13.3 Core Real Estate Mathematics: Areas, Percentages, Rates & Legal Descriptions
Key Takeaways
- One acre contains 43,560 square feet; a section contains 640 acres and one square mile; a township contains 36 sections and 36 square miles.
- Florida is surveyed under the government survey system from the Tallahassee Principal Meridian and its base line, and fractional section descriptions are solved by multiplying the denominators.
- The T-bar relationship — Part equals Total multiplied by Rate — solves commission, profit, loss, capitalization, and interest problems from the same structure.
- Percentage of profit or loss is always computed against the ORIGINAL cost or basis, never against the new sale price.
- Volume equals length times width times height, and cubic yardage requires dividing cubic feet by 27, a step candidates routinely omit.
13.3 Core Real Estate Mathematics: Areas, Percentages, Rates & Legal Descriptions
Core Principle: The DBPR broker outline places Basic Math under Content Area XII, and mathematics also appears inside Financing (Finance Math), Investment (Investment Math), and Closing Transactions (Prorating Procedures). This section covers the arithmetic those other areas assume you already have. Every figure here is memorizable, and the exam is closed book.
1. Constants You Must Know Cold
| Measure | Value |
|---|---|
| 1 acre | 43,560 square feet |
| 1 section | 640 acres = 1 square mile |
| 1 township | 36 sections = 36 square miles |
| 1 mile | 5,280 feet |
| 1 square yard | 9 square feet |
| 1 cubic yard | 27 cubic feet |
| 1 front foot | 1 foot of frontage on the street or water, running the full depth of the lot |
2. Area and Volume
| Shape | Formula |
|---|---|
| Rectangle or square | Area = Length × Width |
| Triangle | Area = ½ × Base × Height |
| Trapezoid | Area = ½ × (Base₁ + Base₂) × Height |
| Volume | Volume = Length × Width × Height |
Worked: Irregular Lot
A lot is a rectangle 180 ft × 240 ft with an adjoining right triangle on one end having a base of 180 ft and a height of 90 ft. How many acres?
- Rectangle: 180 × 240 = 43,200 sq ft
- Triangle: ½ × 180 × 90 = 8,100 sq ft
- Total: 43,200 + 8,100 = 51,300 sq ft
- Acres: 51,300 ÷ 43,560 = 1.18 acres
Worked: Concrete Volume
A slab measures 60 ft × 45 ft and is 6 inches thick. How many cubic yards of concrete are required?
- Convert thickness to feet: 6 in ÷ 12 = 0.5 ft
- Cubic feet: 60 × 45 × 0.5 = 1,350 cu ft
- Cubic yards: 1,350 ÷ 27 = 50 cubic yards
[!IMPORTANT] The ÷ 27 step is the most commonly omitted operation on the exam. Any question phrased in cubic yards requires it. Similarly, any thickness or depth expressed in inches must be converted to feet before multiplying.
3. Legal Descriptions and Section Arithmetic
Florida is surveyed under the government survey (rectangular survey) system, measured from the Tallahassee Principal Meridian and its base line, which intersect near Tallahassee. Townships are numbered north and south of the base line; ranges are numbered east and west of the principal meridian.
Solving a Fractional Description
To find the acreage of a description such as the NW ¼ of the SE ¼ of Section 14, multiply the denominators together and divide 640 by the result.
| Description | Denominators | Acres |
|---|---|---|
| NW ¼ of Section 14 | 4 | 640 ÷ 4 = 160 |
| NW ¼ of the SE ¼ | 4 × 4 = 16 | 640 ÷ 16 = 40 |
| S ½ of the NW ¼ of the SE ¼ | 2 × 4 × 4 = 32 | 640 ÷ 32 = 20 |
| NE ¼ of the NE ¼ of the SW ¼ of the NW ¼ | 4 × 4 × 4 × 4 = 256 | 640 ÷ 256 = 2.5 |
Reading the description backwards is the reliable habit. The S ½ of the NW ¼ of the SE ¼ means: start with the section, take the southeast quarter, then take the northwest quarter of that, then take the south half of that.
Worked: Land Value from a Description
A seller conveys the E ½ of the NW ¼ of Section 22 at $7,500 per acre. What is the price?
- Denominators: 2 × 4 = 8
- Acres: 640 ÷ 8 = 80 acres
- Price: 80 × $7,500 = $600,000
4. The T-Bar: One Structure for Most Problems
The great majority of real estate percentage problems are the same relationship in different clothing.
+-------------------+
| PART | Part / Total = RATE
+---------+---------+ Part / Rate = TOTAL
| TOTAL | RATE | Total x Rate = PART
+---------+---------+
| Problem type | PART | TOTAL | RATE |
|---|---|---|---|
| Commission | Commission dollars | Sale price | Commission rate |
| Annual interest | Interest dollars | Loan principal | Interest rate |
| Capitalization | Net operating income | Property value | Capitalization rate |
| Profit or loss | Gain or loss dollars | Original cost | Percentage of profit or loss |
| Loan-to-value | Loan amount | Value or price, whichever is lower | LTV ratio |
Worked: Working Backwards to Sale Price
A broker receives $14,700 as the listing side of a commission. The total commission was 6%, shared equally between the listing and selling brokerages. What was the sale price?
- The listing side is half the total commission, so the total commission was $14,700 × 2 = $29,400
- Sale price = PART ÷ RATE = $29,400 ÷ 0.06 = $490,000
Worked: Seller's Net
A seller must net $310,000 after paying a 6% commission and $4,200 in other closing costs. What must the property sell for?
The seller keeps 94% of the price after commission, so:
- Required amount before other costs: $310,000 + $4,200 = $314,200
- Sale price = $314,200 ÷ 0.94 = $334,255.32, rounded to $334,256
[!IMPORTANT] Never compute a net-to-seller problem by adding 6% to the net. $314,200 × 1.06 = $333,052, which is wrong, because the commission is charged on the sale price, not on the seller's net. Always divide by (1 − commission rate).
5. Profit, Loss, and Percentage Change
Percentage of profit or loss is always measured against the original cost or basis, never against the sale price.
Worked: Percentage of Profit
An investor purchased a Florida duplex for $285,000 and later sold it for $342,000.
- Gain: $342,000 − $285,000 = $57,000
- Percentage of profit: $57,000 ÷ $285,000 = 20%
Dividing by the $342,000 sale price instead would produce 16.67%, the distractor this problem type always offers.
Worked: Working Backwards From a Percentage
A property sold for $391,000, representing a 15% profit over the original cost. What was the original cost?
- The sale price is 115% of cost, so cost = $391,000 ÷ 1.15 = $340,000
- Check: $340,000 × 0.15 = $51,000 gain; $340,000 + $51,000 = $391,000 ✓
Worked: Price Per Front Foot
A waterfront parcel with 95 feet of frontage sells for $1,187,500.
- Price per front foot: $1,187,500 ÷ 95 = $12,500 per front foot
Front-foot pricing values the frontage regardless of lot depth, which is why it is used for waterfront and commercial corridor property where access, not area, drives value.
A parcel is described as the SW ¼ of the NE ¼ of the SE ¼ of Section 8. If the land is worth $9,200 per acre, what is the value of the parcel?
A seller must net $268,000 after paying a 7% brokerage commission and $3,500 in other closing costs. What is the minimum sale price required?
A contractor is pouring a driveway measuring 90 feet by 24 feet at a thickness of 4 inches. How many cubic yards of concrete are required?
An investor bought a Florida commercial building for $640,000 and later sold it for $528,000. What is the investor's percentage of loss?
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