Real Estate Math on the Arizona Real Estate Salesperson Exam: What's Tested and How to Study It

Seven questions, seven kinds of arithmetic

The Pearson VUE outline splits these seven items into 4 application questions and 3 analysis questions, with no knowledge items at all. Every other area on the General exam includes some questions you can answer by remembering a definition. In these seven you have to produce a number.

A handful of relationships generates almost all of them.

Area and volume

Area starts from length × width. It also needs one conversion you have to carry in your head: 43,560 square feet in an acre. A parcel 330 feet by 660 feet is 217,800 square feet →
217,800 ÷ 43,560 = exactly 5 acres.

Work it in that order every time — multiply to get square feet, then divide by 43,560 to get acres. The other conversion is 5,280 feet in a mile. Neither is given to you at the test center.

Price per square foot is price ÷ area, so a 2,400-square-foot home at $432,000 works out at$ 180 a square foot. Volume adds a third dimension for questions about concrete, fill or storage: a space 30 by 20 by 8 feet holds 30 × 20 × 8 = 4,800 cubic feet.

One acre, shaded, laid over an American football field. The acre is 43,560 square feet; the field including its end zones is 360 by 160 feet, or 57,600. So an acre covers a bit over three-quarters of a field — which is the size intuition worth carrying, along with the 43,560 the test center will not give you.

Valuation and equity

Valuation questions lean on the income approach, which rests on two terms worth pinning down before the arithmetic. Net operating income is what a property earns after operating expenses and before any loan payment: gross income, minus vacancy, minus operating costs. The capitalization rate is the annual return investors expect from a property of that kind — what similar buildings return as a percentage of their price.

Those two and the value form a triangle. Write all three arrangements down once and you own the topic:

value = NOI ÷ cap rate cap rate = NOI ÷ value NOI = value × cap rate

Enter the rate as a decimal, so 7.5% goes in as 0.075. A building throwing off $54,000 of net operating income where similar properties trade at a 7.5% cap rate is worth$ 54,000 ÷ 0.075 = $720,000.

Equity is what remains after debt, so that same $720,000 property carrying a$ 465,000 loan balance leaves $255,000 of equity.

Two lighter valuation tools also appear, and the trap is which rent figure goes into them. The gross rent multiplier is price ÷ gross monthly rent, while the gross income multiplier is price ÷ gross annual income. Take a property selling at $384,000 that brings in$ 4,000 a month, which is $48,000 a year: its gross rent multiplier is$ 384,000 ÷ $4,000 = 96, and its gross income multiplier is$ 384,000 ÷ $48,000 = 8. Same property, two numbers that differ by a factor of twelve, so read which one the question hands you before dividing. You work either in reverse on the exam: take the multiplier from recent comparable sales, then multiply your subject property's rent by it — monthly rent for a gross rent multiplier, annual income for a gross income multiplier — to estimate value. Property tax is figured on assessed value, the figure the county assessor sets rather than the market price. An assessed value of$ 180,000 at a 1.2% rate produces a $2,160 annual bill.

Commission

Commission is a percentage of the sale price, split between the brokerages, and then split again between a brokerage and its agent. Take a $385,000 sale at 5.5% total:

$385,000 × 0.055 =$ 21,175 in total commission →
the two brokerages divide it evenly, $21,175 ÷ 2 =$ 10,587.50 each →
an agent on a 60% split takes $10,587.50 × 0.60 =$ 6,352.50.

The last step is a multiplication. A 60% split takes 60% of the brokerage's half, so it is not a third division.

Loan costs

Loan-to-value is the mortgage over the price: $260,000 ÷$ 325,000 = 80% LTV.

Simple interest is principal × rate × time. A $260,000 loan at 6.5% costs$ 260,000 × 0.065 = $16,900 for a year, which is$ 16,900 ÷ 12 = $1,408.33 in the first month's interest.

Discount points are figured on the loan, not the price. One point is 1% of the loan amount, paid up front to buy the interest rate down, so two points on that $260,000 loan is$ 5,200. Two percent of the $325,000 price would have been$ 6,500.

Prorations

Prorations divide a yearly cost between seller and buyer at closing. The exam is unusually helpful here: the outline states that a proration question will tell you which day-count basis to use, and whether the closing day belongs to the buyer or the seller.

Take an annual property tax bill of $2,190, closing on September 15, a 365-day year, taxes paid in arrears, and the closing day belonging to the buyer. Because the buyer owns the closing day, the seller's last chargeable day is September 14.

$2,190 ÷ 365 =$ 6.00 a day →
count the seller's days, January 1 through September 14: 31+28+31+30+31+30+31+31 = 243 for the eight full months, plus 14 = 257 days →
257 × $6.00 =$ 1,542.

Since the taxes are paid in arrears, the buyer will settle the full bill later, so that $1,542 is a debit to the seller and a credit to the buyer.

Run the same bill on a 360-day year and both halves of the sum change. Every month counts as 30 days, so the seller's share becomes 8 × 30 + 14 = 254 days, and the daily rate becomes $2,190 ÷ 360 =$ 6.0833 recurring. Keep the full decimal, or work it as a fraction to avoid rounding drift:

(254 ÷ 360) × $2,190 =$ 1,545.17.

Both figures look plausible on the page, which is why the basis named in the question is the only thing that settles it.

A calendar year has 365 days; the banker's year used in many prorations has 360. Same tax bill, different daily rate, different answer — which is why each proration question tells you which basis to use.

Closing statements

Closing-statement questions run in two directions. Going forward, you start at the sale price and subtract: that $385,000 sale, minus$ 21,175 of commission, minus a $210,000 loan payoff, minus$ 4,300 of other closing costs, nets the seller $149,525.

Going backward, the question gives you the net the seller must walk away with and asks what to list at. Add the fixed costs to the desired net, then divide by 1 minus the commission rate. A seller wanting $150,000 clear, with that same$ 210,000 payoff and $4,300 of costs at 5.5% commission, needs:

( $150,000 +$ 210,000 + $4,300) ÷ 0.945 =$ 385,502.65.

The buyer's side asks a different question: cash to close. Add the down payment to the buyer's closing costs, then subtract any earnest money — the deposit already sitting in escrow. On that $385,000 purchase with 20% down,$ 6,400 of closing costs and $5,000 of earnest money already in escrow, the buyer brings$ 77,000 + $6,400 −$ 5,000 = $78,400.

Debits and credits only make sense once you know whose column you are in:

A debit on this side	Does this
Seller	Reduces what the seller takes away
Buyer	Increases the cash the buyer must bring

Investment and property management

The exam keeps two ideas apart that look alike. Appreciation measures a change in value against the original value: a property bought at $300,000 and now worth$ 351,000 has gained $51,000, and$ 51,000 ÷ $300,000 = 17%. Dividing by the new value is the most common way to get this wrong.

Return on investment measures annual return against the cash invested, normally the down payment. Someone who put $75,000 down and clears$ 6,750 a year is earning $6,750 ÷$ 75,000 = 9%.

Property management questions turn on vacancy and fees, and they use two different quantities that are easy to run together. Take a 24-unit building renting at $1,450 a month with 3 units empty:

potential annual rent is 24 × $1,450 × 12 =$ 417,600 →
the vacancy loss is 3 × $1,450 × 12 =$ 52,200 in dollars →
the vacancy rate is a separate figure, 3 units of 24, or 12.5%.

A management fee is charged on rent collected rather than rent potential, so start from the 21 occupied units: 21 × $1,450 =$ 30,450 collected each month, and 8% of that is a $2,436 fee.

How it shows up on the exam

The published Pearson VUE outline breaks this area into seven subtopics and assigns one scored question to each:

property area calculations
property valuation
commission and compensation
loan financing costs
settlement and closing costs
investment
property management calculations

No single formula is worth more than any other. Spend an hour on cap rates and nothing on prorations and you still lose a question.

Our four-hour recommendation is our own estimate; the board publishes no study-time guidance. Spread it as short sessions. Our three study topics map onto the outline's seven:

Work it in this order:

Memorize the two conversions first, since nothing else is retrievable without them →
drill the three ratio relationships (value/NOI/cap rate, loan/price/LTV, and gain/original value) until you can rearrange them without thinking →
then practice full closing statements in both directions, which combine several of the others →
finish on prorations, reading each question twice for the day-count basis.

Bring a calculator you are allowed to use. Arizona permits a hand-held calculator: battery-powered, silent, nonprinting, no letter keys. The candidate handbook names acceptable finance models, among them the HP 12C, the HP 10B and the Real Estate Qualifier Plus IIIX. A calculator that fails mid-exam is not grounds for extra time or a challenge to your result, so test it beforehand.

Common questions

How many math questions are on the Arizona real estate exam? Seven of the 80 scored questions on the General exam, about 9%. The Arizona State exam has no separate math area, though closing-cost figures appear inside its contracts questions.

Are formulas or conversions provided at the test center? No. The outline states plainly that 43,560 square feet per acre and 5,280 feet per mile are not available and must be memorized.

Can I use a calculator? Yes, within the limits above. The candidate handbook lists the acceptable real estate finance models by name.

How is a proration calculated — 360 or 365 days? Whichever the question specifies. Each proration item tells you the day-count basis and whether the closing day belongs to buyer or seller.

What is the difference between appreciation and return on investment? Appreciation divides the gain by the original value of the property. Return on investment divides the annual return by the cash actually invested, which is usually the down payment rather than the price.

What score do I need? 75% of the scored questions, and the General and State exams are scored separately.

Write the relationship down on paper before you reach for the calculator — putting the formula in front of you is what stops you dividing when you meant to multiply. Our full question bank and study guides for this exam are free, where the big providers charge for them. Try the free practice questions.