Illinois real estate math
Real estate math formulas to memorize
A formula is useful only if you can choose its inputs. Start with the short list below, then practise the questions that change the base, time period or requested result. Those changes cause more trouble than the multiplication.
Last reviewed:
The short answer
Learn these formula families: percentages and compensation; property taxes and prorations; loan-to-value, equity and mortgage points; area and acreage; gross rent multiplier, net operating income and capitalization; and buyer funds and seller proceeds. Add simple interest only when the problem calls for it. Use decimal rates, matching units and the definitions supplied. A memorized equation cannot choose the right value, debt balance or closing adjustment for you.
Reviewed October 26, 2026. PSI assigns 7 of 100 scored national broker questions to the whole Real Estate Calculations area. This is an original study reference, not an official required-memorization list. Measurement and valuation also appear elsewhere in the outline. The examples use stated teaching assumptions, not a local tax bill, lender approval or closing quote.
Six habits that make a formula easier to recall
- Write the requested answer first. Dollars, square feet, acres, a percentage and a closing debit are different results. Cross out numbers that do not belong in that calculation.
- Name the base. A percentage needs a clearly identified whole: the loan for points, the stated compensation base for a fee, or the original amount for a percentage change. A second split may use a smaller subtotal.
- Write units beside the inputs. Convert percent to decimal, monthly to annual when needed, and all lengths to a common unit before finding area. Do not mix a rate per $100 with a rate per $1,000.
- Derive the reverse form. From part = base x rate, divide part by the nonzero rate to find base, or by the nonzero base to find rate. From annual NOI = value x cap rate, divide by the appropriate nonzero input.
- Carry precision and apply the actual final rule. Ordinary final rounding, a form instruction to round a dollar upward, and a tax rule to raise a partial $500 unit are three different operations.
- Check the answer in the original relationship. Multiply a recovered base by its rate, reconcile both closing ledgers, or convert acres back to square feet. On a missed question, write whether the problem was the formula, base, unit, posting or arithmetic.
| Find | Relationship | Before calculating |
|---|---|---|
| Percentage amount | Part = base x decimal rate | 6% = 0.06 |
| Unknown base or rate | Base = part / rate; rate = part / base | The divisor cannot be zero |
| Agreed compensation and splits | Fee = stated base x rate; share = assigned subtotal x split | No mandatory standard rate |
| Property tax | Supplied taxable assessed base x aggregate decimal rate | Build the stipulated assessment/EAV/exemption chain first |
| A tax rate quoted per $100 or in mills | Tax = base / 100 x quoted dollars; or base / 1,000 x mills | One mill is $1 per $1,000 of taxable value |
| Prorated share | Periodic charge / stated period x responsible time | Use the supplied day count and closing-day allocation |
| LTV percentage | Included loan / required value x 100 | Use the actual program or problem definition |
| Current equity | Stated current value - included current secured debt | Not automatically seller cash or borrowing capacity |
| Mortgage-point charge | Loan amount x number of points / 100 | 1.25 points = 1.25% of the loan |
| PITI | Monthly principal + interest + property taxes + insurance | A separately requested HOA charge is additional |
| Simple interest | Interest = principal x annual decimal rate x fraction of a year | Only the stated simple-interest model and day-count basis |
| Rectangle and triangle area | Rectangle = length x width; triangle = base x perpendicular height / 2 | Area is in square units |
| Acres and nominal survey area | Acres = square feet / 43,560; nominal section = 640 acres | A standard survey township has 36 sections; actual survey controls |
| Gross rent multiplier | GRM = price / gross rent; indicated price = rent x matching GRM | Match monthly or annual rent; do not substitute NOI |
| Annual NOI and capitalization | NOI = effective gross income - defined operating expenses; value = annual NOI / cap rate | Use a decimal cap rate; debt service is separate; reserve treatment follows the method |
| Repeated percentage changes | Ending amount = starting amount x each successive multiplier | A 10% rise and 10% fall do not cancel |
| Buyer cash and seller proceeds | Buyer cash = uses - funding and credits; seller cash = credits - debits | Separate parties and count each deposit, credit and payoff once |
| Illinois state transfer tax | Completed $500 units, including any partial unit, x $0.50 | Start with the determined nonexempt taxable base; other taxes are separate |
The distinctions that matter
| Distinction | How it works | What to notice |
|---|---|---|
| Property value, assessed value and taxable assessed value | They are not interchangeable. Most Illinois property uses a one-third assessment level, but Cook County classifications, farmland and other preferential rules differ. Equalization and exemptions can change the taxable base. Use the supplied facts rather than assigning every parcel one rate or assessment percentage. | Write the tax chain before applying the rate. |
| Loan-to-value and current equity | For ordinary Fannie Mae purchase LTV, the denominator is the lower price or appraisal, subject to specified exceptions. Follow the actual problem's inputs. First-loan LTV and current equity can use different debts or value bases. | A ratio is not automatically the complement of total equity. |
| A point charge and an interest-rate change | One mortgage point costs 1% of the loan amount. That does not promise a one-percentage-point reduction in the interest rate. The pricing effect varies, and some fees described as points do not reduce the rate. | Compute the charge from the loan. |
| Gross rent and net operating income | A gross multiplier uses its defined gross rent or income basis. Direct capitalization uses annual NOI. Build effective gross income, then deduct the operating expenses required by the stated method. Do not deduct loan principal or interest, income tax or tax depreciation to find property NOI. Replacement-reserve treatment depends on the method. | Match income level, period and valuation method. |
| A proration amount and its closing entry | The calculation gives a share, not its debit or credit direction. If the buyer will later pay an accrued bill covering the seller's assigned period, the usual supplied adjustment is a seller debit and buyer credit. A prepaid item can require the opposite direction. The contract and actual facts matter. | Ask who pays the bill and who owes the share. |
| Nearest rounding and two upward rules | For a taxable base of $250,001, the state transfer-tax $500-unit rule requires 501 units. Separately, PTAX-203 Step 2 directs lines 11 through 18 to be rounded up to the next whole dollar. Neither rule is permission to round every intermediate math step upward. | Read which quantity the rule changes. |
Work through the facts
Recover the price, then follow two splits
An agreed 5.5% fee is $19,800. One firm receives 40% of that fee. A licensee receives 65% of that firm's share. These are supplied terms, not standard rates.
- Price: $19,800 / 0.055 = $360,000.
- Firm share: $19,800 x 0.40 = $7,920.
- Licensee share: $7,920 x 0.65 = $5,148.
Answer: The price is $360,000; the licensee share is $5,148. Applying 65% to the original fee would use the wrong base.
Keep the property-tax units visible
A simplified problem supplies $300,000 market value, exactly one-third assessment, a 1.02 equalization factor, a $10,000 exemption and an aggregate rate of $7.50 per $100 of taxable assessed value. The exemption and rate are hypothetical inputs.
- Assessed value: $300,000 / 3 = $100,000. EAV: $100,000 x 1.02 = $102,000.
- Taxable assessed base: $102,000 - $10,000 = $92,000.
- Tax: $92,000 / 100 x $7.50 = $6,900. Equivalently, $92,000 x 0.075 = $6,900.
Answer: The supplied annual tax is $6,900. Do not apply the rate to the original market value or treat the example as a county rule.
Calculate a share before posting it
The question gives a $7,300 annual bill, a 365-day method and 120 days assigned to the seller. The bill is unpaid, and the buyer will pay it after closing.
- Daily amount: $7,300 / 365 = $20.
- Seller's share: $20 x 120 = $2,400.
- Because the buyer will pay this accrued seller obligation, the supplied adjustment debits the seller and credits the buyer.
Answer: Seller debit $2,400 and buyer credit $2,400. The 120 days are given; no universal Illinois closing-day convention is assumed.
Separate points, PITI and an additional charge
The loan is $320,000 with 1.25 points. Monthly principal and interest are given as $1,800, annual property taxes as $3,600 and annual homeowners insurance as $1,200. A separate $150 monthly HOA charge also applies. No other insurance component is supplied.
- Points: $320,000 x 0.0125 = $4,000.
- Monthly PITI: $1,800 + $3,600 / 12 + $1,200 / 12 = $2,200.
- If asked for the stated housing total including HOA: $2,200 + $150 = $2,350.
Answer: Points cost $4,000. PITI is $2,200; the requested total including HOA is $2,350. Simple-interest multiplication is not how the given amortized payment was calculated.
Build annual NOI before capitalizing
Annual potential gross income is $110,000, vacancy and collection loss is $5,000, and other property income is $3,000. Defined operating expenses total $38,000, including every expense and reserve item required by this problem. The stated cap rate is 8%. Debt service is separately $12,000.
- Effective gross income: $110,000 - $5,000 + $3,000 = $108,000.
- NOI: $108,000 - $38,000 = $70,000. Do not subtract the $12,000 debt service to find NOI.
- Indicated value under the supplied direct-cap method: $70,000 / 0.08 = $875,000.
Answer: Annual NOI is $70,000 and the indicated value is $875,000. The 8% rate is a problem input, not a current market recommendation.
Distinguish area from survey assumptions
Find the area of a 220-foot by 198-foot rectangle, a triangle with a 100-foot base and 80-foot perpendicular height, and the nominal NE quarter of the SW quarter of a section. Use a nominal 640-acre section.
- Rectangle: 220 x 198 = 43,560 square feet = 1 acre.
- Triangle: 100 x 80 / 2 = 4,000 square feet.
- Nested quarters: 640 x 1/4 x 1/4 = 40 nominal acres.
Answer: The results are 1 acre, 4,000 square feet and 40 nominal acres. The nested-quarter result is not an acreage guarantee for an actual surveyed parcel.
Raise the statutory unit, not every dollar
The already determined, whole-dollar taxable value for a nonexempt transaction is $250,001. Calculate only Illinois state transfer tax. No county or municipal tax is requested.
- $250,001 / $500 = 500.002 units. Any fraction requires another unit: 501.
- 501 x $0.50 = $250.50.
- At exactly $250,000 taxable value, 500 units x $0.50 would be $250.00.
Answer: State tax is $250.50. A county may impose its authorized additional tax; municipal rules can differ. Do not assume those taxes are included or subtract an ordinary payoff automatically.
Match the gross-rent period
The stated gross rent is $2,500 monthly, or $30,000 annually. One comparable-derived multiplier is 132 on a monthly basis; its equivalent annual-basis multiplier is 11. No adjustments or other income are involved in this simplified comparison.
- Monthly approach: $2,500 x 132 = $330,000.
- Annual approach: $30,000 x 11 = $330,000.
- Using annual rent with 132 would give $3,960,000, twelve times the matching result.
Answer: Both correctly matched approaches indicate $330,000. Do not mix monthly and annual inputs, or treat this simplified multiplier as a complete appraisal.
Apply each percentage to its own base
A hypothetical value starts at $200,000, rises 10%, then falls 10% from the new amount. These are stated changes, not a market forecast or tax-depreciation calculation.
- After the rise: $200,000 x 1.10 = $220,000.
- After the fall: $220,000 x 0.90 = $198,000.
- Net change: -$2,000 / $200,000 x 100 = -1%.
Answer: The ending value is $198,000, a 1% overall decline. Equal percentage changes in opposite directions use different bases.
Balance the buyer and seller ledgers
Price is $300,000. Buyer charges are $9,000; new loan funding is $240,000; a separate $10,000 deposit remains available for this closing. The seller pays a $3,000 credit within those buyer charges, a $200,000 payoff, $15,000 agreed compensation and $4,000 separate seller charges. No other adjustments apply.
- Buyer cash: $300,000 + $9,000 - $240,000 - $10,000 - $3,000 = $56,000.
- Seller proceeds: $300,000 - $200,000 - $15,000 - $4,000 - $3,000 = $78,000.
- External funds: $240,000 loan + $10,000 deposit + $56,000 buyer cash = $306,000.
- Disbursements: $78,000 seller + $200,000 payoff + $15,000 compensation + $4,000 seller charges + $9,000 buyer charges = $306,000.
Answer: The buyer brings $56,000 and the seller receives $78,000. The $3,000 credit shifts cost between the parties; it is not an additional external funding source or a second $3,000 charge.
Mistakes to avoid
- Memorize a letter triangle without naming the inputs. Say which base, rate and period each letter represents.
- Turn every given percentage into a multiplier without checking the task. A price decrease uses 1 - rate for the new price; a fee amount uses the rate itself.
- Use one Illinois tax rate or one assessment percentage everywhere. Follow the supplied classification, equalization, exemptions and aggregate rate.
- Use 360 days, 365 days or a tax-proration percentage by habit. Use the stated convention and contract assumptions.
- Count gross rent as NOI or deduct debt service to find NOI. Build the correct income level before selecting GRM or cap rate.
- Call every housing charge PITI. Name the principal, interest, tax and insurance components, then add separately requested charges.
- Treat 640 acres or 36 sections as proof of every actual parcel. Keep nominal survey arithmetic separate from record and survey evidence.
- Treat a score on these drills as a guaranteed exam result. Use each miss to choose what to practise next. These are original questions, not a validated readiness score.
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 fee is $12,600 at a supplied 3.5% rate. What is its base?
- $12,159
- $441,000
- $360,000
- $36,000
Show answer and explanation
Answer: C. $360,000
$12,600 / 0.035 = $360,000. Check by multiplying $360,000 by 0.035.
2. Total compensation is $24,000. A firm gets 30%, and a licensee gets 60% of that firm share. What does the licensee receive?
- $7,200
- $14,400
- $4,320
- $10,080
Show answer and explanation
Answer: C. $4,320
$24,000 x 0.30 = $7,200; $7,200 x 0.60 = $4,320. The second split uses the firm subtotal.
3. Use a stipulated $125,000 taxable base and 24 mills. What is the tax?
- $24,000
- $300
- $30,000
- $3,000
Show answer and explanation
Answer: D. $3,000
$125,000 / 1,000 x 24 = $3,000. A mill is $1 per $1,000, not per $100.
4. Use a supplied $3,600 annual charge, a 360-day method and 45 responsible days. What share results?
- $450
- $443.84
- $4,500
- $810
Show answer and explanation
Answer: A. $450
$3,600 / 360 = $10 per day; $10 x 45 = $450. This amount alone does not determine the posting direction.
5. A problem directs you to use the lower of $400,000 price and $380,000 appraisal. The included loan is $304,000. What is LTV?
- 76%
- 20%
- 95%
- 80%
Show answer and explanation
Answer: D. 80%
$304,000 / $380,000 x 100 = 80%. The problem explicitly supplies the applicable denominator.
6. Current value is $400,000. Included current debt is a $280,000 first loan and a separate $20,000 drawn junior loan. What is equity?
- $100,000
- $80,000
- $300,000
- $120,000
Show answer and explanation
Answer: A. $100,000
$400,000 - ($280,000 + $20,000) = $100,000. This is not automatically sale cash.
7. A $240,000 loan carries 1.5 points. What is the point charge?
- $2,400
- $3,600
- $36,000
- $1,600
Show answer and explanation
Answer: B. $3,600
$240,000 x 1.5 / 100 = $3,600. The loan amount is the base; no fixed interest-rate reduction follows.
8. An explicitly simple-interest problem uses $400,000 principal, 6% annual interest and 30/360 of a year. What interest accrues?
- $2,000
- $2,400
- $20,000
- $24,000
Show answer and explanation
Answer: A. $2,000
$400,000 x 0.06 x 30/360 = $2,000. This is the supplied simple-interest model, not a fully amortized monthly-payment formula.
9. A triangle has a 90-foot base and 60-foot perpendicular height. What is its area?
- 150 square feet
- 2,700 linear feet
- 5,400 square feet
- 2,700 square feet
Show answer and explanation
Answer: D. 2,700 square feet
90 x 60 / 2 = 2,700 square feet. Perpendicular height and square units matter.
10. A rectangle contains 87,120 square feet. How many acres is that?
- 4 acres
- 2 acres
- 0.5 acre
- 20 acres
Show answer and explanation
Answer: B. 2 acres
87,120 / 43,560 = 2 acres.
11. Annual EGI is $120,000, defined operating expenses are $40,000 and separate debt service is $18,000. What is NOI?
- $160,000
- $102,000
- $80,000
- $62,000
Show answer and explanation
Answer: C. $80,000
$120,000 - $40,000 = $80,000. Debt service is outside property NOI.
12. Annual NOI is $48,000 and the stated direct-cap rate is 8%. What value is indicated?
- $6,000,000
- $384,000
- $3,840
- $600,000
Show answer and explanation
Answer: D. $600,000
$48,000 / 0.08 = $600,000. Gross income is not substituted for NOI.
13. A supplied monthly GRM is 120 and the matching gross monthly rent is $2,200. What price is indicated?
- $18.33
- $3,168,000
- $264,000
- $22,000
Show answer and explanation
Answer: C. $264,000
$2,200 x 120 = $264,000. An annual rent input cannot be multiplied by this monthly GRM.
14. Already determined nonexempt whole-dollar taxable value is $180,001. What is Illinois state transfer tax alone?
- $360.50
- $180.50
- $270.00
- $180.00
Show answer and explanation
Answer: B. $180.50
There are 360.002 $500 units, raised to 361. At $0.50 per unit, state tax is $180.50. Other taxes are not requested.
15. Which describes the identified PTAX-203 Step 2 instruction for lines 11 through 18?
- Round amounts up to the next whole dollar.
- Always round every intermediate formula to two decimals.
- Round each taxable amount to the nearest $500.
- Discard every fraction of a dollar.
Show answer and explanation
Answer: A. Round amounts up to the next whole dollar.
The whole-dollar form instruction differs from the statutory partial-$500-unit ceiling. Neither is a general rule for all exam calculations.
16. An amount rises 10%, then falls 10% from the new amount. Starting at $200,000, where does it end?
- $200,000
- $198,000
- $202,000
- $180,000
Show answer and explanation
Answer: B. $198,000
$200,000 x 1.10 x 0.90 = $198,000. The second change uses the updated base.
17. Given monthly principal and interest $1,500, property tax $300, homeowners insurance $100 and separate HOA $200, what is the specified PITI?
- $1,500
- $2,100
- $1,700
- $1,900
Show answer and explanation
Answer: D. $1,900
$1,500 + $300 + $100 = $1,900. The separate HOA raises a requested housing total to $2,100 but is not a PITI component.
18. A reverse formula calls for dividing by a supplied zero rate. What is the sound response?
- Replace zero with 1%.
- Recognize that this reverse division is undefined; reconsider the inputs and relationship.
- Use the largest dollar amount.
- Report zero as the recovered base.
Show answer and explanation
Answer: B. Recognize that this reverse division is undefined; reconsider the inputs and relationship.
Division by zero does not recover a base. Do not invent a replacement rate or report a numeric answer unsupported by the relationship.
Common questions
Which real estate math formulas should I memorize for Illinois?
Start with the percentage relationship and its reverse forms, then compensation, taxes, prorations, LTV, equity, points, PITI, area, gross rent multiplier, annual NOI and capitalization, and party-specific closing cash. Know simple interest when the question expressly uses it. The table is a study aid, not a PSI-issued required list.
Does PSI say there are seven questions on these formulas?
The June 2026 outline assigns seven of the 100 scored national broker questions to the entire Real Estate Calculations area. That is not seven on each formula or a promise about one particular calculation. Measurement and valuation also appear elsewhere in the outline.
Should I memorize every rearrangement?
Learn the parent relationship and practise solving for each input. For part = base x rate, divide the part by a nonzero rate to recover base, or by a nonzero base to recover rate. Label the input before entering the numbers.
How do I enter a percent in a formula?
Divide the displayed percent by 100: 5.5% becomes 0.055. To express a ratio as a percentage, multiply the decimal ratio by 100. Keep percentage points, mortgage points and decimal multipliers distinct.
Which value belongs in LTV?
Use the defined loan and value. Price and appraisal can differ. Program reporting rules do not establish exam rounding.
What is one mortgage point?
One point is 1% of the loan amount. Fractional points work the same way. This charge does not establish a fixed reduction in the interest rate; use actual pricing terms for that separate question.
Can I calculate a mortgage payment using principal x rate x time?
That is a simple-interest relationship under its stated assumptions. It is not the general formula for a fully amortized periodic principal-and-interest payment. Use the supplied payment, factor or payment method when the problem requires one.
Does PITI include an HOA charge?
PITI stands for principal, interest, taxes and insurance. A separately specified HOA charge can be added when the question asks for a broader housing total, but it does not change what the acronym means. Include the insurance components actually given.
Is there one Illinois property-tax formula with one statewide rate?
The multiplication relationship is straightforward, but the taxable base and aggregate rate depend on the property and taxing districts. Illinois has classification and assessment exceptions. Follow the supplied assessment, equalization, exemptions and rate units.
Is one mill the same as one percent?
No. One mill is $1 per $1,000 of taxable value, or 0.001 as a decimal tax rate. One percent is 0.01. A rate quoted per $100 needs its own conversion.
Do all prorations use a 360-day year?
No. Use the question's stated year, day-count method, responsible period and allocation of the closing day. Real contract assumptions can differ. Determine who pays an accrued or prepaid charge before choosing the debit and credit.
What is the difference between GRM and cap rate?
A gross rent multiplier uses its matching gross-rent period and definition. Cap rate relates annual NOI to value. Gross rent is not NOI, and a monthly multiplier cannot be used with annual rent without the corresponding conversion.
Does NOI subtract the mortgage payment or depreciation?
Property NOI excludes loan principal and interest, income tax and tax depreciation. Use the operating-expense definition supplied. Replacement-reserve treatment is method-dependent; do not memorize a blanket rule that every reserve is excluded.
Should I use exactly 640 acres for every section?
Use 640 acres when a problem expressly assumes a nominal section. A standard survey township contains 36 sections. Actual surveyed sections and parcels can differ, so these teaching quantities do not establish legal parcel acreage.
How do Illinois transfer-tax rounding rules work?
For state tax, each $500 of the determined nonexempt taxable base, including a fraction of a $500 unit, incurs $0.50. The PTAX-203 whole-dollar instructions are a separate earlier step. Do not assume every sale is taxable or that an ordinary mortgage payoff automatically reduces the taxable base.
How do I avoid counting a closing credit twice?
Keep buyer and seller ledgers separate. A seller credit reduces buyer cash and reduces seller proceeds, but it is not new money entering the combined settlement. Count each deposit only to the extent still available and applied, and do not add the same covered charge again. In a complete simplified example, external funds should equal disbursements.
Are these official questions or a guarantee that I am ready?
No. They are original learning questions. After a miss, explain the correct base, units and posting, then solve a new problem without looking back. No percentage on this page is an official readiness threshold or a guaranteed exam result.
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Official sources and scope
The June 2026 PSI outline identifies calculation topics. Illinois law and IDOR guidance govern the tax qualifications discussed here. CFPB, OCC and the identified Fannie Mae topic explain financial definitions; the Fannie topic is dated June 2022, not newly published in 2026. Government measurement and appraisal references support the unit and multiplier examples. These original simplified drills do not replace a tax form, survey, lender policy or settlement statement.
- PSI: national real estate calculations outline
- NIST Guide to the SI: percent represents 0.01
- IDOR: assessment levels and property-tax rates
- IDOR PTAX-1004, January 2026: value chain, extension and tax bills
- Michigan Legislature, For the People, page 33: mill definition only
- CFPB: discount points and lender credits
- Fannie Mae: loan-to-value ratios
- 35 ILCS200/31-10: state transfer-tax unit and base
- IDOR PTAX-203 instructions: form-specific rounding and taxable amount
- 55 ILCS 5/5-1031: county tax and exemption links
- OCC Commercial Real Estate Lending handbook: income and value analysis
- California DRE Reference Book, chapter 15, page 419: gross multipliers
- OpenStax Prealgebra 2e 9.4: rectangles, triangles and trapezoids
- NIST Handbook44,2026 AppendixC: exact length, area and volume units
- BLM 2017 Specifications for Descriptions of Land, printed 8-11: nominal units, description order, lots and government-measurement halves
- FTC: prices must be set independently
- 12 CFR 1026.38: closing costs and buyer/seller ledgers
- CFPB Closing Disclosure explainer
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.