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Illinois real estate math

Real estate math word problems: write the setup, then calculate

A $300,000 sale price and a $240,000 loan can both be relevant, but they answer different questions. Two points use the loan amount. A fee based on the sale price uses the price. Label the amounts before entering them into the calculator.

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The short answer

Start by naming the requested result and its unit. Copy the relevant facts with labels, choose the relationship, align units and time periods, and identify the correct percentage base. Estimate the result, calculate in visible steps, then substitute the answer back or reconcile the totals. Apply a specific ceiling, day-count or reporting rule where it belongs. A correct calculation answers the exact question, not merely a nearby question using the same numbers.

Reviewed for October 2026. PSI places Real Estate Calculations at national XI, with 7 of the 100 scored national broker questions. This is an original solving method and practice set, not a PSI-endorsed workflow or readiness measure. Rates and figures in examples are supplied assumptions, not standard commissions, market cap rates or loan quotes. Live lending, tax and closing calculations require their actual rules and records.

Five steps from words to a checked answer

  1. Name the target. Write net to seller in dollars, monthly PITI, LTV as a percent or area in square feet. Keep state-only, taxable, annual, estimated and other qualifiers attached. They can change which answer is correct.
  2. Label the facts and the base. Record loan amount, sale price, appraised value, income and expenses as different quantities. Ask what the rate applies to. Include preliminary calculations when needed, and explain why a number is irrelevant rather than discarding it at random.
  3. Write the relationship with compatible units. Use a decimal rate for multiplication. Align monthly and annual cash flows, convert linear dimensions before calculating area, and specify the day-count convention. Do not apply a generic annual-divided-by-12 shortcut to every effective interest rate or compound-growth problem.
  4. Calculate with named steps. Estimate the scale, then solve each required subtotal. A loan can legitimately be used for both LTV and points. The error to avoid is charging the same cost twice in one total, not reusing any number in two different relationships.
  5. Check and report. Reverse the formula, reconcile a ledger or compare with an estimate. Keep intermediate precision unless instructed otherwise, apply any special rule at its required stage, and finish with the requested unit and rounding. If no choice fits, inspect the setup before choosing the nearest number.
Choose the relationship before the calculator sequence
Requested resultSetupCheck
Percentage amountStated base x decimal rateWhich amount is the rate applied to?
Base or price from a partPart / decimal rateDoes base x rate recover the part?
LTVLoan / applicable property value x 100The value rule can differ from the sale price
Monthly PITIMonthly P&I + annual taxes / 12 + annual insurance / 12This acronym is not necessarily every housing cost
Defined NOIEGI minus the specified operating deductionsDebt service is a separate step; check reserve convention
Direct-capitalization valueConsistently defined annual NOI / matching decimal cap rateAnnual income and rate definitions must agree
Seller netStated seller credits minus stated seller debitsDo not deduct the same fee twice
ProrationAmount / stated period x responsible unitsCount the stipulated dates and allocate the closing day
Rectangle areaLength x width after unit conversionArea has square units
Illinois state transfer taxCeiling(taxable value / 500) x $0.50Use the taxable base and state-only scope

The distinctions that matter

Real estate math word problems: key distinctions
DistinctionHow it worksWhat to notice
Sale price, value and loanThey can be three different amounts in one problem.A lower-value LTV denominator does not automatically become the purchase price or the points base.
Percent and decimalA percent is per hundred: 0.5% is 0.005, while 5% is 0.05.Write the decimal explicitly rather than relying on an unfamiliar percent-key sequence.
PITI and total housing outlayPITI names principal, interest, taxes and insurance. Actual required payments or expenses can include other items.Answer PITI when asked, while recognizing that mortgage insurance, nonescrowed charges or association costs may require separate treatment.
NOI and cash after debtIncome before debt service is different from cash remaining after financing payments. Reserve conventions also need identification.Do not subtract debt simply because the stem includes it, or assume every source defines reserves identically.
Rounding and ceilingNearest rounding and counting every positive fractional taxable unit are different operations.A value just above an exact $500 unit exposes the difference more clearly than a half unit.
Down payment and cash still dueCash to close can include closing costs and adjustments for amounts already paid and allowed credits.Subtract a paid deposit once rather than adding it again to the remaining amount due.

Work through the facts

Choose the fee base

A home listed for $435,000 sells for $420,000. The agreement supplies a fee of 5% of the sale price. Find that fee.

  1. Unknown: fee dollars. Base: $420,000, not list price.
  2. 5% = 0.05; $420,000 x 0.05 = $21,000.
  3. Reverse: $21,000 / $420,000 = 0.05.

Answer: The stated fee is $21,000. The example does not establish a standard commission rate.

Points use the loan amount

Sale price is $300,000, loan amount is $240,000 and the stated charge is two points. Find the points charge.

  1. One point is 1% of the loan amount.
  2. Two points: $240,000 x 0.02 = $4,800.
  3. Using $300,000 would answer a different percentage-base question.

Answer: The charge is $4,800. It does not mean the interest rate falls by two percentage points.

Put PITI on a monthly basis

Monthly P&I is $1,900. Annual property taxes are $7,800 and annual homeowners insurance is $1,920. Find monthly PITI for these supplied components.

  1. Taxes: $7,800 / 12 = $650 per month. Insurance: $1,920 / 12 = $160 per month.
  2. $1,900 + $650 + $160 = $2,710.
  3. Check: the annual tax and insurance total of $9,720 equals $810 x 12.

Answer: Monthly PITI is $2,710. This is not a complete quote for every possible housing expense.

One loan, two different relationships

Price is $400,000, appraised value $390,000 and loan $312,000. The exercise explicitly uses the lower price-or-value rule for LTV. Find LTV and price-minus-loan down payment before costs and other adjustments.

  1. LTV base is $390,000: $312,000 / $390,000 = 0.80 = 80%.
  2. The purchase price remains $400,000: $400,000 - $312,000 = $88,000.
  3. Reverse: $390,000 x 0.80 = $312,000; $312,000 + $88,000 = $400,000.

Answer: LTV is 80%; the stated price-minus-loan amount is $88,000. The $78,000 difference between appraised value and loan is not the purchase down payment.

Define income before capitalizing it

An exercise supplies collected rent of $3,000 monthly, no other income or losses, annual operating deductions of $12,000, and no separate reserve allowance under its stated NOI convention. A consistent annual cap rate is 8%. Annual debt service is $10,000. Find the direct-capitalization value.

  1. Annual EGI: $3,000 x 12 = $36,000. Defined annual NOI: $36,000 - $12,000 = $24,000.
  2. Value: $24,000 / 0.08 = $300,000. Debt service does not enter this NOI-to-value step.
  3. Reverse: $300,000 x 0.08 = $24,000. Cash after the supplied debt service would be a separate $14,000.

Answer: The formula result is $300,000 under the stated assumptions. It is not an appraisal or a universal reserve treatment.

Convert length before area

A rectangular lot measures 30 yards by 60 feet. Use 3 feet per yard and 43,560 square feet per acre. Find area in square feet and approximate acres.

  1. 30 yards x 3 = 90 feet. Area: 90 feet x 60 feet = 5,400 square feet.
  2. Acres: 5,400 / 43,560 = 15/121, approximately 0.124 acre.
  3. Square yards: 5,400 / 9 = 600, which is a different requested unit.

Answer: The area is 5,400 square feet, approximately 0.124 acre. Do not label 30 x 60 as square feet while one input is still yards.

Write the responsible dates first

Use a supplied $3,650 annual amount and an actual/365 convention for 2026. Closing is March 15; the problem assigns the closing day to the buyer and the seller January 1 through March 14. Find the seller's share.

  1. Days: 31 in January + 28 in February + 14 in March = 73.
  2. Daily amount: $3,650 / 365 = $10. Seller share: $10 x 73 = $730.
  3. Closing-day ownership and the year convention are explicit exercise facts, not universal Illinois closing rules.

Answer: The seller share is $730 under the stated convention. A different assigned date range can produce a different correct answer.

A fraction is not nearest rounding

The already determined taxable value is $248,501 for a nonexempt Illinois transfer. Calculate state tax only at $0.50 for each $500 or fraction.

  1. $248,501 / $500 = 497.002 taxable units.
  2. The fraction rule counts 498 units. $0.50 x 498 = $249.
  3. Nearest-unit rounding to 497 would incorrectly produce $248.50. County or municipal charges are outside this state-only question.

Answer: Illinois state tax is $249 on the supplied taxable base. Do not assume every sale's price is automatically its final taxable base.

Seller net, then the reverse price

Price is $360,000. The supplied fee is 5% of price, payoff is $210,000 and other seller debits total $12,000. No other credits or debits apply. Find net, then check the price needed for that same target.

  1. Fee: $360,000 x 0.05 = $18,000. Total debits: $18,000 + $210,000 + $12,000 = $240,000.
  2. Net: $360,000 - $240,000 = $120,000. Reverse: P - 0.05P - $222,000 = $120,000.
  3. 0.95P = $342,000; P = $342,000 / 0.95 = $360,000.

Answer: Net is $120,000. For the reverse problem, the fee depends on the unknown sale price, not on the target net.

Equal percentage changes use different bases

A hypothetical value of $500,000 rises 10% and then falls 10% from the new amount. Find the ending value.

  1. First change: $500,000 x 1.10 = $550,000.
  2. Second change: $550,000 x 0.90 = $495,000.
  3. The $55,000 decrease is larger than the earlier $50,000 increase because the bases differ.

Answer: The ending value is $495,000, a net 1% decrease from the original. Equal opposite percentages do not automatically cancel.

Cash remaining to close

A simplified exercise supplies $80,000 down payment, $10,000 closing costs, a $20,000 deposit already paid and an allowed $3,000 credit. No other adjustments apply. Find remaining cash due.

  1. Required amounts: $80,000 + $10,000 = $90,000.
  2. Subtract the already-paid deposit and allowed credit: $90,000 - $20,000 - $3,000 = $67,000.
  3. Reconcile: $67,000 new cash + $20,000 deposit + $3,000 credit = $90,000.

Answer: Remaining cash is $67,000 under the stated simplified assumptions. It is different from either down payment alone or total costs before prior payments.

Mistakes to avoid

  • Calculate with the first two numbers. Name the requested result and identify the relationship first.
  • Use price as the base for every percentage. Points, fees, vacancy and other percentages can have different defined bases.
  • Use every number in the final formula. A number can be background, a preliminary input or irrelevant to the requested result.
  • Never use the same number twice. A loan can enter more than one legitimate relationship. Avoid duplicate charges in the same ledger total.
  • Every annual rate can be divided by 12 without further thought. Align periods under the specified method. Effective yields and compound rates need the correct conversion relationship.
  • Round all fractional taxable units to the nearest integer. Apply the actual Illinois state fraction rule, and distinguish it from ordinary rounding.
  • Choose the nearest option when the calculation does not fit. Recheck the base, period, sign, units, date range and rounding instructions.

Check your understanding

These are original teaching questions, not recalled or predicted PSI questions. Answer first, then explain why the other choices fail.

1. What is the best first step when a stem contains several amounts?

  1. Use every amount once
  2. Write the requested result, unit and relevant qualifier
  3. Multiply the first two
  4. Choose the largest answer
Show answer and explanation

Answer: B. Write the requested result, unit and relevant qualifier

The target determines which quantities and preliminary steps matter. Keep monthly, annual, state-only and similar qualifiers attached.

2. A supplied fee is 0.5% of $200,000. What is the fee?

  1. $100,000
  2. $1,000
  3. $100
  4. $10,000
Show answer and explanation

Answer: B. $1,000

0.5% = 0.005. $200,000 x 0.005 = $1,000. Using 0.05 would charge 5%, ten times as much.

3. Price is $240,000 and the loan is $200,000. A stated 2.5-point charge equals what?

  1. $500
  2. $5,000
  3. $50,000
  4. $6,000
Show answer and explanation

Answer: B. $5,000

Points use the loan amount: $200,000 x 0.025 = $5,000. The sale price is not this charge's base.

4. Monthly P&I is $1,700, annual tax $6,000 and annual homeowners insurance $1,200. Find PITI for those components. An additional $90 monthly association cost is separate.

  1. $2,200
  2. $2,390
  3. $8,990
  4. $2,300
Show answer and explanation

Answer: D. $2,300

Taxes are $500 monthly and insurance $100 monthly: $1,700 + $500 + $100 = $2,300. The separate association cost is not one of these PITI components; that does not mean it disappears from actual housing expenses.

5. Use the stipulated lower price-or-value LTV rule. Price is $400,000, appraised value $390,000 and loan $312,000. Which pair is LTV and price-minus-loan amount before other adjustments?

  1. 125% and $88,000
  2. 80% and $88,000
  3. 78% and $88,000
  4. 80% and $78,000
Show answer and explanation

Answer: B. 80% and $88,000

The value denominator is $390,000, giving 80% LTV. The purchase price remains $400,000, so price minus loan is $88,000. The two calculations can reuse the loan while using different other quantities.

6. Define NOI here as annual EGI minus supplied operating deductions, with no separate reserve allowance. EGI is $180,000, operating deductions $60,000 and debt service $45,000. Find NOI.

  1. $180,000
  2. $120,000
  3. $60,000
  4. $75,000
Show answer and explanation

Answer: B. $120,000

The defined NOI is $180,000 - $60,000 = $120,000. The separate after-debt figure would be $75,000. Do not mix the requested income level with another one.

7. A direct-capitalization exercise supplies annual NOI of $36,000 and a consistent 9% cap rate. What value follows?

  1. $324,000
  2. $400,000
  3. $4,000,000
  4. $3,240
Show answer and explanation

Answer: B. $400,000

$36,000 / 0.09 = $400,000. Reverse-check: $400,000 x 0.09 = $36,000. It is a formula exercise, not a market-supported appraisal.

8. A rectangle is 20 yards by 45 feet. Using 3 feet per yard, what is its area?

  1. 8,100 square feet
  2. 2,700 feet
  3. 900 square feet
  4. 2,700 square feet
Show answer and explanation

Answer: D. 2,700 square feet

20 yards = 60 feet. 60 feet x 45 feet = 2,700 square feet. Convert the linear dimension first and keep square units on area.

9. Given a nonexempt taxable value of $250,001, what is Illinois state transfer tax only at $0.50 per $500 or fraction?

  1. $250.50
  2. $375.75
  3. $125.00
  4. $250.00
Show answer and explanation

Answer: A. $250.50

The quotient is 500.002, so the fraction rule counts 501 units. 501 x $0.50 = $250.50. The given amount is already the taxable base, and this is state tax only.

10. Price is $300,000, the supplied fee is 6% of price, payoff $200,000 and other seller debits $7,000. No other items apply. What is net?

  1. $75,000
  2. $57,000
  3. $107,000
  4. $93,000
Show answer and explanation

Answer: A. $75,000

The fee is $18,000. Net = $300,000 - $18,000 - $200,000 - $7,000 = $75,000. Do not omit the fee or deduct it twice.

11. Target seller net is $58,000, fixed debits are $130,000 and a supplied fee is 6% of unknown price P. No other items apply. Which price works?

  1. $191,480
  2. $188,000
  3. $198,000
  4. $200,000
Show answer and explanation

Answer: D. $200,000

P - 0.06P - $130,000 = $58,000, so 0.94P = $188,000 and P = $200,000. A 6% fee is $12,000; $200,000 - $12,000 - $130,000 = $58,000.

12. Use actual/365 for 2026 and an annual amount of $3,650. Buyer owns closing day April 10; seller is assigned January 1 through April 9. What is seller's share?

  1. $990
  2. $1,000
  3. $900
  4. $980
Show answer and explanation

Answer: A. $990

Seller days are 31 + 28 + 31 + 9 = 99. $3,650 / 365 = $10 per day, so the share is $990. Closing-day allocation is stipulated, not a universal Illinois rule.

13. A simplified cash-to-close exercise supplies down payment $60,000, closing costs $9,000, deposit already paid $12,000 and an allowed credit $3,000. No other adjustments apply. What remains due?

  1. $66,000
  2. $54,000
  3. $84,000
  4. $69,000
Show answer and explanation

Answer: B. $54,000

$60,000 + $9,000 - $12,000 - $3,000 = $54,000. The paid deposit reduces remaining cash rather than being added again.

14. Price is $200,000, payoff is $202,000 and all other supplied seller costs total $3,000. No other items apply. What is the arithmetic net?

  1. Negative $5,000, a $5,000 shortfall
  2. $205,000
  3. $5,000 of positive proceeds
  4. $0, because a net cannot be negative
Show answer and explanation

Answer: A. Negative $5,000, a $5,000 shortfall

$200,000 - $202,000 - $3,000 = -$5,000. The sign matters. This is a supplied arithmetic shortfall, not a determination that a private closing or debt arrangement may proceed.

Common questions

What should I write before calculating?

The unknown, unit and qualifier: for example, monthly PITI in dollars or Illinois state transfer tax only. Then label the givens. A nearby number can be useful for another question while irrelevant to the one asked.

How do I find the percentage base?

Complete the sentence the rate applies to this amount. Points use the loan amount. A supplied fee might use sale price, and a management agreement might specify collected rent or EGI. Use the actual definition or stated agreement, rather than assuming every percentage uses price.

Can I use the same amount in two steps?

Yes. Loan amount can be needed for LTV, points and a price-minus-loan calculation in one problem. The safeguard is to avoid charging the same single expense twice in the same total. Reusing an input and duplicating a cost are different things.

How do I select the formula?

Use the requested result and the whole relationship. For an amount, multiply the correct base by the decimal rate. For a base, divide the part by the rate. For net proceeds, reconcile credits and debits. A word such as value does not by itself tell you whether the question needs LTV, capitalization or something else.

Is the LTV denominator always the sale price?

No. The example explicitly supplies a lower price-or-appraised-value rule. Fannie Mae's identified topic distinguishes standard purchase and refinance calculations and includes special transaction and reporting requirements. A classroom calculation is not a universal rule for every loan or the complete lender delivery process.

Is PITI the whole cost of owning the home?

Not necessarily. The acronym covers the stated principal, interest, tax and insurance components. Mortgage insurance, nonescrowed charges, association amounts and other costs can require separate treatment. CFPB distinguishes P&I, total monthly payment and amounts paid outside escrow. Answer the requested measure accurately.

Should annual amounts always be divided by 12?

For a supplied annual tax or insurance amount used in a simplified monthly PITI exercise, divide by 12. For monthly rent used with annual income and expenses, multiply by 12. Effective yields, compound growth and other rate conversions require their own relationships; do not apply a cash-flow shortcut indiscriminately.

Does NOI always use the same reserve treatment?

No. Identify the convention and match it to the capitalization rate or other use. The worked exercise explicitly defines its deductions and reserve assumption. Debt service is a separate financing step. Do not silently mix an income definition from one source with a rate based on another.

How do I set up a proration?

Write the amount, its stated period, the assigned dates and who owns the closing day before counting. Actual/365, actual/366 and a 360-day convention can produce different results. The example's closing-day allocation is supplied; it is not declared the rule for every Illinois transaction.

Why is the Illinois transfer-tax example a ceiling?

The state provision charges $0.50 for each $500 of taxable value or fraction. A quotient of 497.002 therefore counts as 498 units, rather than 497 by nearest rounding. The example starts with an already determined taxable base and excludes county or municipal charges. An actual transfer also needs its applicable base, exemptions and other requirements.

When should I round?

Follow the question or applicable specific rule. Otherwise retain intermediate precision and report the final result at the requested precision. A statutory unit ceiling and a lender's delivery-ratio rules are different from an ordinary nearest-cent instruction. Do not assume one process applies to every output. IDOR separately instructs rounding PTAX-203 Lines 11 through 18 to the next highest whole dollar. These examples begin with an already determined whole-dollar taxable base.

Is seller net the same as buyer cash to close?

No. Seller net reconciles the seller's credits and debits. A simplified buyer cash-to-close calculation can combine down payment and costs, then account for deposits already paid and allowed credits. Keep the parties and payment timing attached to every amount.

What if my result does not match an answer choice?

Check the base, percent-to-decimal conversion, period, sign, units, dates and special rounding. Then reverse the formula or reconcile the total. Do not pick the nearest choice merely because the calculator produced a number; the setup or an unstated assumption may be wrong. Preserve a negative result when it represents a shortfall; do not turn it into positive proceeds or silently replace it with zero.

Are these official PSI questions or a pass predictor?

No. They are original exercises. The whole Real Estate Calculations category has 7 scored national broker questions, not a separate quota for this setup method. Explain the relationship and check a fresh mixed problem. This small set does not establish a validated readiness score.

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Official sources and scope

CFPB distinguishes loan amounts, points, payments and cash remaining to close. Fannie Mae supplies a specific LTV program context, not a universal lender rule. The Illinois statute and IDOR instructions distinguish taxable units and form rounding. NIST supports the identified unit conversions. The OCC and Fannie multifamily definitions show why income and reserve conventions must be matched. The worked figures, rates and solving method are original teaching examples; the PSI outline supplies the exam categories.

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.

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