Real Estate Math Formulas Worked Out Line by Line

Relationship: price x rate = commission dollars. Every split that follows is that dollar figure multiplied by another rate.

Worked problem: a property closes at $420,000 under an agreed rate of 5 percent. Commission dollars = 420,000 x 0.05 = $21,000.

Layered problem: of that $21,000, the listing side is credited half, so 21,000 x 0.50 = $10,500. If the listing agent's agreement gives the agent 60 percent of the brokerage side, the agent is credited 10,500 x 0.60 = $6,300 and the brokerage keeps $4,200.

Setup error: leaving the rate as a whole number. Five percent entered as 5 rather than 0.05 inflates the result a hundredfold, and the giveaway is an answer far larger than the sale price itself.

Two directions exist here. Forward, you subtract: price - commission - other seller charges = proceeds. Backward, you divide, because the commission is a slice of a price you do not yet know.

Forward problem: a sale at $360,000 with a 6 percent rate and $5,200 of seller charges. Commission = 360,000 x 0.06 = $21,600. Proceeds = 360,000 - 21,600 - 5,200 = $333,200.

Backward problem: the seller must walk away with $250,000 and expects $4,000 of charges at a 6 percent rate. Required price = (250,000 + 4,000) / (1 - 0.06) = 254,000 / 0.94 = $270,212.77.

Setup error: treating the backward version as an addition. Adding 6 percent to the target net understates the price, because the commission is computed on the higher figure the sale actually reaches, not on the proceeds.

Three arrangements of one relationship: rate = net operating income / value, value = net operating income / rate, and income = value x rate. Know any two terms and the third falls out.

Worked problem: a small retail building nets $63,000 a year and carries a value of $840,000. Rate = 63,000 / 840,000 = 0.075, stated as 7.5 percent.

Reverse problem: a buyer will not accept less than a 9 percent return and the building nets $81,000. Value = 81,000 / 0.09 = $900,000, the ceiling on what that buyer pays.

Setup error: feeding debt service into the income figure. Net operating income sits after operating expenses and before loan payments, so a question that mentions a mortgage payment is usually testing whether you leave it out.

Relationship: loan / value = ratio, where value means the appraised amount or the contract price, whichever comes in lower. Rearranged, loan = value x ratio.

Worked problem: a $270,000 purchase financed with a $216,000 loan. Ratio = 216,000 / 270,000 = 0.80, or 80 percent, which makes the cash down payment $54,000.

Appraisal-gap problem: contract price $340,000, appraisal $325,000, loan $260,000. The lower figure governs, so the ratio is 260,000 / 325,000 = 0.80, and the buyer covers both the down payment and the $15,000 shortfall in cash.

Setup error: anchoring on the contract price out of habit. When a question supplies two values, it is supplying them because they differ, and the smaller one is the one the lender lends against.

Relationship: annual charge / days in the year = daily amount, then daily amount x days assigned to a party = that party's share. The method has three inputs and a question can vary any of them.

Worked problem, arrears: annual taxes of $4,380 unpaid, closing on March 16, seller charged through the closing date on a 365-day basis. Days = 31 + 28 + 16 = 75. Daily amount = 4,380 / 365 = $12.00. Seller's debit = 75 x 12.00 = $900.00.

Worked problem, prepaid: a $1,800 annual assessment already paid by the seller, closing on October 1, buyer reimbursing for October through December. Days = 31 + 30 + 31 = 92. Daily amount = 1,800 / 365 = $4.9315. Buyer's credit to the seller = 92 x 4.9315 = $453.70.

Setup error: counting days without reading the instruction. Whether the seller owns the closing date, and whether the problem wants a calendar year or a twelve-month banker's year of thirty-day months, is stated in the question. Statutory and customary practice on this varies by jurisdiction, so use the convention the item gives you.

Relationships: a rectangle is length x width, a triangle is half of base x height, and an acre is 43,560 square feet, which is the one raw figure worth committing to memory.

Worked problem: a parcel measuring 220 feet by 396 feet. Area = 220 x 396 = 87,120 square feet. In acres, 87,120 / 43,560 = 2 acres exactly.

Composite problem: a lot shaped like a rectangle with a triangular corner added. The rectangle is 150 by 180, giving 27,000 square feet. The triangle has a 60-foot base and a 40-foot height, giving 0.5 x 60 x 40 = 1,200 square feet. Total = 28,200 square feet.

Setup error: mixing units inside one calculation. Convert yards, rods or miles to feet before multiplying, and convert to acres only at the end, once the square footage is final.

Two steps, and questions often hide the first. Market value x assessment ratio = assessed value. Assessed value x levy = annual tax, where a levy expressed in mills is applied per thousand dollars of assessed value.

Worked problem with mills: market value $350,000, assessment ratio 60 percent, levy 28 mills. Assessed value = 350,000 x 0.60 = $210,000. Tax = 210,000 x 28 / 1,000 = $5,880.

Worked problem with a percentage levy: the same $210,000 assessed value taxed at 2.8 percent yields 210,000 x 0.028 = $5,880, the identical answer stated a different way, which is the point of the comparison.

Setup error: taxing the market value. Assessment ratios, exemptions and levy formats are creatures of local law and differ widely, so the problem will supply whichever ratio it wants used; apply it before the levy, and do not carry a ratio from one question into the next.

Relationship: price / gross rent = multiplier, and multiplier x gross rent = indicated value. It is a rough comparison tool, not a valuation that accounts for expenses.

Worked problem: a duplex sold for $288,000 while collecting $28,800 a year in gross rent. Multiplier = 288,000 / 28,800 = 10.

Applied problem: comparable sales in the same submarket support a multiplier of 11, and the subject collects $2,750 a month. Annualize first: 2,750 x 12 = $33,000. Indicated value = 11 x 33,000 = $363,000.

Setup error: substituting net income for gross rent. The multiplier is built on collections before expenses, and questions bait this by listing operating costs that the formula never touches.

Relationship for a single year: prior value x (1 + rate) for a gain, or prior value x (1 - rate) for a loss. Across several years the factor is applied again to each new balance, not added up.

Worked problem: a $240,000 property gaining 4 percent in a year is worth 240,000 x 1.04 = $249,600. Losing 4 percent instead gives 240,000 x 0.96 = $230,400, and the two are not mirror images of each other over time.

Multi-year problem: 4 percent a year for three years is 240,000 x 1.04 x 1.04 x 1.04 = $269,967.36. Adding twelve percent instead returns $268,800, close enough to look plausible and still wrong.

Cost recovery is a separate idea with a separate formula: depreciable basis / the recovery period set in federal tax rules = the annual straight-line deduction. Recovery periods differ for residential and nonresidential property and land is never depreciated, so take the period from the problem rather than from memory.

Relationship: current market value - outstanding loan balance = equity. Both terms are current figures; neither is historical.

Worked problem: a house presently valued at $395,000 secured by a loan balance of $242,000. Equity = 395,000 - 242,000 = $153,000.

Two-source problem: an owner bought at $310,000 with a $248,000 loan. Years later the value stands at $358,000 and the balance at $221,000, so equity is $137,000. Of the growth, $48,000 came from the rise in value and $27,000 from paying the balance down.

Setup error: subtracting from the purchase price. A question that tells you what the owner paid and what it is worth today is checking which figure you reach for, and equity always uses today's.