National portion · Math
Real Estate Exam Math: Formulas and Worked Examples
The formulas behind the math questions in Pearson VUE’s national salesperson outline, the conventions each one depends on, and worked examples you can check line by line.
On this page 9 sections
Pearson VUE’s national salesperson content outline (effective April 1, 2025) gives real estate math its own section of 7 scored questions: property area, property valuation, commission, loan financing costs, settlement and closing costs, investment, and property management. Related concepts also appear elsewhere in the outline, such as gross rent and gross income multipliers and assessed value under Property Value and Appraisal (section III). The sections below give a formula, its conventions and a worked example for each of these seven areas.
Memorize these. Pearson VUE’s Pennsylvania candidate handbook says 43,560 square feet per acre and 5,280 feet per mile are not provided at the test center. It also says that when a question requires a proration, the question will specify whether to use a 360- or 365-day year and whether the day of closing belongs to the buyer or the seller.
Practice 5 math questionsPractice 10 math questions
A four-step method for math questions
- Find what is being asked. A question about a salesperson’s share is not asking for the total commission.
- Write down the conventions the question gives you: 360 or 365 days, who owns the closing day, monthly or annual figures, how to round.
- Pick the formula, then plug in. Keep full precision until the last step, then round as the question says, or by standard rounding rules if it does not say.
- Sanity-check the size of the answer. A salesperson’s share is smaller than the total commission; a seller’s share of calendar-year taxes at a June closing is less than half the year’s bill.
Area and acreage
Rectangle: length × width
Triangle: ½ × base × height
Acres = square feet ÷ 43,560
Break irregular shapes into rectangles and triangles, then add. Use the same units throughout: convert yards or miles to feet before multiplying.
Acreage and price per acre
A rectangular lot measures 330 feet by 660 feet. How many acres is it, and what is it worth at $18,000 per acre?
- Area = 330 × 660 = 217,800 square feet.
- Acres = 217,800 ÷ 43,560 = 5 acres.
- Value = 5 × $18,000.
Answer: $90,000
Area of an L-shaped floor plan
A one-story house is made of two rectangles: 40 feet by 30 feet and 20 feet by 15 feet. What is its area?
- Split the shape into rectangles: 40 × 30 = 1,200 square feet and 20 × 15 = 300 square feet.
- Add them: 1,200 + 300.
Answer: 1,500 square feet
Common mistakes: multiplying feet by yards, forgetting to divide by 43,560 for acres, and adding the sides of a lot (perimeter) instead of multiplying them (area).
Commission and splits
Commission = sale price × commission rate
each split = the previous share × its percentage
Commission rates and splits are set by agreement and are negotiable; no rate is standard. The examples here state each rate and split as a term of the agreement. Work down the chain in order: total commission, then the listing broker’s side, then the salesperson’s share of that side.
Commission split under a stated agreement
A home sells for $325,000. The listing agreement in this example sets a 6% commission, shared 50/50 between the listing and the cooperating broker. The listing salesperson’s agreement with their broker pays the salesperson 60% of the listing broker’s share. What does the listing salesperson receive?
- Total commission = $325,000 × 0.06 = $19,500.
- Listing broker’s share = $19,500 × 0.50 = $9,750.
- Salesperson’s share = $9,750 × 0.60.
Answer: $5,850
Valuation: NOI, cap rates, multipliers and taxes
NOI = effective gross income − operating expenses
Value = NOI ÷ cap rate (I = R × V)
Value = rent × GRM
Net operating income leaves out mortgage payments (debt service) and income taxes. If you know two of income, rate and value, the third follows from I = R × V. A gross rent multiplier uses monthly rent for houses; a gross income multiplier uses annual income.
Net operating income
An apartment building has potential gross income of $120,000 a year, a 5% vacancy and collection loss, and $42,000 of annual operating expenses. What is its net operating income?
- Effective gross income = $120,000 × (1 − 0.05) = $114,000.
- NOI = $114,000 − $42,000. Debt service is not an operating expense.
Answer: $72,000
Value by income capitalization
Using the NOI above and a capitalization rate of 8%, estimate the property’s value.
- Value = NOI ÷ cap rate = $72,000 ÷ 0.08.
Answer: $900,000
Gross rent multiplier
Similar single-family rentals sell for about 150 times their monthly rent. What value does that suggest for a house renting for $1,800 a month?
- Value = monthly rent × GRM = $1,800 × 150.
Answer: $270,000
Assessed value = market value × assessment ratio
Tax = assessed value × rate (1 mill = $1 per $1,000)
Property tax from mills
A home has a market value of $300,000. The municipality assesses at 50% of market value and the tax rate is 25 mills. What is the annual tax?
- Assessed value = $300,000 × 0.50 = $150,000.
- One mill is $1 per $1,000 of assessed value, so 25 mills = 0.025.
- Tax = $150,000 × 0.025.
Answer: $3,750
Loan math: LTV, interest and points
Loan = value × LTV
Monthly interest = balance × annual rate ÷ 12
1 point = 1% of the loan amount
When a question says the lender uses the lower of the price and the appraised value, base the loan on the lower number; the buyer pays the rest of the price in cash. Points are a percentage of the loan, not the price.
Loan amount from loan-to-value
A buyer agrees to pay $250,000, the appraisal comes in at $240,000, and the lender in this example lends 80% of the lower of the price and the appraised value. What is the maximum loan, and how much cash does the buyer need for the price?
- Lower of price and appraisal = $240,000.
- Loan = $240,000 × 0.80 = $192,000.
- Cash toward the price = $250,000 − $192,000.
Answer: $58,000
First month’s interest and principal
A $200,000 loan at 6% annual interest has a monthly principal and interest payment of $1,200 in this example. How much of the first payment reduces the principal?
- First month’s interest = $200,000 × 0.06 ÷ 12 = $1,000.
- Principal = $1,200 − $1,000.
Answer: $200
Discount points
A lender charges 2 discount points on a $192,000 loan. What do the points cost?
- One point is 1% of the loan amount.
- Cost = $192,000 × 0.02.
Answer: $3,840
Prorations and closing costs
Daily amount = annual amount ÷ 365 (or 360)
Seller’s share = daily amount × the seller’s days
Decide two things first: who has paid the bill, and who owns the day of closing. Taxes paid in arrears mean the buyer will pay the whole bill later, so the seller credits the buyer for the seller’s days. A bill the seller prepaid means the buyer credits the seller for the buyer’s days. With a 360-day year, every month counts as 30 days.
Tax proration, 365-day year
Calendar-year 2026 property taxes of $3,650 have not been paid (they are paid in arrears). Closing is June 15, 2026. Use a 365-day year; the day of closing belongs to the buyer. How much does the seller owe the buyer? Round only the final answer to the nearest cent.
- The seller owns January 1 through June 14: 31 + 28 + 31 + 30 + 31 + 14 = 165 days.
- Daily amount = $3,650 ÷ 365 = $10 a day.
- Seller’s share = 165 × $10 (a credit to the buyer and a debit to the seller, because the buyer will pay the whole bill later).
Answer: $1,650.00
The same proration with a 360-day year
Same facts, but the question says to use a 360-day year (12 months of 30 days).
- Seller’s days = 5 full months × 30 + 14 days = 164 days.
- Daily amount = $3,650 ÷ 360 = $10.1388…
- Seller’s share = 164 × $10.1388… = $1,662.777…, rounded at the end.
Answer: $1,662.78
Net to the seller
A property sells for $300,000. The listing agreement in this example sets a 5% commission; the seller’s other closing costs are $4,500 and the mortgage payoff is $180,000. What does the seller net?
- Commission = $300,000 × 0.05 = $15,000.
- Net = $300,000 − $15,000 − $4,500 − $180,000.
Answer: $100,500
Price needed to net a target amount
A seller wants to net $100,000 after $3,000 of closing costs and a 6% commission (as set in this example’s agreement). What sale price is needed? Round to the nearest cent.
- The seller keeps 94% of the price after commission: price × 0.94 = $100,000 + $3,000.
- Price = $103,000 ÷ 0.94. (Adding 6% to $103,000 would be too low, because the commission is charged on the higher price.)
Answer: $109,574.47
Investment returns, appreciation and depreciation
Return = annual cash flow (or gain) ÷ amount invested
Percent change = (new − old) ÷ old
Annual depreciation = (cost − land) ÷ recovery period
Land is never depreciated. For U.S. income tax purposes, residential rental property is depreciated over 27.5 years and nonresidential real property over 39 years under the general rules (IRS Publication 527 and Publication 946); if a question gives a different period, use the one it gives.
Cash-on-cash return
An investor puts $75,000 of cash into a rental that produces $6,000 of pre-tax cash flow a year. What is the cash-on-cash return?
- Return = annual cash flow ÷ cash invested = $6,000 ÷ $75,000.
Answer: 8%
Straight-line depreciation of a rental building
An investor buys a residential rental for $350,000, of which $75,000 is the value of the land. Using the 27.5-year recovery period for residential rental property, what is a full year’s straight-line depreciation?
- Land is not depreciated: depreciable basis = $350,000 − $75,000 = $275,000.
- Annual depreciation = $275,000 ÷ 27.5.
Answer: $10,000
Property management
Rent collected = units × rent × occupancy
Fee = rent collected × fee rate
Monthly rent = annual rent per square foot × area ÷ 12
Property management fee
A 12-unit building rents for $1,100 a unit per month and averages 95% occupancy. The management agreement in this example pays 8% of rent collected. What is the monthly fee?
- Rent collected = 12 × $1,100 × 0.95 = $12,540.
- Fee = $12,540 × 0.08.
Answer: $1,003.20
Annual rent per square foot
A 2,400-square-foot retail space leases at $18 per square foot per year. What is the monthly rent?
- Annual rent = 2,400 × $18 = $43,200.
- Monthly rent = $43,200 ÷ 12.
Answer: $3,600
Printable formula sheet
Print this page to keep the formulas and worked examples on paper; the list below collects every formula in one place. The site menus, table of contents, print button and related-page cards are left out of the printout.
- Acres = square feet ÷ 43,560; one mile = 5,280 feet
- Rectangle = length × width; triangle = ½ × base × height
- Commission = price × rate; each split = previous share × its percentage
- NOI = effective gross income − operating expenses (no debt service)
- Value = NOI ÷ cap rate; I = R × V
- Value = monthly rent × GRM (houses) or annual income × GIM
- Assessed value = market value × ratio; tax = assessed value × rate; 1 mill = $1 per $1,000
- Loan = lower of price and appraisal × LTV; 1 point = 1% of the loan
- Monthly interest = balance × annual rate ÷ 12
- Daily proration = annual amount ÷ 365 or 360; check who owns the day of closing
- Price to net = (desired net + costs) ÷ (1 − commission rate)
- Return = annual cash flow ÷ cash invested; percent change = (new − old) ÷ old
- Annual depreciation = (cost − land) ÷ recovery period