学习材料

Math is the most learnable part of the exam: the same dozen formulas reappear with different numbers. The secret is a single master relationship and disciplined unit-tracking. Recompute every figure yourself — this chapter shows each one worked all the way through.

The rule: the master formula and the "T" method

Almost every real estate math problem is one relationship:

Part = Rate × Whole (equivalently: Whole = Part ÷ Rate, and Rate = Part ÷ Whole).

Draw a T: put the Part on top, and the Rate and Whole on the bottom. Cover what you want:

• Want the Part? Multiply the two on the bottom (Rate × Whole).
• Want the Whole? Divide Part ÷ Rate.
• Want the Rate? Divide Part ÷ Whole.

Commission, interest, tax, LTV, and profit are all this same T with different labels. Track your units (dollars, square feet, percent-as-decimal) and convert percent to decimal by moving the decimal two places left (6% = 0.06).

Area, acreage, and volume

Core facts: Area of a rectangle = length × width. Area of a triangle = 1/2 × base × height. 1 acre = 43,560 sq ft. 1 section = 640 acres = 1 square mile. 1 square yard = 9 square feet.

Worked — rectangle area. A lot is 150 ft × 200 ft: area = 150 × 200 = 30,000 sq ft.

Worked — square feet to acres. A parcel is 87,120 sq ft: 87,120 ÷ 43,560 = 2 acres. A 130,680-sq-ft parcel: 130,680 ÷ 43,560 = 3 acres.

Worked — a strip lot in acres. A parcel is 200 ft × 435.6 ft: area = 200 × 435.6 = 87,120 sq ft; ÷ 43,560 = 2 acres.

Worked — half a section. One section = 640 acres, so one-half section = 320 acres.

Worked — triangle. Base 100 ft, height 80 ft: area = 1/2 × 100 × 80 = 4,000 sq ft.

Worked — square yards of carpet. A room is 15 ft × 18 ft = 270 sq ft; ÷ 9 = 30 square yards.

Worked — lot yield from a tract. A 5-acre tract loses 20% to streets/drainage: usable = 5 × 0.80 = 4 acres. At 0.25 acre per lot: 4 ÷ 0.25 = 16 lots.

Commission

Commission = Sale price × commission rate. Splits chain through: brokerage split, then agent's split of the brokerage share.

Worked — total commission. $350,000 at 6%: 350,000 × 0.06 = $21,000. A $525,000 sale at 6%: 525,000 × 0.06 = $31,500.

Worked — agent's take-home through two splits. $420,000 at 6% = $25,200 total. Brokerages split 50/50: listing brokerage gets 25,200 × 0.50 = $12,600. The listing agent keeps 60% of that: 12,600 × 0.60 = $7,560. Again with $600,000 at 7% = $42,000; 50/50 → $21,000; agent keeps 70%: 21,000 × 0.70 = $14,700.

Worked — find the rate. A $450,000 sale paid $27,000 commission: Rate = 27,000 ÷ 450,000 = 0.06 = 6%.

Net-to-seller (the "net" price problem)

If a seller wants to net a certain amount after a commission (and no other costs), the sale price is not the net divided by (1 + rate) — it is the net divided by (1 − rate), because commission comes off the top of the sale price.

Sale price = Net ÷ (1 − commission rate).

Worked. Seller wants $190,000 net after a 5% commission: 190,000 ÷ (1 − 0.05) = 190,000 ÷ 0.95 = $200,000. Check: 200,000 × 5% = $10,000 commission; 200,000 − 10,000 = $190,000. ✓ Seller wants $285,000 net after 5%: 285,000 ÷ 0.95 = $300,000.

Loan-to-value, down payment, and points

LTV = loan ÷ value (or price if lower). Down payment = price − loan. 1 point = 1% of the loan.

Worked — down payment. $280,000 with 15% down: 280,000 × 0.15 = $42,000. $360,000 with 10% down: $36,000.

Worked — loan from LTV. 80% LTV on $325,000: 325,000 × 0.80 = $260,000. 90% LTV on $250,000: $225,000.

Worked — LTV from the numbers. Loan $180,000 on a $225,000 home: 180,000 ÷ 225,000 = 0.80 = 80% LTV.

Worked — points. 2 points on $300,000: 300,000 × 0.02 = $6,000. 1.5 points on $320,000: 320,000 × 0.015 = $4,800.

Worked — price from a percentage down payment. A $48,000 down payment is 15% of price: price = 48,000 ÷ 0.15 = $320,000.

Worked — total cash to close. $280,000 home, 20% down, closing costs = 3% of the loan. Down = 280,000 × 0.20 = $56,000; loan = $224,000; closing costs = 224,000 × 0.03 = $6,720; total cash = 56,000 + 6,720 = $62,720.

Interest (simple)

Interest = Principal × Rate × Time. One month = Principal × Rate ÷ 12. You can also solve for the missing piece with the T.

Worked — first month's interest. $200,000 at 6%: annual = $12,000; ÷ 12 = $1,000. $180,000 at 5%: annual = $9,000; ÷ 12 = $750. $120,000 at 4.5%: annual = $5,400; ÷ 12 = $450.

Worked — find the rate. A $180,000 loan generated $10,800 interest in year one: Rate = 10,800 ÷ 180,000 = 0.06 = 6%.

Worked — find the principal. $625 of interest in the first month at 5% annual: monthly rate = 0.05 ÷ 12; Principal = 625 ÷ (0.05 ÷ 12) = 625 × 12 ÷ 0.05 = $150,000.

Worked — principal vs. interest in a payment. $150,000 at 6%, monthly P&I payment $899.33. First month's interest = 150,000 × 0.06 ÷ 12 = $750. Principal portion = 899.33 − 750 = $149.33.

Property tax

Tax rates come two ways: per $100 of assessed value, or in mills (1 mill = $1 per $1,000 = $0.001).

Worked — per $100. Assessed $180,000 at $2.50 per $100: 180,000 ÷ 100 = 1,800 units; × 2.50 = $4,500. Assessed $240,000 at $1.80 per $100: 2,400 × 1.80 = $4,320.

Worked — mills. Assessed $150,000 at 25 mills: 150,000 × 0.025 = $3,750.

Proration

Proration splits a shared expense at closing. Two conventions: the statutory/banker's year (360 days, 12 equal 30-day months) and the 365-day (actual) year. Texas property taxes are paid in arrears (at year-end), so at closing the seller owes their share of the year already occupied, credited to the buyer (who will pay the full bill later).

Worked — 360-day tax proration, mid-year. Annual taxes $4,800, closing July 1, year split evenly. The seller owned Jan 1–Jul 1 = the first half = 4,800 × 6/12 = $2,400 (seller's share).

Worked — 360-day, Sept 1 closing. Annual taxes $3,600, closing Sept 1. Seller owned Jan 1–Sept 1 = 8 months; 3,600 × 8/12 = $2,400 (seller's share, credited to buyer).

Worked — 365-day proration. Annual taxes $3,650 (= $10/day). Seller owned 90 days: 90 × 10 = $900 seller's share (arrears → credited to buyer).

Worked — prepaid item (seller paid ahead). A $1,200 one-year hazard policy paid in advance; seller closes 3 months in. Unused = 9 months; 1,200 × 9/12 = $900 the buyer reimburses the seller.

Worked — rent proration. Seller collected the full $1,800 month's rent; buyer owns from the 21st through the 30th = 10 days of a 30-day month. Buyer's share = 1,800 × 10/30 = $600 credited to the buyer.

Profit, loss, and appreciation

Percent change = (new − old) ÷ old. "Percent profit/loss on cost" uses the original cost as the base.

Worked — appreciation. $250,000 → $300,000: (300,000 − 250,000) ÷ 250,000 = 50,000 ÷ 250,000 = 0.20 = 20%. $400,000 → $460,000: 60,000 ÷ 400,000 = 15%. $250,000 → $325,000: 75,000 ÷ 250,000 = 30%.

Worked — depreciation (decline). $500,000 → $425,000: 75,000 ÷ 500,000 = 15% decrease.

Worked — profit on cost. Buy $200,000, sell $250,000: profit $50,000 ÷ 200,000 = 25%.

Worked — profit on total cost (with repairs). Buy $180,000, spend $20,000 → total cost $200,000; sell $230,000: profit $30,000 ÷ 200,000 = 15%.

Worked — loss on cost. Buy $320,000, sell $272,000: loss $48,000 ÷ 320,000 = 15%.

Worked — two-year compounding. $200,000 appreciates 10%, then 10% on the new value: Year 1 → 200,000 × 1.10 = $220,000; Year 2 → 220,000 × 1.10 = $242,000 (not $240,000 — the second 10% is on the larger base).

Valuation math: cap rate, GRM, NOI, price-per-foot, percentage lease

Value = NOI ÷ cap rate. NOI = effective gross income − operating expenses (NOI excludes debt service and depreciation). Value = rent × GRM. Cap rate = NOI ÷ value.

Worked — NOI. Gross annual income $60,000, 5% vacancy, operating expenses $18,000: effective gross = 60,000 × 0.95 = $57,000; NOI = 57,000 − 18,000 = $39,000.

Worked — value from NOI. NOI $24,000 at 8% cap: 24,000 ÷ 0.08 = $300,000. NOI $36,000 at 9%: $400,000.

Worked — cap rate from a sale. NOI $40,000, sold $500,000: 40,000 ÷ 500,000 = 8%.

Worked — GRM value. GRM of 12 × annual rent, monthly rent $1,500 → annual rent $18,000; value = 18,000 × 12 = $216,000. GRM 11, rent $2,000/mo → annual $24,000 × 11 = $264,000.

Worked — price per square foot. 2,400 sq ft priced $360,000: 360,000 ÷ 2,400 = $150/sq ft. Construction: 40 × 50 = 2,000 sq ft × $120 = $240,000; 60 × 45 = 2,700 sq ft × $150 = $405,000.

Worked — percentage lease. Base rent $2,000/mo (= $24,000/yr) plus 5% of annual sales over $500,000; sales $700,000. Overage = 700,000 − 500,000 = $200,000; 5% = $10,000; total = 24,000 + 10,000 = $34,000.

Qualifying ratios

Lenders cap the front-end (housing) ratio — housing payment (PITI) ÷ gross monthly income.

Worked. Gross monthly income $6,000, 28% front-end ratio: max PITI = 6,000 × 0.28 = $1,680.

The seller's net sheet and buyer's cash-to-close

The exam often disguises arithmetic inside a settlement story. Two standard problems:

Worked — seller's net proceeds. A home sells for $400,000. The seller pays a 6% commission, a $3,200 title-policy contribution, and has a $210,000 loan payoff. Net = 400,000 − (400,000 × 0.06) − 3,200 − 210,000 = 400,000 − 24,000 − 3,200 − 210,000 = $162,800. The method: start at the sale price and subtract every seller cost and every payoff.

Worked — buyer's cash to close. A buyer purchases at $400,000 with an 80% loan, pays 1 point, and has $5,500 of other closing costs; earnest money already deposited is $6,000. Loan = 320,000, so down payment = $80,000. Point = 320,000 × 0.01 = $3,200. Cash needed = 80,000 + 3,200 + 5,500 − 6,000 (earnest-money credit) = $82,700. The method: down payment + closing costs − credits already paid.

Worked — commission split three ways. A $500,000 sale at 6% = $30,000. It splits 50/50 between listing and selling brokerages ($15,000 each). The listing brokerage keeps 30% and pays its agent 70%: agent gets 15,000 × 0.70 = $10,500; the brokerage keeps 15,000 × 0.30 = $4,500. Always resolve the splits in order, largest pool to smallest.

Debits and credits at a Texas closing

On a settlement statement, a debit is money a party owes and a credit is money in that party's favor. Learn the recurring entries:

• Sale price: debit buyer / credit seller (the buyer owes it; the seller receives it).
• Earnest money: credit buyer (already paid toward the price).
• New loan proceeds: credit buyer (financing the buyer brings).
• Seller's loan payoff and commission: debit seller.
• Unpaid taxes accrued to the closing date (arrears): debit seller / credit buyer — because the buyer will pay the full bill at year-end, the seller's accrued share is handed over now as a credit.
• A prepaid item the seller already paid past closing (e.g., a paid-ahead insurance policy that transfers): credit seller / debit buyer.

Worked — is the tax proration a debit or credit to the seller? Annual taxes $3,600, arrears, seller owned 8 months before a 360-day-year closing: seller's share = 3,600 × 8/12 = $2,400. Because Texas is an arrears state, that $2,400 is a debit to the seller and a credit to the buyer — the direction, not just the number, is the point the exam tests.