Quick Reference
Real Estate Math Formulas: A Working Reference
Compensation rates, assessment ratios, millage, proration conventions and closing customs are local and negotiated, and no figure here should be carried into a problem that supplies its own. On the exam, use only what the question gives you. Nothing on this page is drawn from any official examination.
Almost every calculation on the licensing exam is one of three operations: a part taken from a whole, a whole recovered from a part, or a rate recovered from both. The subject headings below change the labels on those three positions. The operations do not change.
Use this as a reference rather than a memory exercise. Learn the one relationship in the first section, then read each group as that relationship wearing different vocabulary. All dollar amounts shown are illustrative figures chosen to divide cleanly.
One Relationship, Three Positions
Write down three positions: a whole, a rate, and a part. The rate times the whole gives the part. The part divided by the rate gives the whole. The part divided by the whole gives the rate. That is the entire arithmetic engine of this subject.
Worked with illustrative numbers: suppose the whole is a price of $300,000 and the rate is 6 percent. The part is 300,000 x 0.06 = $18,000. Recover the whole from the part: 18,000 / 0.06 = $300,000. Recover the rate: 18,000 / 300,000 = 0.06, which is 6 percent.
Two habits keep this from breaking down under time pressure. Convert a percentage to its decimal form as a separate written step before multiplying, and label each of the three positions in words before you compute. A number in the wrong position is the error that dominates this section of the exam, and it produces an answer that appears in the option list.
When a problem gives you a rate that applies to what is left over rather than to what is taken, subtract the rate from 1 and use the result. That single adjustment handles net-to-seller problems, down payment problems, and any question phrased around a remainder.
Compensation
The whole is a price, the rate is a negotiated percentage, and the part is the amount generated. Divisions after that use a smaller whole each time.
Amount generated = sale price x rate. Illustrative: 400,000 x 0.05 = $20,000.
Sale price = amount generated / rate. Illustrative: 20,000 / 0.05 = $400,000.
Rate = amount generated / sale price. Convert the decimal result to a percentage as the last step.
A firm's portion = total amount generated x that firm's share of the cooperation split.
A licensee's portion = the firm's portion x the licensee's contracted share. Note the whole has changed: the licensee's percentage applies to the firm's money, not to the full amount generated.
Off-the-top obligations such as a referral come out of the total before any split. Subtract first, then divide the remainder.
Price needed to leave a seller a target amount = target / (1 - total rate of deductions). Illustrative: a seller needing $188,000 net with deductions of 6 percent requires 188,000 / 0.94 = $200,000. Adding 6 percent to the target instead gives $199,280, which is wrong and is always among the options.
Financing and Value Ratios
Loan figures rest on the lower of appraised value and contract price. Interest here is simple interest, which is all the exam uses.
Loan-to-value = loan amount / value, then multiply by 100 for a percentage.
Loan amount = value x loan-to-value. Illustrative: 250,000 x 0.80 = $200,000.
Cash down on a purchase = contract price - loan amount. On a purchase where the appraisal matches the price, the down payment percentage and the loan-to-value percentage sum to 100.
Combined loan-to-value = the total of all liens secured by the property / value. Use this whenever a second lien appears in the problem.
One year of interest = loan balance x annual rate. One month of interest = that annual figure / 12. Illustrative: a $200,000 balance at 6 percent produces $12,000 for the year and $1,000 for the month.
Principal reduction in a payment = total payment - that month's interest. The next month's interest is computed on the reduced balance, which is why the interest portion shrinks as the loan ages.
One discount point = 1 percent of the loan amount. Cost of points = loan amount x number of points x 0.01. Illustrative: two points on a $180,000 loan cost $3,600. Points are computed on the loan, never on the price.
Recovery period for points = cost of the points / the monthly payment reduction they buy, expressed in months.
Income Property
Income capitalization is the same three positions again: income is the part, the capitalization rate is the rate, and value is the whole.
Effective gross income = potential gross income - vacancy and collection loss + any other income.
Net operating income = effective gross income - operating expenses. Debt service, income taxes and depreciation are not operating expenses and never come out here.
Value = net operating income / capitalization rate. Illustrative: 48,000 / 0.08 = $600,000.
Capitalization rate = net operating income / value. Illustrative: 48,000 / 600,000 = 0.08, which is 8 percent.
Net operating income = value x capitalization rate. Same relationship read in the third direction.
Gross rent multiplier = sale price / gross monthly rent. Price = multiplier x gross monthly rent. Illustrative: a $240,000 price against $2,000 monthly rent gives a multiplier of 120.
Gross income multiplier uses annual gross income in place of monthly rent. Check which one the problem gives you, because the two multipliers differ by a factor of twelve.
Annual debt service = the monthly payment of principal and interest x 12.
Cash-on-cash return = (net operating income - annual debt service) / cash invested, then multiply by 100.
Straight-line annual depreciation for cost problems = depreciable improvement basis / useful life in years. Land is never depreciated.
Taxes, Prorations, and Equity
Proration questions turn on two decisions before any arithmetic: which day convention applies, and whether the item was paid ahead or is owed behind.
Assessed value = market value x the assessment ratio. Taxable value = assessed value - any exemptions the problem states.
Annual tax using a millage rate = (taxable value / 1,000) x the number of mills, since one mill is one dollar per thousand of value.
Annual tax using a percentage rate = taxable value x the rate.
Daily amount, statutory year = annual amount / 360, with every month treated as 30 days. Daily amount, calendar year = annual amount / 365.
Proration = daily amount x the number of days charged to that party. Count the days under the convention the problem or the closing statement specifies, including whether the closing date itself is charged to the seller or the buyer.
Paid in arrears means the seller owes the buyer for the period already elapsed, so the amount is a debit to the seller and a credit to the buyer. Paid in advance means the seller has covered days after closing and is reimbursed, reversing both entries.
Illustrative proration: an annual charge of $3,650 on a 365-day basis is $10 per day, so 90 days is $900.
Equity = current market value - the total of all liens against the property.
Percentage of change = (new figure - old figure) / old figure, then multiply by 100. The denominator is always the earlier figure, and using the later one is the standard error on appreciation items.
Straight-line appreciation or depreciation over a period = original value x (1 plus or minus the annual rate x the number of years).
Measurement and Land
Area problems are usually decomposition problems. Split an irregular shape into rectangles and triangles, compute each, and add.
Rectangular area = length x width. Volume = length x width x height.
Triangular area = 0.5 x base x height, where the height is measured perpendicular to the base.
An acre holds 43,560 square feet. Acres = total square feet / 43,560, and square feet = acres x 43,560.
Square yards = square feet / 9. Cubic yards = cubic feet / 27. Flooring and concrete problems usually want these units.
Front footage is the measurement along the street line. Price per front foot = price / front feet, and it is unrelated to the parcel's area.
Price per square foot = price / finished area. Compare only within one property type and a narrow size band, since the figure falls as units get larger.
Convert every measurement to one unit before computing. Mixing feet and yards inside a single calculation is the most common measurement error on the exam.
Illustrative area problem: a lot 200 feet by 217.8 feet contains 43,560 square feet, which is exactly one acre.
Three Setups Worked End to End
Reverse compensation. Illustrative figures: a seller must clear $235,000 after a 6 percent compensation rate and no other deductions. The rate applies to what is taken, so the seller keeps 94 percent. Price = 235,000 / 0.94 = $250,000. Check by working forward: 250,000 x 0.06 = $15,000, and 250,000 - 15,000 = $235,000.
Capitalization with an expense step. Illustrative figures: a building produces $90,000 of potential gross income, loses $6,000 to vacancy, and carries $30,000 of operating expenses. Effective gross income is 90,000 - 6,000 = $84,000. Net operating income is 84,000 - 30,000 = $54,000. At a capitalization rate of 9 percent, value = 54,000 / 0.09 = $600,000. If the problem also listed annual debt service, it stays out of this calculation entirely.
Proration on a statutory year. Illustrative figures: an annual tax of $2,880 is paid in arrears, and closing falls at the end of the sixth month with the seller charged through the day before closing. Daily amount = 2,880 / 360 = $8. Days charged to the seller = 5 months x 30 days plus 29 days, which is 179 days. Seller's share = 8 x 179 = $1,432, entered as a debit to the seller and a credit to the buyer. Change the convention to a 365-day year and both the daily amount and the day count change, so read the instruction before computing.
Mechanics Worth Settling Before Test Day
What is the T-bar and is it worth using?
It is a drawing of the three positions. Put the part on top and the rate and the whole side by side underneath. Cover the position you are solving for: covering the top leaves rate beside whole, which means multiply, and covering either bottom position leaves the part over the other one, which means divide. It works for compensation, capitalization, loan-to-value and multiplier problems because all of them are the same relationship. Use it if drawing helps you; the rule it encodes is what matters.
How do I know whether to use a 360-day or a 365-day year?
The problem tells you, directly or through the convention it names. A statutory or banker's year means 360 days with uniform 30-day months. A calendar year means actual days. Which convention a real closing uses is set by state practice and by the contract, so learn both procedures rather than one number.
Am I allowed a calculator?
A basic or on-screen calculator is commonly permitted, but what you may bring, what is provided, and what is prohibited are set by the testing vendor and your jurisdiction. Confirm it in the current candidate handbook, and practice with the same tool you will be allowed to use.
When should I round?
Once, at the end. Carry full precision through every intermediate step, then round to match the precision of the answer choices. Rounding mid-problem on a percentage or a daily rate can shift the result past the nearest option and turn correct work into a wrong answer.
How much of the exam is arithmetic?
A minority of items, and the exact share varies by jurisdiction and by exam form. The published content outline in your candidate handbook gives the weighting your state's exam actually uses. It is a small enough portion that skipping it feels safe and a large enough portion that skipping it decides borderline results.
What is the fastest way to raise a weak math score?
Sort your missed items by category rather than reworking them one at a time. Wrong position, wrong decimal conversion, and right work reported from the wrong level are three different failures with three different corrections. Drill the category that appears most often until the setup is automatic, then move to the next.