Real estate math scares many candidates, but almost every problem is one of a handful of patterns. Learn the patterns, always write the formula first, and watch your units (per month vs. per year, per $100 vs. per $1,000). Work each example below with a calculator, then cover the answer and redo it.

The One Formula Behind Most Problems: Part = Total × Rate

Most real estate math is a version of:

Part = Total × Rate

Rearranged: Total = Part ÷ Rate, and Rate = Part ÷ Total. A memory aid is the T-bar: the part sits on top; the total and rate sit on the bottom. Cover the unknown to see whether to multiply or divide. To find the part, multiply; to find total or rate, divide.

Convert percentages to decimals before multiplying (6% = 0.06). This one relationship covers commissions, LTV, cap rate, tax rate, appreciation, and more.

Commissions

Commission = Sale Price × Commission Rate. Splits then divide that commission among brokerages and agents.

Example 10A — Commission with a two-level split. A home sells for $340,000 at a 6% total commission. The listing brokerage receives 50% of the total, and the listing agent receives 60% of the listing brokerage's share. What does the agent earn?

Total commission = $340,000 × 0.06 = $20,400.
Listing brokerage share = 50% × $20,400 = $10,200.
Listing agent = 60% × $10,200 = $6,120.

Example 10B — Find the commission rate. A broker earned $14,400 on a $240,000 sale. Rate?

Rate = Commission ÷ Sale Price = $14,400 ÷ $240,000 = 0.06 = 6%.

Example 10C — Even brokerage split. A $425,000 sale at 6% is split equally between the two brokerages. Each brokerage receives?

Total = $425,000 × 0.06 = $25,500.
Each = $25,500 ÷ 2 = $12,750.

Net-to-Seller ("seller wants to net…")

The commission is charged on the sale price, not on the net, so you cannot just add the rate back to the net. Use:

Sale Price = (Desired Net + Other Costs) ÷ (1 − Commission Rate)

Example 10D — Net with commission only. A seller wants to net $200,000 after a 6% commission and no other costs. Required sale price?

Sale Price = $200,000 ÷ (1 − 0.06) = $200,000 ÷ 0.94 = $212,766 (rounded to the nearest dollar).
Check: $212,766 × 6% = $12,766; $212,766 − $12,766 = $200,000. ✓

Example 10E — Net with commission plus other costs. A seller wants to net $250,000 after a 5% commission and $3,000 in other closing costs. Required sale price?

Sale Price = ($250,000 + $3,000) ÷ (1 − 0.05) = $253,000 ÷ 0.95 = $266,316 (rounded).
Check: $266,316 × 5% = $13,316; $266,316 − $13,316 − $3,000 = $250,000. ✓

Area, Volume, and Land Measurement

Area of a rectangle = length × width.
Area of a triangle = ½ × base × height.
Volume = length × width × height.
Conversions to memorize: 1 acre = 43,560 square feet; 1 mile = 5,280 feet; 1 section = 640 acres = 1 square mile; 1 township = 36 sections; 1 square yard = 9 square feet.

Example 10F — Square feet to acres. A parcel contains 130,680 square feet. How many acres?

Acres = 130,680 ÷ 43,560 = 3 acres.

Example 10G — Rectangular survey fractions. How many acres are in the NW ¼ of the SW ¼ of a section?

Multiply the fractions by 640: ¼ × ¼ × 640 = 1/16 × 640 = 40 acres.

Example 10H — Price per square foot / total price. A lot measures 120 ft × 90 ft and sells for $12 per square foot. Total price?

Area = 120 × 90 = 10,800 sq ft.
Price = 10,800 × $12 = $129,600.

Example 10I — Triangle area. A triangular lot has a base of 200 ft and a height of 150 ft. Area?

Area = ½ × 200 × 150 = ½ × 30,000 = 15,000 sq ft.

Example 10J — Depth from area and frontage. A lot is exactly 0.25 acre with 100 ft of street frontage. Depth?

Area = 0.25 × 43,560 = 10,890 sq ft.
Depth = area ÷ frontage = 10,890 ÷ 100 = 108.9 ft.

Example 10K — Carpet in square yards. A room is 15 ft × 18 ft, carpeted at $22 per square yard (9 sq ft = 1 sq yd). Cost?

Area = 15 × 18 = 270 sq ft.
Square yards = 270 ÷ 9 = 30 sq yd.
Cost = 30 × $22 = $660.

Interest, LTV, Points

Interest (annual) = Principal × Rate. Monthly interest = annual ÷ 12.
LTV = Loan ÷ Value. Down payment = Value − Loan.
One point = 1% of the loan amount.

Example 10L — Monthly interest. One month of simple interest on a $150,000 loan at 5%?

Annual = $150,000 × 0.05 = $7,500. Monthly = $7,500 ÷ 12 = $625.

Example 10M — Max loan from LTV. A property appraises at $320,000; the lender makes an 80% LTV loan. Max loan?

Loan = 0.80 × $320,000 = $256,000.

Example 10N — Value from loan and LTV. A $270,000 loan is 90% LTV. Property value?

Value = Loan ÷ LTV = $270,000 ÷ 0.90 = $300,000.

Example 10O — Points cost. 3 discount points on a $250,000 loan?

Cost = 3 × 1% × $250,000 = 0.03 × $250,000 = $7,500.

Property Tax (per $100, per $1,000, and mills)

Read the units carefully:

"$X per $100 of assessed value": Tax = (Assessed Value ÷ 100) × X.
"$X per $1,000": Tax = (Assessed Value ÷ 1,000) × X.
Mills: 1 mill = $0.001 = $1 per $1,000. Tax = Assessed Value × (mills ÷ 1,000).

Example 10P — Rate per $100. Assessed $220,000 at $1.25 per $100. Annual tax?

Tax = ($220,000 ÷ 100) × $1.25 = 2,200 × $1.25 = $2,750.

Example 10Q — Mills. Assessed $180,000 at 20 mills. Annual tax?

Tax = $180,000 × (20 ÷ 1,000) = $180,000 × 0.02 = $3,600.

Example 10R — Assessment ratio then rate. Market value $300,000, assessed at 40% of market, tax rate $3.00 per $100. Annual tax?

Assessed value = 40% × $300,000 = $120,000.
Tax = ($120,000 ÷ 100) × $3.00 = 1,200 × $3.00 = $3,600.

Appreciation, Profit, and Equity

Percent change = (New − Old) ÷ Old.
Equity = Value − Loan balance.

Example 10S — Appreciation. A home bought for $250,000 is now worth $325,000. Total appreciation percent?

Change = $325,000 − $250,000 = $75,000.
Percent = $75,000 ÷ $250,000 = 0.30 = 30%.

Example 10T — Original price from a profit percent. A property sold for $230,000, a 15% profit over what the seller paid. Original price?

Sale = Original × (1 + 0.15) → Original = $230,000 ÷ 1.15 = $200,000. (Don't subtract 15% of the sale price — that's the classic error.)

Example 10U — Equity. Value $400,000, mortgage balance $250,000. Equity?

Equity = $400,000 − $250,000 = $150,000.

Prorations at Closing

Prorations split shared costs (taxes, interest, rent) by the number of days each party owns the property. Determine the daily amount, count the days, and decide the direction (credit/debit) based on who has paid and who owes. Watch whether the problem uses a 365-day year or a 360-day "banker's" year (12 months of 30 days). Taxes are often paid in arrears (after the fact) or in advance — read carefully.

Example 10V — Taxes paid in arrears; seller owes the buyer. Annual taxes $3,650, paid in arrears, 365-day year. Closing is on the 90th day; the seller is responsible through the day before closing (89 days). Seller's share credited to the buyer?

Daily = $3,650 ÷ 365 = $10 per day.
Seller's share = 89 × $10 = $890 credited to the buyer (because the buyer will later pay the whole bill).

Example 10W — Taxes prepaid; buyer reimburses the seller. Taxes of $3,650 for the year were already paid in full by the seller; the seller owned the property 100 days before closing (365-day year). Buyer's reimbursement?

Daily = $3,650 ÷ 365 = $10.
Days the buyer will own = 365 − 100 = 265 days.
Buyer reimburses = 265 × $10 = $2,650 to the seller (who prepaid the whole year).

Example 10X — Prepaid interest at closing (360-day year). A $200,000 loan at 6% closes on the 21st of a 30-day month; the buyer prepays interest from closing through month-end (10 days), using a 360-day year. Prepaid interest?

Daily interest = $200,000 × 0.06 ÷ 360 = $12,000 ÷ 360 = $33.333… per day.
Prepaid = 10 × $33.333… = $333.33.

Example 10Y — Transfer tax per $500. A state transfer tax is $0.50 per $500 of sale price. On a $180,000 sale?

Units of $500 = $180,000 ÷ 500 = 360.
Tax = 360 × $0.50 = $180.

Example 10Z — Documentary stamp rounded up per $1,000. A stamp of $1.00 per $1,000, rounding the price up to the next full $1,000, on a $349,400 sale?

Round $349,400 up to $350,000.
Units of $1,000 = 350.
Tax = 350 × $1.00 = $350.

### Key Concept Nearly every problem reduces to Part = Total × Rate — multiply to find the part, divide to find the total or the rate. The two most-missed patterns are net-to-seller (divide by 1 − rate, never add the rate back) and prorations (find the daily amount, count the days, then set the direction). Always convert the percent to a decimal and check your units — per month vs. per year, per $100 vs. per $1,000, 360- vs. 365-day year.

Common Traps

Net-to-seller: divide the desired net (plus costs) by (1 − commission rate). Adding the rate to the net is the #1 wrong answer.
Original-price-from-profit: divide by (1 + profit %); don't subtract the percent from the sale price.
Per $100 vs. per $1,000 vs. mills. A rate "per $100" divides the value by 100; "per $1,000" and mills divide by 1,000. Misreading the unit gives an answer off by 10×.
Monthly vs. annual. Divide annual interest or tax by 12 for a monthly figure; a percentage lease and rent proration usually work in months/days.
360 vs. 365 days. Use the day-count the problem specifies; mixing them changes the daily amount.
Cap rate is inverse to value. Value = NOI ÷ Rate — a higher rate means a lower value.
Survey fractions multiply by 640. "¼ of ¼ of a section" = 1/16 × 640 = 40 acres; multiply all the fractions, then by 640.
Adjust units before multiplying: convert square feet to acres (÷ 43,560) or to square yards (÷ 9) as the problem requires.

Check Yourself

1. A home sells for $380,000 at a 6% commission. The listing and selling brokerages split it evenly, and the listing agent keeps 70% of the listing brokerage's half. What does the listing agent earn?

A) $7,980
B) $11,400
C) $8,000
D) $22,800

2. A seller wants to net $180,000 after paying a 6% commission and no other costs. Rounded to the nearest dollar, the sale price must be:

A) $190,800
B) $191,489
C) $180,000
D) $189,000

3. How many acres are in the S ½ of the NW ¼ of a section?

A) 20 acres
B) 40 acres
C) 80 acres
D) 160 acres

4. A property is assessed at $250,000 and taxed at 15 mills. The annual tax is:

A) $375
B) $3,750
C) $37,500
D) $1,667

5. Annual taxes of $3,650 are paid in arrears (365-day year). Closing is on day 100, and the seller is responsible through day 99. What is the seller's share credited to the buyer?

A) $1,000
B) $990
C) $3,650
D) $365

Answers: 1 — A. Total = $380,000 × 0.06 = $22,800; listing brokerage half = $11,400; agent = 70% × $11,400 = $7,980. 2 — B. Sale Price = $180,000 ÷ 0.94 = $191,489 (rounded). Check: × 6% = $11,489; $191,489 − $11,489 = $180,000. 3 — C. ½ × ¼ × 640 = 1/8 × 640 = 80 acres. 4 — B. $250,000 × (15 ÷ 1,000) = $250,000 × 0.015 = $3,750. 5 — B. Daily = $3,650 ÷ 365 = $10; seller's 99 days × $10 = $990 credited to the buyer.

Real Estate Math