Subject 10 of 19
The calculations named in official Subject 10, plus the math tested across finance, closing, valuation, taxation, and investment questions.
2 chapters 9 lessons 5 free practice questions
Read the rule, try the recall check, then continue to the next lesson.
Chapter 1
Build every core exam calculation from the correct base, formula, time period and unit, then check the answer before moving on.
Exam rule
A percent means parts per hundred. Change 6 percent to 0.06 before multiplying. Every percent problem has a whole, a rate and a part. Part equals whole times rate. Whole equals part divided by rate. Rate equals part divided by whole. First identify the whole. Commission uses the sale price unless the facts name another base. Gross commission equals sale price times commission rate. A salesperson's share is then found from the gross commission, not from the sale price. Net after commission equals sale price minus gross commission when no other costs are included. Required sale price equals desired net divided by one minus the commission rate, again when other costs are ignored. Profit or loss equals selling price minus original cost. Percentage change equals the change divided by the original value. A later value after appreciation equals original value times one plus the rate. A later value after depreciation equals original value times one minus the rate. To work backward, divide the later value by the correct factor. Always label the base before using the calculator.
Why it matters
Choosing the base is the essential step. Label the sale price, commission and salesperson share before applying their percentages.
New York scenario
A Syracuse home sells for $360,000 at 5 percent commission. The gross commission is $18,000. If the salesperson receives 60 percent of that commission, the share is $10,800. The split is not taken from the sale price.
Common misconception: A 10 percent rise followed by a 10 percent fall does not cancel out. The second percentage works on the new value.
Check your recall
Divide the part by the whole, then express the result as a percent.
Ignoring other costs, divide the desired net by one minus the commission rate.
Use the property's value at the start of the period.
Exam rule
Simple interest is calculated on the original principal. Unlike compound interest, previously earned interest is not added to that principal before the next period is calculated. The formula is interest equals principal times annual rate times time in years, often written with I, P, R and T. Principal = interest / (annual rate x time in years). Annual rate = interest / (principal x time in years). Time in years = interest / (principal x annual rate). Keep rate and time together in the denominator when rearranging the formula. If a problem gives months, divide them by 12 to convert them to years; if it gives days, follow its stated day-count method. For a 360-day year, divide days by 360. For an actual-year problem, use the stated 365 or 366 days. A point equals 1 percent of the loan amount, not the sale price. Points cost equals loan amount times the number of points times 0.01. An origination point is a lender charge. A discount point is paid in return for a reduced interest rate. Do not mix the one-time point charge with interest earned or paid over time.
Why it matters
Interest questions often hide the unknown. Points questions test the correct base. Writing the formula before the numbers handles both.
New York scenario
A buyer borrows $320,000 and pays two points. The points cost $6,400. If the same principal earns 6 percent simple interest for three months, the interest is $4,800 because the time is 3 divided by 12.
Common misconception: Not every point is a discount point. Also, an annual rate must be adjusted when the problem covers less than one year.
Check your recall
Interest equals principal times annual rate times time in years.
Divide interest by the product of the annual decimal rate and time in years: P = I / (R x T).
One point is 1 percent of the loan amount.
Exam rule
Keep linear units separate from square units. Perimeter is the distance around a shape and stays in feet. Rectangle area equals length times width. Triangle area equals base times height divided by two. Trapezoid area equals the sum of the parallel sides times the height divided by two. For an L-shaped or irregular lot, split it into simple shapes. Add the pieces, or find a large rectangle and subtract the missing part. One acre is 43,560 square feet. One hectare is exactly 10,000 square meters and is about 2.471 acres. A front foot is one foot along the street or other stated frontage. Front-foot pricing equals price per front foot times frontage. It is not an area calculation. Price per square foot equals price divided by square feet. Price equals square feet times price per square foot. Square feet equals price divided by price per square foot. Write the unit beside every answer.
Why it matters
A correct number with the wrong unit is still wrong. The unit tells you whether to add lengths, multiply dimensions or use a conversion.
New York scenario
A Queens lot is 40 feet wide and 100 feet deep. Its area is 4,000 square feet, and its perimeter is 280 feet. At $1,500 per front foot, its front-foot value is $60,000. That figure does not use the lot's depth.
Common misconception: Do not confuse perimeter with area or front-foot pricing with price per square foot. Each answers a different question.
Check your recall
One acre contains 43,560 square feet.
Add the parallel sides, multiply by the height and divide by two.
It applies a price to each foot of stated frontage, not to the parcel's area.
Exam rule
Loan-to-value, or LTV, equals loan amount divided by the value used by the lender. Loan amount equals value times LTV. Value equals loan amount divided by LTV. In a purchase question, use the lower of the price or appraised value when the facts say the lender follows that rule. If price and value match, down payment percent plus LTV equals 100 percent. Mortgage qualification usually starts with gross monthly income. Divide annual income by 12. The front ratio equals the proposed monthly housing expense divided by gross monthly income. Housing expense often includes principal, interest, property taxes and insurance, called PITI. Include association dues or another housing cost when the problem says to. The back ratio equals total monthly debt divided by gross monthly income. To find limits, multiply gross monthly income by each supplied ratio. Subtract existing monthly debt from the back-ratio limit. The allowed housing payment is the lower result. Ratio limits vary by loan and lender, so use the figures in the question.
Why it matters
The exam may ask for a ratio or a maximum payment. The lower of the front and adjusted back limits controls the answer.
New York scenario
A buyer earns $96,000 a year, or $8,000 a month. With supplied limits of 30 percent and 40 percent, the front limit is $2,400. If other debt is $900, the back limit for housing is $2,300. The lower $2,300 figure controls.
Common misconception: Use gross income, not take-home pay. The back ratio includes the proposed housing payment and recurring debts named in the facts.
Check your recall
Divide the loan amount by the value used for the loan.
Principal, interest, property taxes and homeowners insurance.
Use the lower of the front-ratio limit and the adjusted back-ratio limit.
Exam rule
A proration divides income or an expense as of closing, and the first question is whether the item is accrued or prepaid. An accrued expense is owed but not yet paid. The seller is usually debited for the seller's share, and the buyer is credited because the buyer will pay later. A prepaid expense was already paid by the seller. The buyer is usually debited for the buyer's future share, and the seller is credited. Rent collected in advance can reverse the direction because the seller may owe the buyer the rent for the buyer's ownership period. Next, the day-count rule. A statutory or actual method uses the actual days named in the problem. A 360-day method uses twelve 30-day months. Some questions assign the closing day to the buyer. Others assign it to the seller. Count it only as directed. Find the daily or monthly amount, multiply by the assigned period and keep debit and credit equal.
Why it matters
The amount and its placement both matter. Decide who paid, who benefits and who still owes before calculating.
New York scenario
Annual taxes are $7,300, the problem uses 365 days and the seller owes 90 days. The seller's share is $1,800. The seller receives a $1,800 debit and the buyer receives an equal credit.
Common misconception: Do not assume a 360-day year or who owns the closing day. The problem must supply the convention.
Check your recall
Debit the seller and credit the buyer for the seller's unpaid share.
It assumes twelve months of 30 days each.
The seller collected money that may belong to the buyer's ownership period.
Chapter 2
Calculate New York property taxes, transfer taxes and mortgage recording taxes, then use rental income to solve for value and rates.
Exam rule
Real property tax is based on taxable assessment, not directly on sale price. In the usual exam model, assessed value equals market value times the level of assessment, or LOA. Taxable assessment equals assessed value minus allowed exemptions. A tax levy is the amount a taxing jurisdiction must raise after subtracting other revenue from its budget. A rate per $1,000 equals the levy divided by total taxable assessed value, then multiplied by 1,000. A parcel's tax equals its taxable assessment times the rate, divided by 1,000. Add town, county, school and other supplied rates before using a combined rate. New York generally requires a uniform percentage within an assessing unit. New York City and Nassau County are special assessing units that may use different fractional levels within property classes. An equalization rate compares a municipality's total assessed value with its total market value. It helps compare assessing units. For exam arithmetic, estimated full value equals assessed value divided by the equalization rate. That estimate is not an appraisal of one parcel.
Why it matters
This single chain connects value, assessment, exemptions, the public budget and the final bill. Learn the order and every tax problem becomes shorter.
New York scenario
A property has a $285,000 assessment and a $10,000 exemption. Its taxable assessment is $275,000. At a combined rate of $24 per $1,000, the tax is $6,600.
Common misconception: Do not subtract an exemption from market value unless the question expressly says to. Exemptions normally reduce the assessment used for tax.
Check your recall
Subtract allowed exemptions from the assessed value.
Divide the levy by total taxable assessed value, then multiply by 1,000.
It compares a municipality's total assessed value with its total market value.
Exam rule
New York State real estate transfer tax applies once consideration exceeds $500. The base rate is $2 for each $500, or fractional part of $500. Divide consideration by $500, round any fraction up and multiply by $2. A conveyance for exactly $500 owes no State transfer tax because the statute applies only when consideration exceeds $500. Although the base rate is often called 0.4 percent, rounding a fractional $500 unit upward can make the exact charge slightly higher. The grantor, usually the seller, pays the base tax and any additional base tax, and if the seller fails to pay or is exempt, the buyer must pay instead. An additional 1 percent mansion tax applies to qualifying residential property at $1 million or more. The buyer pays that additional tax unless the buyer fails to pay or is exempt. For certain New York City conveyances, New York State imposes additional transfer taxes, subject to the July 1, 2019 start date and the exception for a binding written contract entered into by April 1, 2019. The additional base tax starts at $3 million for residential property and $2 million for other property. A supplemental tax on residential property starts at $2 million, with rates from 0.25 percent through 2.9 percent. Use the current rate supplied in a problem because the property type, location and price matter.
Why it matters
Transfer-tax questions test thresholds, fractional $500 units and the party charged. Those details can change an otherwise correct answer.
New York scenario
A home outside New York City transfers for $410,250. That is 820.5 units of $500, so use 821 units. The state base tax is $1,642. A simple 0.4 percent shortcut would miss the rounding rule.
Common misconception: Do not apply tax at exactly $500. Do not treat the mansion tax as part of the seller's ordinary base tax.
Check your recall
$2 for each $500 of consideration, or fractional part, when consideration exceeds $500.
The seller normally pays the base tax, and the buyer normally pays the mansion tax.
A fractional $500 unit must be rounded up before the tax is calculated.
Exam rule
Mortgage recording tax is charged for recording a mortgage on New York real property, and it follows the mortgage debt or obligation secured rather than the purchase price. The statewide pieces are a basic tax of 50 cents per $100 and a special additional tax of 25 cents per $100. Another additional tax is 25 cents per $100, or 30 cents per $100 in the Metropolitan Commuter Transportation District. That additional tax does not apply where it has been suspended. New York City, Yonkers and various counties add local tax of their own. Current combined rates therefore vary by location and, in New York City, by property type and mortgage amount. Use the combined rate supplied in the question. If the rate is quoted per $100, divide the debt by $100 and multiply by the rate. For a one-family or two-family residence, subtract the first $10,000 of principal only when computing the additional tax. Do not subtract it from the basic or special additional tax.
Why it matters
Start with the loan, not the price. Use the stated rate. Check whether the $10,000 deduction applies to the tax part you are finding. Work out each part before adding the totals.
New York scenario
A buyer records a $400,000 mortgage where the problem supplies a combined rate of $1.05 per $100. There are 4,000 units of $100, so the tax is $4,200 before any stated adjustment.
Common misconception: Do not use the sale price. Do not apply the one-family or two-family $10,000 deduction to every part of the tax.
Check your recall
The mortgage debt or obligation secured, not the property's purchase price.
State components, local taxes, location, property type and mortgage amount can affect it.
It applies only to the additional tax for a qualifying one-family or two-family residence.
Exam rule
Start an income statement with potential gross income, or PGI, which assumes both full occupancy and full collection. Add other property income when the facts include it. Subtract vacancy and collection loss to get effective gross income, or EGI. Subtract operating expenses to get net operating income, or NOI. Operating expenses can include management, repairs, utilities paid by the owner, insurance and property tax. Debt service, depreciation, capital improvements and the owner's income tax are not operating expenses in this formula. Use the IRV relationship for direct capitalization. Income equals rate times value. Value equals NOI divided by the capitalization rate. Rate equals NOI divided by value. A gross rent multiplier, or GRM, uses gross rent and ignores expenses. Value equals rent times GRM. GRM equals value divided by rent. Match the period. Monthly rent needs a monthly GRM, and annual rent needs an annual GRM. Vacancy rate equals vacant units or lost rent divided by the possible total. Operating expense ratio equals operating expenses divided by EGI.
Why it matters
Build the income statement one line at a time. Then select capitalization or GRM according to whether the facts give net income or gross rent.
New York scenario
A building has $150,000 of potential rent, $10,000 of other income, $8,000 of vacancy loss and $52,000 of operating expenses. EGI is $152,000 and NOI is $100,000. At an 8 percent cap rate, value is $1,250,000.
Common misconception: Do not subtract mortgage payments when finding NOI. Do not use a GRM with NOI, because GRM uses gross rent and ignores expenses.
Check your recall
Add other income, subtract vacancy and collection loss, then subtract operating expenses.
Divide net operating income by the capitalization rate.
The rent period and multiplier period must match, such as monthly with monthly.
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Free web sample
Answer one selected question at a time. These web samples are not the complete subject bank. The mobile app contains all 109 questions for Real Estate Mathematics.
Question 1 of 5
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Worked examples
Choose an answer to see how the rule applies, or reveal the explanation to study the example.
1. Identify
Name the legal, financial or factual issue the question is testing.
2. Apply
Use the controlling rule. Ignore facts that do not change that rule.
3. Conclude
Choose the answer that follows the rule without adding assumptions.
Choose an answer to see how the rule applies, or reveal the explanation to study the example.
1. Identify
Name the legal, financial or factual issue the question is testing.
2. Apply
Use the controlling rule. Ignore facts that do not change that rule.
3. Conclude
Choose the answer that follows the rule without adding assumptions.
Scenario lab
Scenario 1
A Syracuse home sells for $360,000 at 5 percent commission. The gross commission is $18,000. If the salesperson receives 60 percent of that commission, the share is $10,800. The split is not taken from the sale price.
A percent means parts per hundred. Change 6 percent to 0.06 before multiplying. Every percent problem has a whole, a rate and a part. Part equals whole times rate. Whole equals part divided by rate. Rate equals part divided by whole. First identify the whole. Commission uses the sale price unless the facts name another base. Gross commission equals sale price times commission rate. A salesperson's share is then found from the gross commission, not from the sale price. Net after commission equals sale price minus gross commission when no other costs are included. Required sale price equals desired net divided by one minus the commission rate, again when other costs are ignored. Profit or loss equals selling price minus original cost. Percentage change equals the change divided by the original value. A later value after appreciation equals original value times one plus the rate. A later value after depreciation equals original value times one minus the rate. To work backward, divide the later value by the correct factor. Always label the base before using the calculator.
Scenario 2
Annual taxes are $7,300, the problem uses 365 days and the seller owes 90 days. The seller's share is $1,800. The seller receives a $1,800 debit and the buyer receives an equal credit.
A proration divides income or an expense as of closing, and the first question is whether the item is accrued or prepaid. An accrued expense is owed but not yet paid. The seller is usually debited for the seller's share, and the buyer is credited because the buyer will pay later. A prepaid expense was already paid by the seller. The buyer is usually debited for the buyer's future share, and the seller is credited. Rent collected in advance can reverse the direction because the seller may owe the buyer the rent for the buyer's ownership period. Next, the day-count rule. A statutory or actual method uses the actual days named in the problem. A 360-day method uses twelve 30-day months. Some questions assign the closing day to the buyer. Others assign it to the seller. Count it only as directed. Find the daily or monthly amount, multiply by the assigned period and keep debit and credit equal.
Scenario 3
A building has $150,000 of potential rent, $10,000 of other income, $8,000 of vacancy loss and $52,000 of operating expenses. EGI is $152,000 and NOI is $100,000. At an 8 percent cap rate, value is $1,250,000.
Start an income statement with potential gross income, or PGI, which assumes both full occupancy and full collection. Add other property income when the facts include it. Subtract vacancy and collection loss to get effective gross income, or EGI. Subtract operating expenses to get net operating income, or NOI. Operating expenses can include management, repairs, utilities paid by the owner, insurance and property tax. Debt service, depreciation, capital improvements and the owner's income tax are not operating expenses in this formula. Use the IRV relationship for direct capitalization. Income equals rate times value. Value equals NOI divided by the capitalization rate. Rate equals NOI divided by value. A gross rent multiplier, or GRM, uses gross rent and ignores expenses. Value equals rent times GRM. GRM equals value divided by rent. Match the period. Monthly rent needs a monthly GRM, and annual rent needs an annual GRM. Vacancy rate equals vacant units or lost rent divided by the possible total. Operating expense ratio equals operating expenses divided by EGI.
Keep practicing
Use the web guide to understand the rules. Use the mobile question bank to build speed, diagnose weak areas and repeat the material until it sticks.
Keep these within reach
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Start here
Official Subject 10: percentages, commission, interest, appreciation, depreciation, and points
Official Subject 10: area, perimeter, acreage, hectares, irregular lots, and price per square foot
Official Subject 10: basic mortgage qualifying, transfer tax, mortgage recording tax, and property tax
Math across other subjects: loan-to-value ratio and down payment
Math across other subjects: prorations and closing adjustments
Math across other subjects: NOI, capitalization rate, gross rent multiplier, and investment returns
Write the units beside each value before choosing a formula.
Know which calculations are listed in official Subject 10 and which are taught across finance, closing, valuation, tax, and investment subjects.
State the day-count and closing-day convention in every proration problem.
Use net operating income, not debt service or income tax, in a basic capitalization-rate calculation.
The calculations named in official Subject 10, plus the math tested across finance, closing, valuation, taxation, and investment questions. This guide covers 9 lessons with New York scenarios, common mistakes, documents, worked examples, selected web practice and direct links to the sources used.
Subject vocabulary
Open any term for a direct definition, the exam cue, a New York example, the common mix-up and links to the source material.
Exam traps
Percentages, commissions and changing values
A 10 percent rise followed by a 10 percent fall does not cancel out. The second percentage works on the new value.
Area, perimeter, front feet and price per square foot
Do not confuse perimeter with area or front-foot pricing with price per square foot. Each answers a different question.
Prorations, days and closing debits and credits
Do not assume a 360-day year or who owns the closing day. The problem must supply the convention.
New York real estate transfer taxes
Do not apply tax at exactly $500. Do not treat the mansion tax as part of the seller's ordinary base tax.
Gross income, NOI, capitalization and GRM
Do not subtract mortgage payments when finding NOI. Do not use a GRM with NOI, because GRM uses gross rent and ignores expenses.
Forms and records
The exam often gives you a document and asks who uses it, what it proves or when it belongs in the transaction.
Keeps the known values, requested value, units and formula in one place.
Exam cue: Write the units before calculating. They often reveal the correct operation.
Provides the figures used for prorations, transfer charges and cash-to-close questions.
Exam cue: Follow the day-count and closing-day convention stated in the problem.
Supplies assessed value, exemptions, tax rate or levy data.
Exam cue: Use taxable assessed value when the question gives exemptions.
Supplies loan amount, points, rate and projected costs for common finance calculations.
Exam cue: One point equals one percent of the loan amount, not the sale price.
Primary sources
These are the government, statutory and other authoritative materials cited in the lessons and questions above.
12 CFR Part 1026: Truth in Lending, Regulation Z
Electronic Code of Federal Regulations
SI Units: Area
National Institute of Standards and Technology
Real Estate Salesperson 77-Hour Curriculum
New York Department of State
How Property Taxes Are Calculated
New York State Department of Taxation and Finance
Mortgage Recording Tax
New York State Department of Taxation and Finance
Overview of the Assessment Roll
New York State Department of Taxation and Finance
Real Estate Transfer Tax
New York State Department of Taxation and Finance
Real Property System Glossary of Terms
New York State Department of Taxation and Finance
Uniform Assessment Standards: Valuation Standards
New York State Department of Taxation and Finance
Separate required course
1 of 77 course hour
New York assigns this time within its required pre-licensing course. It is not the time needed to use this exam-prep guide or mobile app.
9
lessons
109
mobile questions
5
free web samples
9
sources
Editorial record
Built from the mobile app's reviewed New York lesson and question bank, then checked against the official curriculum and cited sources.
Scope note
This guide supports exam study and issue recognition. It does not replace an approved 77-hour course or advice from an attorney, tax professional, lender, engineer, inspector or other qualified specialist.
