Part, Rate and Whole for New York Real Estate Math

The arithmetic is simple after the three roles are correct. The real skill is deciding which number is the base, converting the percentage accurately and resisting a familiar formula when the words describe a different relationship.

What are the three formulas?

Start with:

Part = Rate × Whole

Rearrange only when the missing value changes:

Use letters only as labels:

• P is Part
• R is Rate
• W is Whole

Then:

P = R × W

R = P ÷ W

W = P ÷ R

Do not choose a formula from the size of the numbers. Choose it from what the question asks.

Official source map

The New York State Department of State 77-hour curriculum lists percentages under Subject 10, Real Estate Mathematics. It expressly connects percentages with commissions, interest, appreciation, depreciation and points. The curriculum also includes mortgage qualifying, transfer tax, mortgage recording tax and real property tax rates.

The Department of State salesperson page says the written examination is multiple choice, is based on the 77-hour curriculum and allows 1 1/2 hours after instructions. It does not publish a math-question count or a subject distribution.

Official New York tax materials show the same relationships in practice. The Department of Taxation and Finance explains that a community's level of assessment is a percentage of full value, while its property-tax calculation guide explains taxable assessment and rates per $1,000. These sources show why identifying the base and the unit matters. They do not turn every tax problem into a simple percentage exercise.

How do you identify the whole?

The whole is the base to which the rate applies. It often appears after the word of.

• 6% of the sale price means the sale price is the whole.
• 80% of the value means the value is the whole.
• 2 points on the loan amount means the loan amount is the whole.
• assessed at 40% of market value means market value is the whole.

That shortcut is useful, but read the entire sentence. In a multi-step commission problem, 70% may apply to the brokerage's share, not to the sale price. The nearest dollar amount is not automatically the base.

Ask:

Percentage of what?

Whatever correctly completes that sentence is usually the whole for that step.

How do you identify the part?

The part is the amount created by applying the rate to the whole.

Examples:

• total commission is part of the sale price
• down payment can be part of the purchase price
• salesperson compensation can be part of the brokerage's commission share
• interest is part of the principal for the stated time
• appreciation amount is part of the original value
• assessed value can be part of full market value

The part is not necessarily small in everyday terms. A $36,000 commission is still the part when it is 6% of a $600,000 sale price.

How do you identify the rate?

The rate tells how much of the whole becomes the part. It may appear as:

• a percentage, such as 6%
• a decimal, such as 0.06
• a fraction, such as 6/100
• a verbal relationship, such as “six percent of”

Convert the rate to a decimal before multiplying or dividing unless the calculator handles a percent key in a way you have deliberately verified.

Move the decimal point two places left when changing a percentage to a decimal. Move it two places right when changing a decimal to a percentage.

Why do rates below 1% cause mistakes?

A percentage sign already means “per hundred.” A rate of 0.5% is one-half of one percent, not 50% and not 5%.

0.5% = 0.5 ÷ 100 = 0.005

Example: What is 0.5% of $800,000?

$800,000 × 0.005 = $4,000

Two common wrong answers reveal the conversion error:

• $400,000 comes from treating 0.5 as 50%
• $40,000 comes from treating 0.5% as 5%

Before calculating, estimate. One percent of $800,000 is $8,000, so one-half of one percent must be $4,000.

How do you solve for the part?

Use Rate × Whole when the question asks for the amount produced by a known percentage.

Example 1: Total commission

A property sells for $540,000. The problem states a 5% total commission. What is the commission?

• Whole: $540,000 sale price
• Rate: 5% = 0.05
• Part: unknown commission

$540,000 × 0.05 = $27,000

The commission is $27,000.

This is arithmetic using a stated rate. Real-world commission rates are negotiable; do not import a customary rate into a problem.

Example 2: Loan amount

A lender will make a loan equal to 80% of a $475,000 value used in the problem. What is the loan amount?

• Whole: $475,000
• Rate: 80% = 0.80
• Part: loan amount

$475,000 × 0.80 = $380,000

The loan amount is $380,000.

Example 3: Points

A borrower pays 1.5 points on a $320,000 loan. One point is 1% of the loan amount.

• Whole: $320,000 loan
• Rate: 1.5% = 0.015
• Part: cost of points

$320,000 × 0.015 = $4,800

The points cost $4,800.

How do you solve for the rate?

Use Part ÷ Whole when the question gives an amount and asks what percentage it represents.

Example 4: Commission rate

A stated commission is $31,500 on a $700,000 sale. What rate does the amount represent?

• Part: $31,500 commission
• Whole: $700,000 sale price
• Rate: unknown

$31,500 ÷ $700,000 = 0.045

0.045 × 100 = 4.5%

The rate is 4.5%.

Example 5: Loan-to-value ratio

A loan is $357,000 and the value used in the question is $420,000. What is the loan-to-value ratio?

$357,000 ÷ $420,000 = 0.85 = 85%

The loan-to-value ratio is 85%.

The problem must identify the value basis. Do not substitute a different purchase price, appraised value or lender rule that the facts avoid using.

Example 6: Level of assessment

A simplified exam problem gives a $240,000 assessed value and a $600,000 market value. What percentage of market value is assessed?

$240,000 ÷ $600,000 = 0.40 = 40%

The level of assessment in the problem is 40%.

This illustrates the arithmetic relationship. Actual New York assessment administration has jurisdiction-specific rules and exceptions, including special assessing units, so current official facts control a real property.

How do you solve for the whole?

Use Part ÷ Rate when the question gives the percentage amount and asks for the original or full base.

Example 7: Sale price from commission

A $24,750 commission equals 4.5% of the sale price. What was the sale price?

• Part: $24,750
• Rate: 4.5% = 0.045
• Whole: unknown sale price

$24,750 ÷ 0.045 = $550,000

The sale price was $550,000.

Check:

$550,000 × 0.045 = $24,750

Example 8: Market value from assessed value

A simplified question says $180,000 is 30% of market value. What is the market value?

$180,000 ÷ 0.30 = $600,000

The market value is $600,000.

The answer must be larger than the part because 30% is less than the full amount.

What changes when value increases or decreases?

An appreciation or depreciation question can ask for either the amount of change or the new value.

Find the change first

A property with an original value of $480,000 appreciates by 7.5%.

Change = $480,000 × 0.075 = $36,000

The appreciation amount is $36,000.

Then find the new value

$480,000 + $36,000 = $516,000

The new value is $516,000.

The shortcut is:

$480,000 × 1.075 = $516,000

The multiplier 1.075 represents the full original value plus the 7.5% increase.

For a 12% decrease, 88% remains:

1.00 - 0.12 = 0.88, so 88% remains

A $500,000 value after a 12% decrease would be:

$500,000 × 0.88 = $440,000

Do not subtract 0.12 from the dollar amount. Subtract the dollar value of 12%, or multiply by the percentage that remains.

How do you calculate percentage change?

Use the original amount as the denominator:

Percentage change = Change ÷ Original amount

Example 9: Increase

A value rises from $400,000 to $460,000.

Change = $460,000 - $400,000 = $60,000

$60,000 ÷ $400,000 = 0.15 = 15%

The value increased 15%.

Dividing by $460,000 would answer a different question: what portion of the ending value the change represents.

Example 10: Decrease

A value falls from $625,000 to $550,000.

Change = $625,000 - $550,000 = $75,000

$75,000 ÷ $625,000 = 0.12 = 12%

The value decreased 12%.

Why are percentage increases and decreases not symmetrical?

A rate applies to its own base.

Suppose a value falls 20% from $500,000:

$500,000 × 0.80 = $400,000

An increase of 20% from the new $400,000 base gives:

$400,000 × 1.20 = $480,000

It does not return to $500,000. To rise from $400,000 to $500,000, the change is $100,000 and the new base is $400,000:

$100,000 ÷ $400,000 = 25%

The decrease was 20%; the recovery increase is 25% because the bases differ.

How do multi-step percentage problems work?

Complete one relationship at a time and carry forward the actual dollar result.

Example 11: Brokerage and salesperson split

A property sells for $640,000 with a stated 5% commission. A cooperating brokerage receives 50% of the commission. Its salesperson receives 70% of that brokerage share. What does the salesperson receive?

Step 1: Total commission

$640,000 × 0.05 = $32,000

Step 2: Cooperating brokerage's share

$32,000 × 0.50 = $16,000

Step 3: Salesperson's share

$16,000 × 0.70 = $11,200

The salesperson receives $11,200 under the facts.

The sale price is the whole only in Step 1. The whole changes to $32,000 in Step 2 and $16,000 in Step 3.

What words signal multiplication or division?

These are clues, not substitutes for reading.

Write P, R and W beside the facts before touching the calculator.

What are the most common wrong-answer patterns?

Using the percentage as a whole number

Multiplying by 5 instead of 0.05 makes the result 100 times too large.

Dividing in the wrong order

Rate is Part ÷ Whole, not Whole ÷ Part.

Choosing the ending value as the percentage-change base

Percentage change normally uses the original amount unless the question defines another relationship.

Applying every rate to the sale price

In a split problem, later percentages often apply to a commission or brokerage share.

Giving the change when the question asks for the new total

An appreciation amount and an appreciated value are different answers.

Importing an outside rate

Use the rate stated in the problem. Commission rates are negotiable, lending standards vary and tax rules can change by law or location.

Ignoring units

A rate per $1,000 is not entered like a plain percentage without conversion. The New York property-tax guide, for example, describes local rates commonly stated per $1,000 of taxable assessed value. Follow the units in the problem.

How do you check an answer quickly?

Use three checks.

Size check

If the decimal rate is below 1.00, a positive part should be smaller than the whole.

Reverse check

Put the answer back into the original relationship.

If $24,750 is 4.5% of a $550,000 sale price:

$550,000 × 0.045 = $24,750

Estimate check

Round to a nearby easy rate.

If 4.5% of $550,000 is claimed to be $247,500, compare it with 5% of $550,000, which is only $27,500. The claimed result cannot be reasonable.

Can you use a formula triangle?

Yes. Write P above R and W:

       P
    -------
     R   W

Cover the unknown:

• cover P and see R × W
• cover R and see P ÷ W
• cover W and see P ÷ R

The triangle helps recall the operation. It does not identify which fact belongs in each role. The sentence still controls the setup.

Original practice set

Try these without looking at the solutions.

Question 1

A stated commission is 4% of a $735,000 sale price. What is the commission?

Question 2

A $38,400 amount represents 6% of a sale price. What is the sale price?

Question 3

A $22,750 commission was paid on a $650,000 sale. What rate does it represent?

Question 4

A lender makes a $408,000 loan on a $480,000 value used in the problem. What is the loan-to-value ratio?

Question 5

A property value of $720,000 decreases by 8%. What is the new value?

Question 6

A property rises from $560,000 to $602,000. What is the percentage increase?

Question 7

A $900,000 sale has a stated 4.8% commission. A brokerage receives 55% of the total, and its salesperson receives 65% of that brokerage share. What does the salesperson receive?

Question 8

An assessed value of $252,000 represents 42% of market value in a simplified problem. What is the market value?

Practice solutions

Solution 1

$735,000 × 0.04 = $29,400

Solution 2

$38,400 ÷ 0.06 = $640,000

Solution 3

$22,750 ÷ $650,000 = 0.035 = 3.5%

Solution 4

$408,000 ÷ $480,000 = 0.85 = 85%

Solution 5

Remaining rate = 1.00 - 0.08 = 0.92, or 92%

$720,000 × 0.92 = $662,400

Solution 6

Change = $602,000 - $560,000 = $42,000

$42,000 ÷ $560,000 = 0.075 = 7.5%

Solution 7

$900,000 × 0.048 = $43,200 total commission

$43,200 × 0.55 = $23,760 brokerage share

$23,760 × 0.65 = $15,444 salesperson share

Solution 8

$252,000 ÷ 0.42 = $600,000

Frequently asked questions

What is the part, rate and whole formula?

Part equals Rate times Whole. To find the rate, divide Part by Whole. To find the whole, divide Part by Rate.

What does “of” mean in a percentage problem?

It commonly signals multiplication and points to the whole. In “5% of the sale price,” the sale price is the whole. Confirm the relationship from the full sentence.

How do I turn 4.5% into a decimal?

Divide by 100 or move the decimal two places left: 4.5% becomes 0.045.

Is 0.5% the same as 0.5?

No. A decimal of 0.5 equals 50%. A rate of 0.5% equals 0.005 as a decimal.

How do I know whether to multiply or divide?

If the part is missing, multiply Rate × Whole. If the rate or whole is missing, divide the known part by the other known value.

What is the denominator in a percentage increase?

Use the original amount for ordinary percentage change: change divided by original amount.

Why does the whole change in a commission-split problem?

Each percentage applies to a stated base. The total commission rate applies to sale price, the brokerage split applies to total commission, and the salesperson split applies to that brokerage's share.

Are New York real estate commission rates fixed?

No. Commission rates are negotiable. A math problem should state the rate or provide facts from which it can be calculated.

Does New York publish how many percentage questions are on the exam?

The public salesperson page identifies the format, curriculum basis and time limit but does not publish a math-question count or subject distribution.

Can I use a calculator on the New York salesperson exam?

The Department permits a calculator only if it is silent, battery- or solar-powered, does not contain an alphabetic keyboard and does not print. Recheck the current salesperson page before the exam.

What to do next

Use the New York real estate math formula map to place this relationship beside interest, area, taxes, prorations and investment formulas. Then open the Real Estate Mathematics study guide and complete mixed questions where the unknown changes from part to rate to whole.

For practice, write P, R and W before calculating. Remove the labels only after you can identify the base accurately in multi-step scenarios.

Sources and verification notes

This article was checked on August 27, 2026. The formulas are standard arithmetic. New York sources define the curriculum scope, exam rules and examples of how percentages and rate units operate in real property contexts. A problem's stated facts and current official law control any time-sensitive rate.

1. New York State Department of State, Real Estate Salesperson 77-Hour Curriculum. Subject 10 coverage of percentages, commissions, interest, appreciation, depreciation, points, mortgage qualifying and tax rates.
2. New York State Department of State, Become a Real Estate Salesperson. Multiple-choice format, 77-hour curriculum basis, 1 1/2-hour limit, result reporting and calculator restrictions.
3. New York State Department of Taxation and Finance, The Locally Stated Level of Assessment. Official explanation of level of assessment as a percentage of full value and New York assessment context.
4. New York State Department of Taxation and Finance, How Property Taxes Are Calculated. Taxable assessment, local tax-rate calculation and rates commonly expressed per $1,000.
5. New York State Department of Taxation and Finance, Real Estate Transfer Tax. Current example of transaction rates and thresholds that must be read from official rules rather than reduced to one generic percentage formula.

This article provides educational exam preparation, not tax, legal, lending or brokerage advice. It uses original examples and does not reproduce state-exam questions.