Illinois real estate exam math
The T-bar method: choose the base before you calculate
The T-bar helps you remember which numbers to multiply or divide. Its weak spot is the label underneath the line. A correctly remembered formula still gives the wrong answer if you use the sale price where the question calls for the loan amount, or reuse the original total after a commission split.
Last reviewed:
The short answer
The T-bar arranges part = base × rate with the part above the horizontal line and the base and decimal rate below it. Find a missing part by multiplying base by rate. Find a missing base by dividing part by a nonzero rate. Find a missing rate by dividing part by a nonzero base. Convert a percentage to its decimal value before using it. The layout organizes the equation; the question determines the correct base.
Reviewed for October 2026. The T-bar, formula triangle and formula circle are teaching layouts for algebra, not separate calculation requirements named in the current PSI outline. National broker XI covers calculations within seven of the 100 scored national questions; percentage applications also appear in other topics. That is not a T-bar question quota. These exercises supply their assumptions and do not establish exam-day permission to use a drawing, notes or a particular tool.
Label it, solve it, then reverse-check it
- Name the three quantities. Write the amount being calculated as the part, the reference amount as the base, and the relationship as the rate. The base is not automatically the biggest number. A rate above 100% can produce a part larger than its base.
- Place the part above the line. Place base and rate beside each other below it. Covering the missing part leaves multiplication; covering either lower quantity leaves the top quantity divided by the other lower quantity. The relationship is part = base × rate.
- Convert only percentage rates to decimals: 5% becomes 0.05, 0.5% becomes 0.005 and 125% becomes 1.25. A dollar-per-acre unit price is already a different kind of rate; do not divide it by 100 as if it were a percentage.
- Choose the correct base for this step. Commission uses the stated commission base; points use the stated loan amount; LTV uses the applicable value basis; direct capitalization uses the supplied consistent annual NOI. In a split or a sequence of changes, the result of one step can become the next base.
- Solve without premature rounding. Division requires a nonzero denominator. Apply an express rounding instruction or a specific statutory rounding step when one applies. Do not treat an ordinary final-money rounding convention as permission to ignore a required ceiling.
- Put the result back into the original equation and label the unit. A base recovered from a fee should reproduce that fee when multiplied by its rate. A rate of 0.80 is 80%, not 0.80%. An acreage result is acres, not dollars or percent.
| Unknown or application | Calculation | What must be checked |
|---|---|---|
| Part, above the line | Base × decimal rate | The rate applies to this base |
| Base, below the line | Part ÷ decimal rate | Rate must not be zero |
| Rate, below the line | Part ÷ base | Base must not be zero; convert to percent if asked |
| Commission amount | Stated commission base × stated fee rate | The exercise supplies the agreement and rate |
| LTV ratio | Loan amount ÷ applicable value base | Identify the correct value basis |
| Annual NOI | Value × annual NOI capitalization rate | Use compatible income and rate definitions |
| Value by direct capitalization | Annual NOI ÷ cap rate | Cap rate is nonzero; not gross rent or a loan interest rate |
| Net after a percentage deduction | Gross × (1 − deduction rate) | Only the stated percentage deduction is included |
| Total price from a unit price | Quantity × price per unit | A dollar-per-unit rate is not a percentage |
The distinctions that matter
| Distinction | How it works | What to notice |
|---|---|---|
| Percentage rate and percentage points | A percentage rate is a ratio expressed per hundred. Percentage points describe the difference between two percentage rates. | A rise from 6% to 7% is one percentage point. Relative to the original 6%, the rate increased by 1/6, about 16.67%, not 1%. |
| Original base and a new base | A second percentage may apply to the amount remaining or the amount allocated by the first step. | Half of a fee followed by 70% of that half is 35% of the original fee, not 120% of it. |
| Commission rate and the net fraction | If the only deduction is a stated 6% fee, the seller retains 94% of the price. | To recover the price from that net, divide by 0.94. Dividing the net by 0.06 would use the fee fraction instead of the retained fraction. |
| Algebra and eligibility rules | The equation describes a relationship; it does not select the lawful tax base, appraisal income definition or mortgage-program inputs. | A standard Fannie Mae purchase LTV generally uses the lower purchase price or current appraised value, with program-specific cases. A simple classroom ratio does not establish eligibility or every lender's rule. |
| A unit rate and a percentage | Quantity × dollars per unit produces dollars. Base dollars × a percentage produces part dollars. | Both use multiplication, but $7,500 per acre is not 7,500% and should not be converted to 75 before multiplying. |
Work through the facts
Find a fee, then recover its base
An exercise specifies a $420,000 commission base and a negotiated 4.5% fee. It asks for the total fee before any splits.
- Convert 4.5% to 0.045.
- Fee = $420,000 × 0.045 = $18,900.
- Reverse-check: $18,900 ÷ 0.045 = $420,000.
Answer: The fee is $18,900 under the supplied agreement. No standard commission rate is implied.
Find the missing sale-price base
A stated fee is $15,000 at 5% of the sale price. No minimum fee, cap or other charge is included.
- The fee is the part; the unknown price is the base.
- Price = $15,000 ÷ 0.05 = $300,000.
- Check: $300,000 × 0.05 = $15,000.
Answer: The supplied price base is $300,000. Multiplying the fee by 5% would calculate a different amount.
Solve the LTV rate
The question supplies a loan amount of $288,000 and an applicable value base of $360,000. It asks for the mathematical LTV, with no program reporting adjustment.
- LTV = $288,000 ÷ $360,000 = 0.80.
- Convert the ratio to a percentage: 0.80 × 100 = 80%.
- Check: $360,000 × 0.80 = $288,000.
Answer: The mathematical LTV is 80%. The question has supplied the value basis; this result alone does not approve a loan.
Recover value from annual NOI
An exercise supplies annual NOI of $105,000 and an applicable annual NOI capitalization rate of 7%. It asks for the direct-capitalization indication.
- Value = annual NOI ÷ decimal cap rate.
- $105,000 ÷ 0.07 = $1,500,000.
- Check: $1,500,000 × 0.07 = $105,000.
Answer: The supplied indication is $1,500,000. Gross rent, debt-service interest and an investor's personal cash return are not substitutes for these inputs.
Reset the base after a split
A $20,000 total fee is split equally between two brokerages. Under the stated compensation arrangement, a licensee receives 70% of the receiving brokerage's half.
- Receiving brokerage share = $20,000 × 0.50 = $10,000.
- Licensee share = $10,000 × 0.70 = $7,000.
- Combined fraction of the original fee = 0.50 × 0.70 = 0.35.
Answer: The licensee receives $7,000 under the supplied arrangement. The second base is $10,000, not the original $20,000.
Work backward from a net amount
A seller wants $282,000 after only a stated fee of 6% of the price. The exercise excludes mortgage payoff, taxes, concessions and every other cost.
- The seller keeps 1 − 0.06 = 0.94 of the price.
- Price = $282,000 ÷ 0.94 = $300,000.
- Fee = $300,000 × 0.06 = $18,000; price minus fee returns the $282,000 target.
Answer: The required exercise price is $300,000. This is not a complete seller net sheet or a recommended fee.
Use a unit price without percent conversion
A parcel costs $225,000 and contains 30 acres. The question asks for the average price per acre, with no allocation to separate improvements.
- Unit price = total price ÷ quantity.
- $225,000 ÷ 30 acres = $7,500 per acre.
- Check: 30 acres × $7,500 per acre = $225,000.
Answer: The average is $7,500 per acre. Do not divide $7,500 by 100; it is a dollar-per-acre unit rate, not a percentage.
Recognize a statutory ceiling outside the shortcut
For a nonexempt Illinois transfer, the exercise supplies an already determined whole-dollar taxable value of $250,001. It asks for state transfer tax only at $0.50 per $500 or fraction of $500.
- Divide the supplied taxable value by $500: $250,001 ÷ $500 = 500.002 units.
- Because a fraction counts as a unit, use 501 units.
- State tax = 501 × $0.50 = $250.50.
Answer: The state-only answer is $250.50. A raw percentage shortcut would miss the required unit ceiling. County or municipal taxes, exemptions and the actual taxable-base determination are outside this example.
Mistakes to avoid
- The base must be the largest number. The base is the reference quantity. At 125%, the part is 1.25 times the base and can be larger.
- Five percent means multiply by five. Use 0.05. Keep a percent label separate from the decimal used in the equation.
- The two lower quantities should be added. They are multiplied to produce the top quantity. For an unknown lower quantity, divide the top by the other lower quantity.
- The same base applies to every step. Read the split or change instructions again. A second percentage often applies to a new subtotal.
- Equal percentage increases and decreases cancel. They use different bases. A 10% increase followed by a 10% decrease multiplies the original by 1.10 × 0.90 = 0.99, leaving a 1% net decrease.
- A zero rate lets me divide by zero to find the base. Division by zero is undefined. If both the part and rate are zero, the equation does not identify a unique base. A nonzero part with a zero rate is inconsistent.
- A formula diagram chooses the tax or mortgage base for me. It cannot. Identify the applicable rule or use the base explicitly supplied in the exercise before calculating.
- Round at every line to make the numbers easier. Carry precision until the requested final result unless a specific rule requires an earlier step, such as the Illinois state transfer-tax ceiling.
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 stated $350,000 commission base carries a 6% fee before any splits. What is the fee?
- $21,000
- $350,006
- $5,833.33
- $2,100
Show answer and explanation
Answer: A. $21,000
$350,000 × 0.06 = $21,000. The rate in the equation is 0.06, not 6 or 0.006.
2. Annual NOI is $24,000 and the supplied annual NOI cap rate is 8%. What direct-capitalization value follows?
- $3,000,000
- $1,920
- $192,000
- $300,000
Show answer and explanation
Answer: D. $300,000
$24,000 ÷ 0.08 = $300,000. Check: $300,000 × 0.08 = $24,000.
3. A loan is $270,000 and the exercise supplies an applicable value base of $360,000. What is the mathematical LTV?
- 0.75%
- 133.33%
- 90%
- 75%
Show answer and explanation
Answer: D. 75%
$270,000 ÷ $360,000 = 0.75, or 75%. Dividing in the opposite direction does not calculate LTV.
4. A $12,000 fee is 4% of the supplied price base. Which calculation recovers that base?
- $12,000 ÷ 0.04
- $12,000 × 4
- $12,000 × 0.04
- $12,000 ÷ 4
Show answer and explanation
Answer: A. $12,000 ÷ 0.04
The unknown is the base, so divide the part by the nonzero decimal rate. $12,000 ÷ 0.04 = $300,000.
5. A brokerage receives 50% of a $20,000 fee. A licensee receives 70% of that brokerage share. What is the licensee amount under these supplied terms?
- $10,000
- $24,000
- $7,000
- $14,000
Show answer and explanation
Answer: C. $7,000
$20,000 × 0.50 × 0.70 = $7,000. Adding the percentages ignores their different bases.
6. The only deduction is a stated 6% fee. What price leaves the seller $282,000 after that fee?
- $298,920
- $300,000
- $265,080
- $4,700,000
Show answer and explanation
Answer: B. $300,000
The retained fraction is 0.94. $282,000 ÷ 0.94 = $300,000; the $18,000 fee then leaves $282,000. The exercise excludes all other costs.
7. A stated fee is 0.5% of a $200,000 base. What is the fee?
- $100
- $10,000
- $100,000
- $1,000
Show answer and explanation
Answer: D. $1,000
0.5% is 0.005, so $200,000 × 0.005 = $1,000. Half a percent is not five percent.
8. An amount equals 125% of a $40,000 reference base. What is the amount?
- $32,000
- $50,000
- $5,000,000
- $500
Show answer and explanation
Answer: B. $50,000
125% is 1.25; $40,000 × 1.25 = $50,000. The reference base need not be larger than the resulting part.
9. The rate and part in part = base × rate are both zero. What base can be recovered from those facts alone?
- Any positive amount, uniquely determined
- Exactly one
- No unique base
- Exactly zero
Show answer and explanation
Answer: C. No unique base
Every numerical base multiplied by zero gives zero, so these facts alone do not identify a unique base. Dividing zero by zero is undefined.
10. A parcel costs $225,000 for 30 acres. What is the supplied average price per acre?
- $75
- $7,500
- $750,000
- $6,750,000
Show answer and explanation
Answer: B. $7,500
$225,000 ÷ 30 acres = $7,500 per acre. Dollars per acre are a unit rate, not a percentage to divide by 100.
11. An exercise supplies Illinois state transfer-tax taxable value of $250,001. At $0.50 per $500 or fraction, what is state tax only?
- $250.00
- $250.001
- $501.00
- $250.50
Show answer and explanation
Answer: D. $250.50
The quotient is 500.002, so the statutory fraction rule requires 501 units. 501 × $0.50 = $250.50. No county or municipal tax is included.
12. A $100,000 amount rises 10%, then the new amount falls 10%. What is the final amount?
- $101,000
- $90,000
- $99,000
- $100,000
Show answer and explanation
Answer: C. $99,000
$100,000 × 1.10 = $110,000; $110,000 × 0.90 = $99,000. The decrease applies to the new base, so the changes do not cancel.
13. A stated rate changes from 6% to 7%. Which description is correct?
- It decreased by about 16.67%.
- It increased by one percentage point, about 16.67% relative to the original rate.
- It increased by seven percentage points.
- It increased by 1% relative to the original rate.
Show answer and explanation
Answer: B. It increased by one percentage point, about 16.67% relative to the original rate.
The absolute difference is 7% − 6% = one percentage point. The relative increase is (0.07 − 0.06) ÷ 0.06 = 1/6, about 16.67%. The original rate is the base for the relative-change calculation.
Common questions
What is the T-bar method for real estate math?
It is a visual arrangement of part = base × rate. Put the part above a horizontal line and the base and decimal rate below it. The layout helps you select multiplication or division once you have correctly labeled the quantities.
Is a T-bar different from a formula triangle or circle?
They are alternative teaching layouts for the same algebraic relationship. Whichever layout you use, the facts determine the base, units and rate. A mnemonic does not establish exam-day tool permission or guarantee that a question fits a simple percentage equation.
How do I know whether to multiply or divide?
Multiply base by decimal rate to find the part. Divide part by a nonzero rate to find the base. Divide part by a nonzero base to find the rate. Substitute the answer back into part = base × rate to check it.
Which number is the base?
The base is the reference amount to which the rate applies. A fee can use a stated sale-price base, points can use a loan amount, and a commission split can use a brokerage share. Read the wording rather than assuming that the largest number is the base.
How do I convert the percentage?
Divide the percentage number by 100. Thus 5% = 0.05, 0.5% = 0.005 and 125% = 1.25. To express a computed decimal ratio as a percentage, multiply it by 100 and add the percent sign.
Can I use the method for points on a loan?
Yes, when the problem supplies the loan amount and number of points. One point is 1% of that loan amount; fractional points are possible. For example, 1.5 points on a $280,000 loan equals $4,200. That fee relationship does not tell you a fixed interest-rate reduction or whether buying points is worthwhile.
Why does a seller net-price problem use 94% instead of 6%?
When a stated 6% fee is the only deduction, the seller retains 94% of the price. The supplied net is therefore price × 0.94. Divide the net by 0.94 to recover the price. With payoffs, fixed costs, concessions or other deductions, build the full equation rather than treating the simplified example as a complete net sheet.
What happens when the base or rate is zero?
A zero denominator cannot be used for division. With a zero base, the part is zero but the rate cannot be uniquely recovered. With a zero rate, a zero part does not determine a unique base; a nonzero part contradicts the stated equation. Check the inputs instead of reporting division by zero as zero.
Why do a 10% increase and a 10% decrease not cancel?
The decrease applies to the increased amount, not the original amount. Starting at $100,000, a 10% rise gives $110,000 and a 10% fall from that new base gives $99,000. The overall factor is 1.10 × 0.90 = 0.99, a 1% net decrease.
Can the shortcut handle tax rounding?
The basic relationship may help organize the numbers, but a specific tax rule can require an additional step. Illinois state transfer tax uses $0.50 per $500 or fraction of the applicable taxable value. PTAX-203 also gives separate form-specific rounding instructions for lines 11 through 18. These are different steps, not a universal instruction to round every real estate calculation up.
Does an 80% LTV answer mean the loan qualifies?
No. A mathematical ratio uses its stated inputs. An actual mortgage program determines its value base, loan inputs, reporting conventions and eligibility requirements. Fannie Mae's standard purchase rule generally uses the lower purchase price or current appraised value and has specific cases and reporting instructions. Those instructions are not a general rounding rule for every exam calculation.
Is the cap rate the same as the mortgage interest rate?
No. In the direct-capitalization relationship here, annual NOI = value × the applicable annual NOI capitalization rate. Mortgage interest is a financing cost. Keep the stated income definition and annual period consistent rather than inserting gross rent or a loan interest rate into the NOI equation.
How should I practise this method?
Cover one quantity, identify its unit, solve, then recover one of the supplied inputs. Mix missing-part, missing-base and missing-rate problems. Review a wrong answer by finding the mistaken label, decimal conversion, changing base or rounding step. These original questions are teaching exercises, not a validated readiness threshold or released PSI material.
Keep practising
Your next practice step
Put real estate math into practice
Set up the calculation, keep the units consistent, and check the worked answer.
These web resources are free. The app has a separate free starter tier and optional paid full access.
Official sources and scope
NIST supports the meaning of percent; the CFPB explains points as a percentage of the loan amount. Fannie Mae and OCC materials apply in their stated lending contexts, and Illinois law and IDOR instructions supply the separate transfer-tax steps. Their publication dates differ from this page's October 2026 review date. All prices, fee arrangements and operating assumptions in the exercises are supplied teaching facts, not current market quotes or individual tax, lending or appraisal determinations.
- NIST Guide to the SI: percent represents 0.01
- Fannie Mae B2-1.2-01: LTV bases and program reporting
- 35 ILCS200/31-10: state transfer-tax unit and base
- IDOR PTAX-203 instructions: form-specific rounding and taxable amount
- OCC Commercial Real Estate Lending handbook: income and value analysis
- CFPB: discount points and lender credits
- PSI June24,2026: Real Estate Calculations XI
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.