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Chapter 21 · National foundations

Real Estate Math Workshop

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Quiz this chapter

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  1. A method that survives unfamiliar numbers
  2. 1. Percentage, part, and whole
  3. 2. Percentage change
  4. 3. Successive changes
  5. 4. Rectangular area and acreage
  6. 5. Triangles and irregular parcels
  7. 6. Trapezoid area
  8. 7. Fractional sections
  9. 8. Price per front foot
  10. 9. Cubic volume
  11. 10. Commission and split
  12. 11. Price needed for a target seller net
  13. 12. Loan-to-value and down payment
  14. 13. Discount points
  15. 14. Simple annual and monthly interest
  16. 15. Interest for a stated number of days
  17. 16. First-payment amortization
  18. 17. Payment using a supplied factor
  19. 18. Debt-to-income qualification
  20. 19. Property tax with an assessment ratio and mills
  21. 20. Unpaid-tax proration using 360 days
  22. 21. Unpaid-tax proration using actual days
  23. 22. Prepaid-expense proration
  24. 23. Prepaid-rent proration
  25. 24. Buyer cash to close
  26. 25. Seller proceeds
  27. 26. Net operating income
  28. 27. Capitalization
  29. 28. Gross rent multiplier
  30. 29. Cost-approach depreciation
  31. 30. Percentage rent and natural breakpoint
  32. 31. Cash-on-cash and debt-service coverage
  33. 32. Absorption and months of supply
  34. Twenty independent math drills
  35. Math-drill solutions

A method that survives unfamiliar numbers

Read the requested quantity before calculating. Write the units next to each number. Identify the base of every percentage. Separate intermediate results from the final answer. Round at the end unless the question gives another instruction. Check whether the result is economically reasonable.

Every rate and amount in this workshop is hypothetical. These exercises do not state current interest rates, required commissions, tax rates, or universal underwriting standards. The exam bulletin and question control permitted calculators, rounding, day counts, and closing-day allocation. S01

1. Percentage, part, and whole

The basic relationships are part = whole × rate, rate = part ÷ whole, and whole = part ÷ rate. Convert 6% to 0.06 before multiplying. If $18,000 is 6% of a price, the price is $18,000 ÷ 0.06 = $300,000. Dividing by 6 instead of 0.06 produces a result one hundred times too small.

2. Percentage change

A property rises from $250,000 to $275,000. Increase is $25,000. Divide by the original $250,000: 10%. The denominator is the starting value, not the new value.

3. Successive changes

A $200,000 value rises 10% and then falls 10%. Calculate $200,000 × 1.10 × 0.90 = $198,000. Equal percentage increases and decreases do not cancel because their bases differ.

4. Rectangular area and acreage

A rectangular parcel measures 150 feet by 290.4 feet. Area is 150 × 290.4 = 43,560 square feet, or 1 acre. Multiplying length by width yields square units, not linear feet.

5. Triangles and irregular parcels

A triangular parcel has a base of 200 feet and perpendicular height of 150 feet. Area is 200 × 150 ÷ 2 = 15,000 square feet. Divide an irregular parcel into supported geometric shapes; do not use a sloping side as the perpendicular height without justification.

6. Trapezoid area

Parallel sides are 100 and 140 feet, separated by a perpendicular distance of 200 feet. Area is (100 + 140) ÷ 2 × 200 = 24,000 square feet. Average the parallel sides, then multiply by height.

7. Fractional sections

Find the acreage in the east half of the northwest quarter of the southeast quarter of a standard Section A standard one-square-mile unit in the rectangular survey system, subject to actual survey variations. See Chapter 3. Glossary. Multiply 640 × 1/2 × 1/4 × 1/4 = 20 acres. The fractions multiply regardless of the verbal reading order. Real survey irregularities require actual records; this problem assumes a standard section.

8. Price per front foot

A parcel has 120 feet of frontage and sells for $360,000. Price per front foot is $360,000 ÷ 120 = $3,000. Do not divide by its square-foot area when the requested unit is a front foot.

9. Cubic volume

A concrete slab is 30 feet long, 18 feet wide, and 6 inches thick. Convert thickness to 0.5 foot. Volume is 30 × 18 × 0.5 = 270 cubic feet. Divide by 27 cubic feet per cubic yard: 10 cubic yards, before any Waste Conduct or neglect improperly harming another's interest in property, such as the future interest after a life estate. See Chapter 4. Glossary allowance.

10. Commission and split

Assume a negotiated fee of 4% on a $500,000 sale. Total fee is $20,000. Under the hypothetical agreement, the brokerage receives 60% of that fee: $12,000. The affiliated agent receives 70% of the brokerage's amount: $8,400. Apply each split to the correct intermediate base; the agent does not receive 70% of the entire sale price.

11. Price needed for a target seller net

The seller wants $100,000 after a $180,000 payoff, $20,000 of other seller costs, and a hypothetical 5% fee on price. Let price be P. Then P − 0.05P − $180,000 − $20,000 = $100,000. Therefore 0.95P = $300,000, and P = $315,789.47, rounded to cents. Adding 5% to $300,000 is not the same calculation.

12. Loan-to-value and down payment

A $400,000 purchase has an appraised value of $380,000. The lender permits 80% of the lower amount. Loan is $380,000 × 0.80 = $304,000. Cash toward price is $400,000 − $304,000 = $96,000, before closing costs and other credits.

13. Discount points

A loan is $240,000 and points are 1.5. Points cost $240,000 × 0.015 = $3,600. The base is the loan amount, not the price, unless the problem expressly defines another charge.

14. Simple annual and monthly interest

A $180,000 interest-only loan carries a hypothetical 6% annual rate. Annual interest is $10,800. Monthly interest is $10,800 ÷ 12 = $900. This is not the monthly payment on a fully amortizing loan.

15. Interest for a stated number of days

A $100,000 balance accrues 6% simple interest for 45 days using a 360-day year. Interest is $100,000 × 0.06 × 45/360 = $750. A 365-day convention would give a different answer.

16. First-payment amortization

Beginning Principal In agency, the represented client; in finance, the loan balance apart from interest. See Chapter 8, Chapter 15. Glossary is $200,000, annual rate is 6%, and the monthly principal-and-interest payment is $1,199.10. First month's interest is $200,000 × 0.06 ÷ 12 = $1,000. Principal reduction is $199.10. New balance is $199,800.90. Taxes and insurance, when included in a total housing payment, do not reduce the note balance.

17. Payment using a supplied factor

The question gives a monthly payment factor of $6.00 per $1,000 borrowed. For a $225,000 loan, calculate 225 × $6.00 = $1,350 monthly Principal In agency, the represented client; in finance, the loan balance apart from interest. See Chapter 8, Chapter 15. Glossary and interest. The factor already incorporates Assumption Taking responsibility for an existing loan under the governing arrangement. See Chapter 15. Glossary about rate and term; do not apply the interest rate again.

18. Debt-to-income qualification

Assume gross monthly income of $8,000, an allowed housing ratio of 28%, an allowed total-debt ratio of 36%, and $900 of other monthly debt. Housing-ratio limit is $2,240. Total-debt limit is $2,880 minus $900 = $1,980 available for housing. The binding limit is $1,980. The ratios are Assumption Taking responsibility for an existing loan under the governing arrangement. See Chapter 15. Glossary, not universal lender requirements.

19. Property tax with an assessment ratio and mills

Market value An opinion of value under a defined market-value premise, not necessarily asking price or cost. See Chapter 14. Glossary is $300,000. The stated Assessment A valuation or charge under a tax or improvement system; context determines meaning. See Chapter 20. Glossary ratio is 40%, and an allowed exemption reduces taxable assessed value by $20,000. Taxable value is $300,000 × 0.40 − $20,000 = $100,000. At 25 Mill One dollar of tax per $1,000 of taxable value. See Chapter 20. Glossary, tax is $100,000 × 0.025 = $2,500.

20. Unpaid-tax proration using 360 days

Annual unpaid taxes are $3,600. Use twelve thirty-day months. Closing is June 16, and the buyer owns closing day. Seller days are five full months plus fifteen days: 165. At $10 per day, seller share is $1,650: seller Debit A charge to a party on a settlement statement. See Chapter 17. Glossary, buyer credit.

21. Unpaid-tax proration using actual days

Annual unpaid taxes are $3,650 for a nonleap calendar year. Closing is July 11, and the buyer owns closing day. Seller days are 31 + 28 + 31 + 30 + 31 + 30 + 10 = 191. Daily tax is $10. Seller share is $1,910: seller Debit A charge to a party on a settlement statement. See Chapter 17. Glossary, buyer credit. Use actual month lengths when instructed.

22. Prepaid-expense proration

Assume a transferable annual association charge of $3,600 was paid by the seller. Use a 360-day year. Closing is October 1, and the buyer owns that day. The buyer receives the benefit for the final ninety assumed days. Reimbursement is $900: buyer Debit A charge to a party on a settlement statement. See Chapter 17. Glossary, seller credit. This example assumes the charge is properly transferable; do not assume every insurance policy is assignable.

23. Prepaid-rent proration

The seller collected $1,800 rent for a thirty-day June. Closing is June 11, with rent belonging to the buyer from closing day. Buyer receives twenty days: $1,800 ÷ 30 × 20 = $1,200: seller Debit A charge to a party on a settlement statement. See Chapter 17. Glossary, buyer credit. The buyer receives rent income for their period; they are not reimbursing a seller-paid expense.

24. Buyer cash to close

Price is $350,000; buyer costs and prepaids are $8,000; loan proceeds are $280,000; deposit already held is $10,000; allowed seller credit is $3,000. Cash due is $350,000 + $8,000 − $280,000 − $10,000 − $3,000 = $65,000. Include additional Proration Allocation of a recurring amount between parties or periods. See Chapter 17. Glossary only when given.

25. Seller proceeds

Price is $450,000. Assume a $270,000 payoff, a negotiated 4% fee, other seller costs of $7,000, and a $2,000 seller Debit A charge to a party on a settlement statement. See Chapter 17. Glossary for unpaid taxes. Fee is $18,000. Net proceeds are $450,000 − $270,000 − $18,000 − $7,000 − $2,000 = $153,000. This is not necessarily taxable gain.

26. Net operating income

Potential annual rent is $120,000. Vacancy and collection loss are 5% of that rent. Other income is $3,000, and operating expenses are $42,000. Effective gross income Potential income less vacancy and collection loss plus applicable other income. See Chapter 14. Glossary is $120,000 − $6,000 + $3,000 = $117,000. NOI Net operating income before debt service, income taxes, and accounting depreciation under the conventional framework. See Chapter 14. Glossary is $75,000. Do not subtract the owner's mortgage payment when asked for NOI.

27. Capitalization

NOI Net operating income before debt service, income taxes, and accounting depreciation under the conventional framework. See Chapter 14. Glossary is $75,000 and cap rate is 7.5%. Value is $75,000 ÷ 0.075 = $1,000,000. To find the cap rate instead, divide NOI by value. To find NOI, multiply value by rate.

28. Gross rent multiplier

A Comparable A sufficiently relevant property used as evidence in market analysis or valuation. See Chapter 14. Glossary sells for $288,000 and rents for $2,000 monthly. Monthly GRM Gross rent multiplier, using a clearly identified rent period. See Chapter 14. Glossary is 144. Applying that multiplier to a sufficiently comparable property's $2,200 monthly rent gives $316,800. Do not multiply a monthly GRM by annual rent.

29. Cost-approach depreciation

Assume Replacement cost Current cost of equivalent utility using modern materials and methods. See Chapter 14. Glossary of $300,000, Effective age An appraisal estimate reflecting condition and utility rather than chronological age alone. See Chapter 21. Glossary of ten years, total economic life of fifty years, straight-line age-life Depreciation In appraisal, loss in improvement value; in tax, allocation of qualifying basis under prescribed rules. See Chapter 14, Chapter 20. Glossary, and land value of $80,000. Depreciation ratio is 10/50 = 20%, or $60,000. Indication is $300,000 − $60,000 + $80,000 = $320,000. Effective age can differ from actual chronological age.

30. Percentage rent and natural breakpoint

Base annual rent is $36,000 and percentage rate is 6%. A natural breakpoint, when the lease uses that concept, is $36,000 ÷ 0.06 = $600,000. If rent equals base rent plus 6% of sales above that breakpoint, and sales are $900,000, total rent is $36,000 + $18,000 = $54,000. A lease may specify a different breakpoint; use its terms.

31. Cash-on-cash and debt-service coverage

NOI Net operating income before debt service, income taxes, and accounting depreciation under the conventional framework. See Chapter 14. Glossary is $80,000, annual debt service is $50,000, and cash invested is $300,000. Assuming no other deductions in the problem, before-tax cash flow is $30,000, Cash-on-cash return Annual defined cash flow divided by cash invested. See Chapter 20. Glossary is 10%, and Debt-service coverage NOI divided by annual debt service under the stated convention. See Chapter 20. Glossary is 1.60. These results use different denominators and cannot be substituted for one another.

32. Absorption and months of supply

A market has sixty relevant active listings and averages fifteen Comparable A sufficiently relevant property used as evidence in market analysis or valuation. See Chapter 14. Glossary closed sales monthly. Simplified months of supply is 60 ÷ 15 = 4 months, assuming the sales pace persists and no new listings enter. This is a descriptive calculation, not a guaranteed sellout forecast.

Twenty independent math drills

Work these without looking at the solutions. Assume no unstated fees, taxes, or adjustments.

  1. A parcel measures 220 by 396 feet. How many acres?
  2. How many acres are in one quarter of one quarter of a standard Section A standard one-square-mile unit in the rectangular survey system, subject to actual survey variations. See Chapter 3. Glossary?
  3. A $320,000 price includes a negotiated 3% brokerage fee. What is the fee?
  4. An agent receives 65% of a brokerage's $12,000 compensation. What does the agent receive?
  5. A seller requires $80,000 net after $190,000 debt, $15,000 fixed costs, and a hypothetical 5% fee on price. What price is required?
  6. Price is $375,000, Appraisal An opinion of value developed for a defined assignment. See Chapter 14. Glossary is $360,000, and the loan is 75% of the lower amount. What is the loan?
  7. What is the cost of 2 points on a $270,000 loan?
  8. What is one month's simple interest on $240,000 at a hypothetical 5% annual rate?
  9. What is sixty days' simple interest on $150,000 at 6%, using 360 days?
  10. A $100,000 beginning balance bears 6% interest and has a $700 monthly principal-and-interest payment. What is the balance after the first payment?
  11. Taxable assessed value is $180,000 and the rate is 20 Mill One dollar of tax per $1,000 of taxable value. See Chapter 20. Glossary. What is the tax?
  12. Unpaid annual taxes are $7,200, the convention is 360 days, and the seller is responsible for ninety days. What is the seller's Debit A charge to a party on a settlement statement. See Chapter 17. Glossary?
  13. A seller collected $2,400 rent for a thirty-day month. The buyer is entitled to the last twelve days. What credit does the buyer receive?
  14. Price is $300,000, loan proceeds $240,000, deposit held $6,000, buyer costs $7,000, and an allowed seller credit $2,000. What cash is due from the buyer?
  15. Potential income is $96,000, vacancy loss $4,800, other income $1,800, and operating expenses $33,000. What is NOI Net operating income before debt service, income taxes, and accounting depreciation under the conventional framework. See Chapter 14. Glossary?
  16. NOI is $60,000 and cap rate 8%. What is value?
  17. Price is $330,000 and monthly gross rent is $2,500. What is monthly GRM Gross rent multiplier, using a clearly identified rent period. See Chapter 14. Glossary?
  18. Building Replacement cost Current cost of equivalent utility using modern materials and methods. See Chapter 14. Glossary is $250,000, Depreciation In appraisal, loss in improvement value; in tax, allocation of qualifying basis under prescribed rules. See Chapter 14, Chapter 20. Glossary $40,000, and land value $70,000. What is the cost indication?
  19. NOI is $72,000 and annual debt service is $48,000. What is the Debt-service coverage NOI divided by annual debt service under the stated convention. See Chapter 20. Glossary ratio?
  20. A triangular parcel has base 180 feet and perpendicular height 100 feet. What is its area?

Math-drill solutions

  1. 2 acres. 220 × 396 = 87,120; divide by 43,560.
  2. 40 acres. 640 × 1/4 × 1/4.
  3. $9,600. $320,000 × 0.03.
  4. $7,800. $12,000 × 0.65.
  5. $300,000. ($80,000 + $190,000 + $15,000) ÷ 0.95.
  6. $270,000. $360,000 × 0.75.
  7. $5,400. $270,000 × 0.02.
  8. $1,000. $240,000 × 0.05 ÷ 12.
  9. $1,500. $150,000 × 0.06 × 60/360.
  10. $99,800. Interest is $500; Principal In agency, the represented client; in finance, the loan balance apart from interest. See Chapter 8, Chapter 15. Glossary reduction is $200.
  11. $3,600. $180,000 × 0.020.
  12. $1,800. $7,200 × 90/360; the buyer receives the corresponding Proration Allocation of a recurring amount between parties or periods. See Chapter 17. Glossary credit.
  13. $960. $2,400 ÷ 30 × 12; seller Debit A charge to a party on a settlement statement. See Chapter 17. Glossary and buyer credit.
  14. $59,000. $300,000 + $7,000 − $240,000 − $6,000 − $2,000.
  15. $60,000. $96,000 − $4,800 + $1,800 − $33,000.
  16. $750,000. $60,000 ÷ 0.08.
  17. 132. $330,000 ÷ $2,500; this is a monthly multiplier.
  18. $280,000. $250,000 − $40,000 + $70,000.
  19. 1.50. $72,000 ÷ $48,000.
  20. 9,000 square feet. 180 × 100 ÷ 2.