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Real estate math guide

Convert the rate, then choose the right base

A 0.5% fee on $420,000 is $2,100. Entering 0.5 instead of 0.005 would produce $210,000. Write the conversion before touching the dollar amount, then check whether the answer is the right size.

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The short answer

To convert a percent to a decimal, divide by 100: 4.75% = 0.0475. To convert a decimal to a percent, multiply by 100: 0.045 = 4.5%. Divide a fraction's numerator by its denominator: 1/8 = 0.125 = 12.5%. To turn a percent into a fraction, divide the printed number by 100 and reduce. Keep repeating fractions exact when possible, identify the base, and follow the question's rounding instruction.

Reviewed October 25, 2026. These conversions support National XI, Real Estate Calculations, in the June 24, 2026 PSI broker outline. The whole domain has 7 of 100 scored national questions; this is not a separate seven-question conversion quota. Rates and dollar amounts below are invented teaching inputs. A conversion tells you how to use a given rate, not which fee, assessment level or loan term actually applies.

A five-step routine for percentage problems

  1. Name the base. A percentage must apply to something: the loan amount, sale price, original value or a stated subtotal. Mortgage points use the loan amount. A second commission split may use the brokerage share rather than the original commission.
  2. Write the rate in the form you need. Percent to decimal means divide by 100; decimal to percent means multiply by 100. For a fraction, divide the top number by the bottom number. Keep 1/3 as an exact fraction rather than replacing it with 0.33.
  3. Choose the missing quantity. Part = base x decimal rate. Base = part / decimal rate. Decimal rate = part / base. If the question asks for percent, multiply that last quotient by 100 and include the percent sign.
  4. Calculate each stage on its own base. For percentage change, divide the amount of change by the original amount. For successive increases, decreases or splits, label the new subtotal before applying the next rate.
  5. Check the size and final unit. Half of 1% must be smaller than 1%. A rate of 125% produces 1.25 times the base. Retain working precision and use the specified final rounding or other calculation rule; a special program rule can differ from ordinary rounding.
Common conversions and the form to enter
Printed quantityDecimal value or multiplierEquivalent form
4.75%0.04754.75 / 100
0.5%0.005One-half of 1%
1/80.12512.5%
1/8%0.001250.125%
1/40.2525%
1/30.333... repeatingExactly 33 1/3%
2/30.666... repeatingExactly 66 2/3%
3/40.7575%
1/50.220%
3/80.37537.5%
5/80.62562.5%
7/80.87587.5%
6 1/4%0.06256.25% = 1/16
150% of a base1.5One and one-half times the base
150% increase2.5 final-value multiplierOriginal 100% plus 150%
1 3/8 mortgage points0.01375 of loan amount1.375% of loan amount
1 basis point0.00010.01 percentage point

The distinctions that matter

percent, decimal, and fraction conversions: key distinctions
DistinctionHow it worksWhat to notice
A fraction vs. a fractional percent1/8 is 12.5%. Adding a percent sign changes the quantity: 1/8% is 0.125%, or 0.00125 as a decimal.Read the percent sign before dividing.
Mortgage points vs. basis pointsOne mortgage point costs 1% of the loan amount. One basis point describes a rate difference of 0.01 percentage point.A point charge does not promise a fixed interest-rate reduction.
Percentage points vs. relative percent changeMoving from 6% to 6.5% adds 0.5 percentage point. Relative to the original 6% rate, the increase is about 8.33%.Rate subtraction and relative change answer different questions.
Percent of a value vs. percent increase150% of a value is 1.5 times that value. A 150% increase adds 1.5 times the original, producing 2.5 times the original.Look for the word increase.
Exact fraction vs. rounded decimal1/3 is not exactly 0.33. On $600,000, using 0.33 instead of 1/3 understates the result by $2,000.Keep repeating values exact until the requested final step.
One-third assessment example vs. every Illinois parcelIDOR says most property uses a 33 1/3% assessment level, with different treatment for Cook County classifications, farmland and certain preferential assessments.Use the level given; do not assume every parcel is assessed at one-third.

Work through the facts

Find a stated percentage fee

An invented fee is 4.75% of a $360,000 sale price.

  1. 4.75 / 100 = 0.0475.
  2. $360,000 x 0.0475 = $17,100.
  3. Check: 5% would be $18,000, so this answer should be a little lower.

Answer: The stated fee is $17,100. This is a math input, not a standard commission rate.

Keep a fractional percent separate from a fraction

Compare 1/8 of $320,000 with 1/8% of $320,000.

  1. 1/8 = 0.125, so the fraction of the base is $40,000.
  2. 1/8% = 0.125 / 100 = 0.00125.
  3. $320,000 x 0.00125 = $400.

Answer: The answers are $40,000 and $400. The percent sign makes a hundredfold difference.

Calculate fractional mortgage points

A $320,000 loan has a charge of 1 3/8 points.

  1. 1 + 3/8 = 1.375 points.
  2. 1.375% = 0.01375.
  3. $320,000 x 0.01375 = $4,400.

Answer: The point charge is $4,400. The interest-rate reduction, if any, depends on the actual pricing.

Use exactly one-third

A simplified problem expressly gives a 33 1/3% assessment level and a $600,000 market value.

  1. 33 1/3% = 1/3 exactly.
  2. $600,000 / 3 = $200,000.
  3. Using 0.33 would give $198,000, which is $2,000 too low.

Answer: The assessed value is $200,000 under the stated assumption, before any other given tax steps.

Recover the base or the rate

A $9,000 amount represents 2.5% of an unknown base. Separately, what rate does $9,000 represent out of $360,000?

  1. Base: $9,000 / 0.025 = $360,000.
  2. Decimal rate: $9,000 / $360,000 = 0.025.
  3. Percent rate: 0.025 x 100 = 2.5%.

Answer: The missing base is $360,000; the reverse calculation gives 2.5%.

Report three different rate comparisons

A quoted rate rises from 6% to 6.5%.

  1. Difference: 6.5 - 6 = 0.5 percentage point.
  2. Basis points: 0.5 x 100 = 50 basis points.
  3. Relative increase: (6.5 - 6) / 6 x 100 = 8.333...%.

Answer: The rate rises 0.5 percentage point, or 50 basis points. Its relative increase is about 8.33%.

Apply successive shares to the right subtotals

For an invented split, a brokerage receives 60% of a $10,000 total and an agent receives 70% of that brokerage share.

  1. Brokerage share: $10,000 x 0.60 = $6,000.
  2. Agent share: $6,000 x 0.70 = $4,200.
  3. Combined multiplier: 0.60 x 0.70 = 0.42.

Answer: The agent receives $4,200, or 42% of the original total, under these stated terms.

Do not cancel unequal dollar changes

A $400,000 value first rises 10%, then falls 10% from its new level.

  1. After the rise: $400,000 x 1.10 = $440,000.
  2. After the fall: $440,000 x 0.90 = $396,000.
  3. Combined multiplier: 1.10 x 0.90 = 0.99.

Answer: The final value is $396,000, a 1% decrease from the original amount.

Mistakes to avoid

  • Entering 0.5 for 0.5% Use 0.005. The decimal 0.5 means 50%.
  • Reading 1/8% as 12.5% 1/8 is 12.5%, but 1/8% is 0.125%.
  • Calling one mortgage point one basis point A mortgage point is 1% of the loan amount; a basis point is 0.01 percentage point of a rate.
  • Using the ending value for ordinary percent change Divide the change by the original value, unless another comparison is expressly requested.
  • Adding sequential split percentages Multiply their decimal forms when each applies to the preceding share.
  • Rounding one-third to 0.33 before multiplying Retain 1/3 or full calculator precision.
  • Assuming a 150% increase means a 1.5 multiplier The final multiplier is 2.5: the original 100% plus the 150% increase.

Check your understanding

These are original teaching questions, not recalled or predicted PSI questions. Answer first, then explain why the other choices fail.

1. What is 0.625% as a decimal?

  1. 0.0625
  2. 0.00625
  3. 6.25
  4. 0.625
Show answer and explanation

Answer: B. 0.00625

Divide by 100: 0.625% = 0.00625. The decimal 0.625 would instead mean 62.5%.

2. What percent and fraction equal the decimal 0.125?

  1. 12.5% and 1/4
  2. 0.125% and 1/8
  3. 12.5% and 1/8
  4. 125% and 5/4
Show answer and explanation

Answer: C. 12.5% and 1/8

0.125 x 100 = 12.5%. As a fraction, 125/1000 reduces to 1/8.

3. What is 1/8% of $320,000?

  1. $25,600
  2. $4,000
  3. $400
  4. $40,000
Show answer and explanation

Answer: C. $400

1/8% = 0.125% = 0.00125. Multiply $320,000 by 0.00125 for $400. The $40,000 answer ignores the percent sign.

4. A $300,000 loan has a charge of 0.75 points. What is the charge?

  1. $22,500
  2. $2,250
  3. $225,000
  4. $225
Show answer and explanation

Answer: B. $2,250

0.75 points = 0.75% of the loan = 0.0075 x $300,000 = $2,250. Use the loan amount, not the purchase price.

5. A problem expressly gives a 33 1/3% assessment level on $450,000. What assessed value results?

  1. $135,000
  2. $148,500
  3. $300,000
  4. $150,000
Show answer and explanation

Answer: D. $150,000

33 1/3% is exactly 1/3, so divide $450,000 by 3 for $150,000. Using 0.33 would give the incorrect $148,500.

6. A rate rises from 7% to 7.75%. What is the increase in basis points?

  1. 10.71 basis points
  2. 7.75 basis points
  3. 0.75 basis point
  4. 75 basis points
Show answer and explanation

Answer: D. 75 basis points

The difference is 0.75 percentage point. Multiply by 100 for 75 basis points. About 10.71% would describe the relative rate increase, not basis points.

7. A property rises from $400,000 to $460,000. What is the percentage increase?

  1. 60%
  2. 115%
  3. 13.04%
  4. 15%
Show answer and explanation

Answer: D. 15%

The $60,000 change divided by the original $400,000 is 0.15, or 15%. Dividing by the ending value answers a different comparison.

8. Under stated split terms, a brokerage receives 60% of $10,000 and an agent gets 70% of that share. What does the agent receive?

  1. $4,200
  2. $13,000
  3. $7,000
  4. $6,000
Show answer and explanation

Answer: A. $4,200

$10,000 x 0.60 x 0.70 = $4,200. The second rate applies to $6,000, not the original $10,000.

9. An amount increases by 150%. What multiplier gives the final amount?

  1. 2.5
  2. 0.5
  3. 1.5
  4. 0.15
Show answer and explanation

Answer: A. 2.5

A 150% increase adds 1.5 times the original to the original itself: 1 + 1.5 = 2.5. By contrast, 150% of the original uses 1.5.

10. A $9,000 amount is 2.5% of an unknown base. What is that base?

  1. $3,600
  2. $225,000
  3. $22,500
  4. $360,000
Show answer and explanation

Answer: D. $360,000

Divide the part by the decimal rate: $9,000 / 0.025 = $360,000. Check: $360,000 x 0.025 = $9,000.

Common questions

How do I convert a percent to a decimal?

Divide by 100. For example, 4.75% becomes 0.0475 and 0.5% becomes 0.005. Writing the equality before multiplying helps prevent a hundredfold entry error.

How do I convert a decimal to a percent?

Multiply by 100 and add the percent sign. The decimal 0.045 is 4.5%, while 1.25 is 125%. A decimal greater than one represents more than 100%.

How do I turn a fraction into a decimal or percent?

Divide numerator by denominator for the decimal. Multiply that quotient by 100 for percent. Thus 3/8 = 0.375 = 37.5%. Keep an exact fraction when division repeats, as it does for 1/3.

How do I turn 12.5% into a fraction?

Write 12.5/100. Clear the decimal by multiplying numerator and denominator by ten: 125/1000. Reduce to 1/8. Do not drop the division by 100. For a mixed percent such as 6 1/4%, first combine it to 6.25%, then divide by 100: 0.0625 = 1/16.

What is the difference between 1/8 and 1/8%?

1/8 = 0.125 = 12.5%. But 1/8% means one-eighth of one percent: 0.125/100 = 0.00125. On $320,000, those quantities produce $40,000 and $400 respectively.

How do I calculate a rate smaller than 1%?

Use the same divide-by-100 rule. A 0.5% charge on $420,000 is $420,000 x 0.005 = $2,100. Compare with 1% of the base, $4,200, to check the size.

How do fractional mortgage points work?

One point is 1% of the loan amount. For 1 3/8 points, convert the mixed number to 1.375%, then to 0.01375. On a $320,000 loan that costs $4,400. Actual rate reductions vary by lender and loan; one point does not guarantee a particular reduction.

What is a basis point?

A basis point is 0.01 percentage point, equivalent to 0.0001 as a decimal rate. There are 100 basis points in one percentage point. A rate movement from 6% to 6.5% is 50 basis points.

How are percentage points different from percent change?

Subtract two percentage rates to get the percentage-point difference. Divide that difference by the original rate for relative percent change. From 6% to 6.5%, the difference is 0.5 percentage point and the relative increase is about 8.33%.

Does 33 1/3% equal 0.33?

No. It equals exactly 1/3, or 0.333... repeating. On $600,000, one-third is $200,000 but 0.33 gives $198,000. Do not replace an exact repeating fraction with an insufficient approximation.

Is every Illinois property assessed at one-third of market value?

No. IDOR describes one-third as the level for most property and identifies different treatment for Cook County classifications, farmland and certain preferential assessments. Read the parcel type and the level stated in a problem. This page does not calculate an actual tax bill.

What are the part, base and rate formulas?

Part = base x decimal rate. Base = part / decimal rate. Decimal rate = part / base. For a percentage answer, multiply the decimal rate by 100. These inverse calculations require a nonzero divisor.

What base should I use for percentage change?

Use the original amount for ordinary increase or decrease. A rise from $400,000 to $460,000 is $60,000 / $400,000 = 15%. A later return from $460,000 to $400,000 has a different original base and is a decrease of about 13.04%. An original amount of zero makes this ordinary relative-change formula undefined; do not divide by zero.

Do successive percentages add together?

Not when the second applies to the result of the first. A 60% share followed by a 70% share is 0.60 x 0.70 = 42% of the original. A 10% rise followed by a 10% fall leaves 99% of the original, because the bases differ.

Is 150% of an amount the same as a 150% increase?

No. For a $100,000 starting amount, 150% of it is $150,000. A 150% increase adds $150,000 to the original and gives $250,000. The final multipliers are 1.5 and 2.5 respectively.

When should I round a conversion?

Keep exact fractions or full calculator precision during the working steps. Apply the question's requested final unit and rounding rule. Do not assume a general nearest-cent rule overrides a specific loan-program calculation rule. A conversion also does not decide which contractual fee or assessment base applies.

Are these official exam questions or passing predictions?

No. They are original teaching questions aligned with the cited outline. A practice result can reveal a conversion mistake, but it is not a validated estimate of your exam score.

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Official sources and scope

Checked October 25, 2026. The cited PSI outline, NIST percentage definition, CFPB points explanation and IDOR assessment guidance were read; the SEC basis-point definition was checked earlier in this review sequence. NIST supplies the numeric meaning of percent, not a required Illinois exam writing style. Examples use invented numbers and do not set actual fees, loan pricing or parcel assessment levels.

This is exam preparation, not advice for a specific transaction. State law and the applicable facts control. Our editorial and corrections process explains how to report an issue.

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