Illinois exam math practice
Appreciation and depreciation: choose the right calculation
A house can rise in market value while its owner takes a tax depreciation deduction. A well-maintained older building can also have an effective age below its actual age. These are different measurements. Before reaching for a formula, identify what the question means by depreciation.
Last reviewed:
The short answer
For market value change, subtract original value from new value, then divide by original value for the percentage change. For one period, multiply by 1 + the appreciation rate or 1 - the decline rate. Compound those factors only when the problem calls for compounding. Appraisal age-life depreciation uses effective age divided by total economic life. Tax depreciation concerns cost recovery and basis, not a direct estimate of market value.
Reviewed October 25, 2026. PSI lists the cost approach at National III.B.3 and real estate calculations in Domain XI. These are supporting skills, not a separate appreciation question quota or a tax-return syllabus. All amounts below are hypothetical. The IRS publications identified in the sources are the editions available when checked, not a claim that a complete 2026 tax-return guide has been issued. Use the method and rounding instruction stated in the problem.
Label the question, then choose the base
- Market change: find the starting value, ending value and period. Dollar change is ending value minus starting value. Divide that signed change by the starting value for the percentage change. A negative result describes a decline; a positive result describes an increase. A zero starting value does not support this percentage formula.
- Forward or reverse: to find a new value, multiply the old value by the stated factor. To recover an old value, divide the new value by that factor. For a decline of d, the factor is 1 - d, with d written as a decimal. Subtracting the same percentage from the new value does not undo an earlier increase.
- Simple or compound: simple change repeatedly uses the original base. Compound change applies each period's factor to the updated value. If rates vary, multiply their separate factors in order. Neither method predicts an actual property's future price.
- Appraisal: in the simplified age-life model, effective age divided by total economic life estimates accrued improvement depreciation. If only effective age and remaining economic life are supplied, total economic life is their sum. Keep land separate. If the question also supplies component or other depreciation adjustments, identify what the age-life estimate already covers before subtracting anything again.
- Tax and sale arithmetic: use the applicable basis and stated capital adjustments. Tax depreciation is not the percentage fall in a property's market price. In a simplified sale problem, amount realized less adjusted basis gives realized gain or loss. That result is separate from closing cash and from the amount ultimately taxable.
- Check the units and assumptions: convert 7.5% to 0.075, distinguish annual from total-period rates, and retain precision until the requested rounding step. A mathematically correct answer still depends on the question giving the right market, appraisal or tax inputs.
| Task | Formula | Check first |
|---|---|---|
| Signed dollar change | New value - original value | Preserve a negative sign for a decline |
| Signed percentage change | (New - original) / original x 100% | Original value must be nonzero |
| One-period appreciation | Original x (1 + r) | r is the decimal increase |
| One-period market decline | Original x (1 - d) | d is the decimal decline |
| Recover original value | New / the applicable factor | Divide by the nonzero factor; do not subtract the same percentage |
| Constant-rate compound change | Original x (1 + r)^n | A decline uses a negative signed r |
| Age-life ratio | Effective age / total economic life | Remaining life alone is not the denominator |
| Simplified adjusted basis | Applicable starting basis + increases - reductions | Use the stated qualifying adjustments |
| Simplified realized gain or loss | Amount realized - adjusted basis | Not the same as cash after mortgage payoff |
The distinctions that matter
| Distinction | How it works | What to notice |
|---|---|---|
| Market depreciation and appraisal depreciation | A market decline compares values across dates. Appraisal depreciation measures a loss from improvement cost new at the valuation date. | A changing sale market and an age-life estimate are different questions. |
| Appraisal and tax depreciation | The first is a valuation concept; the second is tax cost recovery under applicable rules. They need not be the same amount. | Do not plug a tax deduction into an appraisal merely because both use the same word. |
| Actual age and effective age | Actual age is elapsed time. Effective age reflects the condition and utility used in the appraisal model. | Use the age the stated method requires. |
| Total and remaining economic life | Total economic life includes the effective age already used plus the estimated life remaining in this simplified model. | Effective age / (effective age + remaining life), not effective age / remaining life. |
| Land and improvements | Land is valued separately in the cost approach. Bare land is not tax-depreciable, although certain land improvements can be. Land can still lose market value. | Not depreciable for tax does not mean guaranteed appreciation. |
| Basis and market value | Basis follows the applicable tax rules and adjustments; market value follows the relevant market evidence. | A rise or fall in an asking price does not by itself reset tax basis. |
| Amount realized and closing cash | The simplified sale-price-minus-selling-expenses amount differs from what remains after paying a mortgage. | A loan payoff does not become adjusted basis or a selling expense merely because it is paid at closing. |
| Realized and taxable gain | A simplified gain calculation does not settle exclusions, recognition, depreciation-related tax treatment or the final tax bill. | Do not label every calculated gain fully taxable. |
Work through the facts
Calculate an increase once
A property value rises from $320,000 by 7.5% over the stated period.
- $320,000 x 0.075 = $24,000 increase.
- $320,000 + $24,000 = $344,000.
- Equivalently, $320,000 x 1.075 = $344,000.
Answer: New value is $344,000; the increase is $24,000.
Work backward from an appreciated value
After one 5% increase, value is $367,500. What was the original value?
- $367,500 / 1.05 = $350,000.
- Check: $350,000 x 1.05 = $367,500.
Answer: The original value was $350,000.
Compare compound and simple growth
Start with $300,000. Compare two annual 4% compound increases with a stated simple 4% per year on the original base for two years.
- Compound: $300,000 x 1.04 x 1.04 = $324,480.
- Simple: $300,000 x (1 + 0.04 x 2) = $324,000.
- The compound result is $480 higher because the second increase uses the updated base.
Answer: Use $324,480 for the compound case and $324,000 for the stated simple case.
Recover a value before a decline
After one 20% market decline, value is $280,000.
- The remaining-value factor is 1 - 0.20 = 0.80.
- $280,000 / 0.80 = $350,000.
- Adding 20% to $280,000 produces only $336,000, so it does not reverse the decline.
Answer: The original value was $350,000.
Equal percentages do not cancel
A $200,000 value falls 10%, then rises 10% from its reduced level.
- $200,000 x 0.90 = $180,000.
- $180,000 x 1.10 = $198,000.
- The final value is $2,000 below the start, a 1% net decline.
Answer: A 10% fall followed by a 10% rise does not restore the starting value.
Find the increase needed to recover a loss
A value fell from $250,000 to $200,000. What percentage increase from $200,000 restores $250,000?
- The recovery amount is $50,000.
- $50,000 / $200,000 = 0.25 = 25%.
- The earlier decline was $50,000 / $250,000 = 20%; the bases differ.
Answer: A 25% increase from the reduced value reverses the earlier 20% decline.
Keep basis, gain and closing cash separate
The facts supply $300,000 applicable starting basis, $50,000 qualifying capital additions and $30,000 required depreciation reduction. Sale price is $450,000; qualifying selling expenses are $25,000. A separate mortgage payoff is $200,000. No other basis, consideration or closing adjustments apply.
- Adjusted basis: $300,000 + $50,000 - $30,000 = $320,000.
- Amount realized: $450,000 - $25,000 = $425,000.
- Realized gain: $425,000 - $320,000 = $105,000.
- Cash after the stated expenses and payoff: $450,000 - $25,000 - $200,000 = $225,000.
Answer: Realized gain is $105,000, while this simplified closing cash is $225,000. The facts do not establish the final taxable amount.
Use total economic life
Improvement cost new is $600,000. Effective age is 10 years and total economic life is 50 years. Land value is $150,000. The simplified model includes no separate adjustments.
- Age-life ratio: 10 / 50 = 20%.
- Improvement depreciation: $600,000 x 0.20 = $120,000.
- Depreciated improvement: $600,000 - $120,000 = $480,000.
- Cost-approach indication: $150,000 + $480,000 = $630,000.
Answer: The simplified cost-approach indication is $630,000, with land valued separately.
Convert remaining life to total life
Improvement cost new is $400,000. Effective age is 12 years and remaining economic life is 48 years. Apply the simplified age-life model.
- Total economic life: 12 + 48 = 60 years.
- Ratio: 12 / 60 = 20%, not 12 / 48 = 25%.
- Depreciation: $400,000 x 0.20 = $80,000; remaining improvement value is $320,000.
Answer: The model indicates $80,000 accrued depreciation, not $100,000.
Compound a repeated decline
A $500,000 value declines 10% per year for two years under the stated compound assumption.
- $500,000 x 0.90 x 0.90 = $405,000.
- Total decline is $95,000, or 19% of the starting value.
- Simply subtracting 20% would produce $400,000, which belongs to a different, simple-change assumption.
Answer: The compound result is $405,000, a 19% total decline.
Mistakes to avoid
- Dividing a dollar increase by the new value. Use the original base for the stated percentage change.
- Reversing appreciation by subtracting the same percentage. Divide the new value by the growth factor.
- Calling two annual increases a simple total without reading the method. Compound updated values only when the problem specifies that treatment.
- Dividing effective age by remaining life. Add effective age to remaining life to find the total-life denominator.
- Depreciating the land allocation with the building. Keep the land and improvement measurements separate.
- Using an unclaimed deduction as a reason to ignore a basis reduction. IRS basis rules account for allowable depreciation as well as depreciation taken.
- Calling gain the cash left after paying the loan. Compute amount realized and basis separately from mortgage payoff.
- Turning a hypothetical rate into a market forecast. The rate is given for arithmetic, not promised future appreciation.
Check your understanding
These are original teaching questions, not recalled or predicted PSI questions. Answer first, then explain why the other choices fail.
1. A $400,000 value rises 6% once. What is the new value?
- $466,000
- $424,000
- $406,000
- $376,000
Show answer and explanation
Answer: B. $424,000
$400,000 x 1.06 = $424,000. The $24,000 increase is not the requested final value.
2. Value rises from $300,000 to $345,000. What is the percentage increase?
- 115%
- 45%
- 15%
- 13.04%
Show answer and explanation
Answer: C. 15%
The $45,000 increase divided by the original $300,000 is 0.15, or 15%. Dividing by the new value uses the wrong base.
3. After one 20% decline, a value is $280,000. What was it before the decline?
- $224,000
- $336,000
- $350,000
- $300,000
Show answer and explanation
Answer: C. $350,000
$280,000 / 0.80 = $350,000. Adding 20% to the reduced value does not reverse a 20% decline from a different base.
4. A $300,000 property has two stated annual 4% compound increases. What is its final value?
- $324,000
- $324,480
- $348,000
- $312,000
Show answer and explanation
Answer: B. $324,480
$300,000 x 1.04 squared is $324,480. The second increase applies to $312,000, not to the original base.
5. A $200,000 value falls 10% and then rises 10% from the lower amount. What is the result?
- $202,000
- $200,000
- $198,000
- $180,000
Show answer and explanation
Answer: C. $198,000
The factors multiply: $200,000 x 0.90 x 1.10 = $198,000. Equal percentage changes use different dollar bases.
6. A value fell from $250,000 to $200,000. What increase from the reduced value restores the start?
- 80%
- 25%
- 20%
- 5%
Show answer and explanation
Answer: B. 25%
The required $50,000 increase divided by the new $200,000 base is 25%. The earlier loss was 20% of $250,000.
7. A question supplies effective age of 12 years and remaining economic life of 48 years. What is the simplified age-life ratio?
- 12 / 60 = 20%
- 48 / 60 = 80%
- 48 / 12 = 400%
- 12 / 48 = 25%
Show answer and explanation
Answer: A. 12 / 60 = 20%
Total economic life is 12 + 48 = 60 years. The ratio uses effective age divided by that total, not remaining life alone.
8. Cost new is $600,000, effective age is 10 years and total economic life is 50 years. What is the simplified improvement depreciation?
- $150,000
- $480,000
- $60,000
- $120,000
Show answer and explanation
Answer: D. $120,000
10 / 50 = 20%, and $600,000 x 20% = $120,000. The $480,000 figure is the remaining improvement value, not depreciation.
9. Starting basis is $250,000, qualifying additions are $40,000 and the required depreciation reduction is $25,000. With no other adjustments, what is adjusted basis?
- $315,000
- $265,000
- $235,000
- $290,000
Show answer and explanation
Answer: B. $265,000
$250,000 + $40,000 - $25,000 = $265,000. Use the qualifying facts supplied rather than a market estimate or loan balance.
10. A simplified sale has $425,000 amount realized and $320,000 adjusted basis. What does the $105,000 difference establish?
- Cash after every loan payoff
- The final tax bill
- A guaranteed exclusion of the entire gain
- Realized gain on the supplied facts
Show answer and explanation
Answer: D. Realized gain on the supplied facts
Amount realized minus adjusted basis is $105,000 realized gain. Further facts are needed for taxable treatment, and mortgage payoff is a separate closing-cash calculation.
11. An otherwise applicable depreciation deduction was not claimed. Which statement is sound?
- The omitted deduction turns land into a depreciable building
- Basis can never change
- Market appreciation automatically replaces basis
- Allowable depreciation can still require a basis reduction
Show answer and explanation
Answer: D. Allowable depreciation can still require a basis reduction
IRS Publication 551 addresses depreciation that could have been deducted as well as deductions taken. Failure to claim an allowable amount is not automatically permission to keep a higher basis.
12. Which statement correctly separates land, tax and market value?
- Bare land is not tax-depreciable, but its market value can fall
- Tax depreciation is the same as market decline
- Bare land always rises in market value
- Every land-related cost is necessarily nondepreciable
Show answer and explanation
Answer: A. Bare land is not tax-depreciable, but its market value can fall
The land rule concerns tax cost recovery, not a guarantee about prices. Certain land improvements can have separate depreciation treatment; identify the actual asset and applicable rules.
Common questions
What is the real estate appreciation formula?
For one stated increase, new value = original value x (1 + r), where r is the decimal rate. At 6%, use 0.06 and multiply by 1.06. For a dollar increase alone, multiply the original value by r. Label which result the question asks for.
How do I calculate a percentage decline?
Subtract new value from original value for a positive loss amount, then divide by original value. Alternatively, the signed-change formula (new - original) / original gives a negative rate. Be consistent about whether you report a positive decline magnitude or a signed change.
Why do I divide by the original value?
The original value is the base against which the change is being measured. A rise from $300,000 to $345,000 is $45,000 / $300,000 = 15%. Dividing by $345,000 answers a different comparison.
How do I find value before appreciation or depreciation?
Divide the new value by the factor used to reach it. For one 5% increase, divide by 1.05. For one 20% decline, divide by 0.80. With repeated compound periods, divide by the full compound factor. The divisor must be nonzero.
Do appreciation and depreciation at the same percentage cancel?
Usually no. The second percentage applies to the changed amount. A 10% fall followed by a 10% rise multiplies value by 0.90 x 1.10 = 0.99, leaving it 1% below the start. Reversing a loss requires a larger percentage increase from the smaller base.
When should I use compound growth?
Use it when the question specifies compounding or successive percentage changes on updated values. For constant signed periodic rate r over n periods, use original x (1 + r)^n. A stated simple rate on the original base uses a different calculation. Do not silently treat a total-period rate as an annual rate.
What if the percentage rate changes each year?
Multiply the separate period factors. For an increase use 1 + the decimal rate; for a decline use 1 - the decline rate. Do not assume that averaging the quoted rates gives the same final compound value. Use the periods actually supplied.
What are the three meanings of depreciation here?
A market decline compares values over time. Appraisal depreciation concerns loss from improvement cost new at the appraisal date. Tax depreciation recovers qualifying cost under tax rules. A single property can show different results under all three measurements.
What is the age-life depreciation formula?
In the simplified model, accrued improvement depreciation = cost new x effective age / total economic life. If effective age and remaining economic life are given, add them to obtain the denominator. Separate component adjustments require checking what is already included, so the same loss is not counted twice. This teaching shortcut does not replace market-supported appraisal analysis.
Is effective age the same as actual age?
No. An older building may function and present like a younger one, or the reverse. Use effective age when the method calls for it. Renovation does not automatically make the entire property new or prove a particular market value.
Can land lose value if it cannot be tax-depreciated?
Yes. A tax classification does not protect land from market declines. Keep bare land separate from improvements; some land improvements have their own depreciation treatment. In a cost-approach problem, do not apply a building age-life percentage to the land value.
Can a property appreciate while tax depreciation is taken?
Yes. Market value and tax cost recovery measure different things. Eligibility for a deduction still depends on the asset, use and applicable tax rules; a home used solely for personal activities is not depreciable merely because it gets older.
What is adjusted basis in these exam examples?
Use the applicable starting basis plus qualifying increases minus required reductions supplied in the problem. The examples deliberately identify those inputs. An actual gift, inheritance, conversion or complex sale can require different starting figures or adjustments; purchase price is not a universal answer.
Does an unclaimed depreciation deduction leave basis unchanged?
Not necessarily. IRS Publication 551 requires accounting for depreciation that could have been deducted as well as depreciation taken. Check the applicable amount rather than assuming that failing to claim it preserves basis.
Is realized gain the seller's cash after paying the mortgage?
No. The simplified gain formula compares amount realized with adjusted basis. A mortgage payoff affects closing cash. Exclusions, recognition and other applicable tax rules then determine tax treatment; a calculated gain is not itself a final tax bill.
How much should I round on the Illinois exam?
Follow the actual item's instruction. For these exercises, keep intermediate precision and round only the requested final result. This page does not claim a universal PSI rounding rule or establish which calculator or writing tools are enabled for a particular delivery method.
Does the page predict future appreciation or the number of exam questions?
No. Rates and values are fictional teaching inputs. PSI identifies relevant valuation and calculation content, but it does not promise a separate question count for this page. These original practice results are not a validated readiness score.
What should I write before solving a question?
Name the requested result, starting base, period and type of depreciation. Then write the formula with labels before substituting numbers. Afterward, ask whether an increase went up, a decline went down and a reverse calculation reproduces the supplied result.
Keep practising
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Put real estate math into practice
Set up the calculation, keep the units consistent, and check the worked answer.
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Official sources and scope
Checked October 25, 2026. PSI supplies the topic scope. The DRE reference illustrates age-life arithmetic; the current Fannie Mae supplement separates improvement depreciation from site value. IRS Publications 551 (12/2025), 946 (2025) and 523 (2025) support the limited tax distinctions. These editions are not labeled as full 2026 tax-return instructions. No example is a property appraisal, investment forecast or individual tax calculation.
- PSI June 24, 2026 outline: valuation and calculations
- IRS Publication 551: adjusted basis and depreciation reductions
- IRS Publication 946: qualifying use and land
- IRS Publication 523: sale arithmetic before tax treatment
- California DRE reference chapter 15: effective-age and remaining-life examples
- Fannie Mae UAD 3.6 supplement: improvement depreciation and site value
- NIST Guide to the SI: percent represents 0.01
This is exam preparation, not advice for a specific transaction. State law and the applicable facts control. Our editorial and corrections process explains how to report an issue.