Texas real estate math: the formulas you need, with worked examples

Math is 7 of the 80 scored questions on the national portion of the Texas real estate exam, one for each math topic on the outline. That's not a lot, but these are questions you can get right every time if you know the method, and the national portion needs 56 of 80 to pass. Here are the formulas, each with a worked example.

1. Area and acreage

The Pearson VUE candidate handbook tells you to memorize two numbers: 43,560 square feet in an acre and 5,280 feet in a mile. A few more you'll use:

• 1 square yard = 9 square feet; 1 cubic yard = 27 cubic feet
• 1 section = 1 square mile = 640 acres; 1 township = 36 sections

For shapes: a rectangle is length x width, a triangle is 1/2 x base x height, and a trapezoid is the average of the two parallel sides times the distance between them. Break odd-shaped lots into rectangles and triangles and add them up.

Example: a lot is 330 feet by 660 feet. 330 x 660 = 217,800 square feet, and 217,800 / 43,560 = 5 acres.

Two traps: convert inches to feet before you multiply (6 inches is 0.5 feet), and divide square feet by 9, not 3, to get square yards.

2. Commission and splits

Commission = sale price x commission rate. When two brokerages share a deal, split the total between the brokerages first, then split each brokerage's share between the broker and the agent.

Example: a $400,000 sale at 6% is $24,000. Split 50/50, each brokerage gets $12,000. A listing agent on a 70% split gets $12,000 x 0.70 = $8,400, and the listing broker keeps $3,600.

The formula runs backward too. If the total commission was $18,000 at 6%, the sale price was $18,000 / 0.06 = $300,000.

Tiered rates apply each rate only to its own slice. At 7% on the first $100,000 and 3% on the rest, a $350,000 sale pays $7,000 + (3% x $250,000) = $7,000 + $7,500 = $14,500.

Traps: apply the agent's percentage to the brokerage's share, not the total commission. And remember the rates in these problems are only examples. Commissions are negotiable and no rate is set by law. The real estate commission calculator shows each step if you want to check your practice answers.

3. Cap rate and the income approach

The income approach values a property by its net operating income (NOI): NOI = effective gross income minus operating expenses. Then:

• Value = NOI / cap rate
• Cap rate = NOI / value
• NOI = value x cap rate

Example: NOI of $100,000 at an 8% cap rate is $100,000 / 0.08 = $1,250,000. Going the other way, $45,000 NOI on a $600,000 price is a 7.5% cap rate.

Traps: divide by 0.08, not 8, or your value comes out 100 times too small. Use NOI, not gross income. And debt service (the mortgage payment) is not an operating expense.

4. Prorations, with a Texas property tax example

A proration splits a yearly or monthly cost between buyer and seller at closing. Exam questions tell you two things you need: whether to use a 360-day or 365-day year, and who owns the day of closing. Always follow what the question says. If it doesn't say, the usual convention is that the seller owns the closing day.

The method has three steps:

1. Find the daily rate: the annual amount / 365 (actual days in each month) or / 360 (every month counts as 30 days, so a 31st counts as the 30th).
2. Count the days each party owns the item.
3. Multiply.

Then get the direction right. An item paid in arrears (not yet paid) is a debit to the seller and a credit to the buyer for the seller's days. An item paid in advance is a credit to the seller and a debit to the buyer for the buyer's days.

Texas property taxes are normally prorated as an arrears item. The year's bill is due by January 31 of the next year, so at a mid-year closing nobody has paid the current year yet, and the buyer will pay the whole bill when it comes. The seller pays the buyer for the seller's share of the year.

Example: annual taxes are $7,300, closing is June 30, use a 365-day year, and the seller owns the closing day.

• Daily rate: $7,300 / 365 = $20
• Seller's days, January 1 through June 30: 31 + 28 + 31 + 30 + 31 + 30 = 181
• Seller's share: 181 x $20 = $3,620, debited to the seller and credited to the buyer

The same idea on a 360-day year: annual taxes of $3,600 are $10 a day. With an April 15 closing, the seller owns 3 full months of 30 days plus 15 days, or 105 days, for $1,050. If the buyer owns the closing day instead, the seller has 104 days, or $1,040.

The TREC resale contract prorates current-year taxes through the Closing Date, and if the actual bill turns out different, the parties adjust later. The property tax proration calculator does the day count for any date and shows its work.

5. Loan-to-value and points

Loan-to-value (LTV) = loan amount / property value, where value is the lower of the sale price and the appraisal.

Example: the price is $310,000, the appraisal comes in at $300,000, and the lender allows 80% LTV. The loan is $300,000 x 0.80 = $240,000, because the lender uses the lower number.

One point is 1% of the loan amount, never the sale price. Two points on a $250,000 loan is $5,000.

Interest questions are simpler than they look: annual interest = loan x rate, and monthly interest = annual interest / 12. A $200,000 loan at 6% is $12,000 a year, or $1,000 for the first month.

6. What the seller walks away with

Net to seller = sale price minus everything the seller pays (commission, loan payoff, closing costs, prorations owed) plus any credits to the seller.

The reverse question is the one people miss: what does the house have to sell for to net the seller a target amount? Add up the desired net, the payoff and the seller's costs, then divide by (1 - commission rate). To net $80,000 with a $150,000 payoff, $5,000 in costs and a 6% commission: $235,000 / 0.94 = $250,000. Multiplying $235,000 by 1.06 gives $249,100, which falls short, because the commission is a percentage of the sale price, not of the net.

One Texas note: Texas has no state real estate transfer tax, so a Texas closing has no transfer tax line.

How to practice

• Write the formula before the numbers. Most wrong answers come from using the right numbers in the wrong formula.
• Estimate first. If a 5-acre lot at $3 a square foot comes out under $100,000, you've skipped the 43,560.
• Do problems, not definitions. Knowing that value = NOI / cap rate earns nothing until you've solved a dozen of them.

For what else is on the exam, see what's on the Texas real estate exam. Encodr's free Texas real estate course has a full math unit covering all of these question types.