method, find the depreciation charge D in year 2.

cost = €45 000life = 4salvage = €4 500year = 2

Find D (€)

Show hints
  1. Sum of the years’ digits: SYD = n(n + 1)/2 = 10.
  2. Remaining life at the start of year 2: 3 years.
  3. Depreciable amount: €40 500.
Show the answer and the step-by-step solution

Depreciation spreads the cost of an asset over its useful life. Straight-line: (cost − salvage value)/life every year. Double-declining balance: 2/life times the book value at the start of each year, so the charge falls each year (salvage is not subtracted, but the book value never goes below it). Sum-of-the-years’-digits: (cost − salvage) × remaining life/(1 + 2 + … + n).

Fraction for year 2

(4 − 2 + 1)/10 = 3/10

Depreciation

D = 40 500·3/10 = €12 150.00

Answer

D = €12 150.00

Questions and answers

What is Depreciation?

A company buys a delivery van for €20 000. It will use the van for 5 years and then sell it for about €2000.

Charging the full €20 000 as a cost in the year of purchase would make that year look terrible and the next four look too good. The van helps earn revenue in all five years.

How do I solve Depreciation problems step by step?

Depreciation spreads the cost of an asset over its useful life. Straight-line: (cost − salvage value)/life every year. Double-declining balance: 2/life times the book value at the start of each year, so the charge falls each year (salvage is not subtracted, but the book value never goes below it). Sum-of-the-years’-digits: (cost − salvage) × remaining life/(1 + 2 + … + n).

What are common mistakes in Depreciation?
  • That is the first year’s charge. Each year the rate applies to the book value at the start of that year, so the charge falls.
  • Declining balance applies the rate to the book value; the salvage value is not subtracted first.
  • The fraction counts down: year 1 uses n/SYD, year 2 uses (n − 1)/SYD, and so on.
What do the Depreciation exercises cover?
  • Easy: straight-line
  • Medium: double-declining balance
  • Hard: sum of the years’ digits

Every problem has new values, so you learn the method instead of an answer. Stuck? Open a small hint first, and the full explanation only when you need it.

How practising works

  • Every problem has new values, so you learn the method instead of an answer.
  • Stuck? Open a small hint first, and the full explanation only when you need it.
  • Your progress, mistakes and saved problems are kept, so you can review before the exam.

Calculators

Depreciation calculator

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Printable worksheet

Depreciation worksheet with answers
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