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Table of Contents; Chapter I: Real Property; Chapter II: Legal Ownership; Chapter III: Agency & Ethics; Chapter IV: Contracts; Chapter V: Real Estate Mathematics; Chapter VI: Financing; Chapter VII: Mortgage Insurance; Chapter VIII: Appraisal; Chapter IX: Transfers of Real Estate; Chapter X: Property Management; Chapter XI: Land Control; Chapter XII: Taxation; Chapter XIII: Fair Housing Laws; Chapter XIV: Macroeconomics; Chapter XV: Legal Professional Requirements; Chapter XVI: Notarial Law; Chapter XVII: Selling Real Estate; Chapter XVIII: Trust Funds Handling; Glossary; Index.

Chapter V: Real Estate Mathematics


Educational Objectives: Learn about Interest Computation and Tables, Three-Variable Formulas, Prorations, R. E. TERMS GLOSSARY, INDEX.


(The following is reprinted by permission from the CalBRE Reference Book, p. 263-269, 783-792)


The practice of real estate requires almost daily calculations of mortgage interest and principal, monthly payments, prorations, square footage, yields and similar basic math computations. Although most computations today are done with computers through the use of specially designed software programs or with a hand-held calculator/computer while using convenient charts showing needed tables and formulas, the real estate licensee should know how to solve common problems relating to real estate without benefit of a calculator, since he or she may encounter situations requiring math solutions when no electronic assistance is available.




Simple interest computation involves multiplying the principal (amount of note) by the selected interest rate and the product or result is the interest for one year. Remember, the interest rate (.06 or .09 for example) is a decimal and two points are to be marked off from the right.


There are 12 months in a year or 365 days. This latter figure makes for an awkward denominator. As a result, as an acceptable business practice, we assume 12 months of 30 days each, or 360 days to a year.


To avoid long computations which may involve cumbersome fractions, it is common to use prepared computations in the form of interest tables which show the base as $1, $100 or $1,000 for a variety of interest and time periods. From the interest table, we determine the factor and multiply it by the amount involved if it exceeds or is less than the base of the table. Let us illustrate the different methods and short cuts:


Long Conventional Method


a. What is the interest on $4,650 for 75 days at 10 percent?


b. What is the interest on $4,650 for 1 year, 4 months and 10 days at 10 percent?




(a) 4650 X .10 X 75/360 = $96.88




(b) 4650 X .10 X               |120| or 490/360 = $632.92


                                          | 10|


(Proper fraction for periods if less than 1 year)


(Improper fraction for periods if more than 1 year)


Use of Interest Tables (Method)


(a) Same problem


Look in table for 30 days at 10 percent


factor is                                      8.3340 = 30


factor is                                      8.3340 = 80


factor for 15 days                       4.1670 = 15


                                                20.835075 days


$4,650 = $4.65 per $1,000


                                   Multiply $4.65 X       20.8350 = $96.88


(b) Table not complete to show higher factors, but it could be done this way:


30 day factor                                                                   8.3340


X16 (16 months)


                                                                                   133.3440 +


                                                                                      2.7780 (10 days)


                       136.1220 factor for 1 year, 4 months and 10 days at $1,000




X4.65 (to get additional thousands)


632.92 Answer




4%: Multiply the principal by number of days; cut off right- hand figure and divide by 9.


5%: Multiply by number of days and divide by 72.


6%:        Multiply by number of days; cut off right-hand figure and divide by 6.


7%: Compile the interest for 6% and add 1/6.


8%: Multiply by number of days and divide by 45.


9%:        Multiply by number of days; cut off right-hand figure and divide by 4.


10%: Multiply by number of days and divide by 36.




Bankers 12%-30 day/6%-60 day interest computation method (Using 360 day


year). To find interest on any principal amount for 30 days at 12%, or for 60 days


at 6%, simply move the decimal point in the principal amount two places to left.


What is the interest amount on $8432.67 at 12% for 30 days?


Answer: $84.33


What is the interest amount on $8432.67 at 6% for 60 days?


Answer: $84.33


(Since 12% per annum is 1% a month, and 1% of any number is the hundredth


part of it, then by pointing off two places from the right of a number, it is in effect divided by 100.)


What is the interest on $7,397.64 at 9% for 69 days?


    Interest @ 6% for 60 days =                     $73.98 


(move decimal two places to left)


Interest @ 3% for 60 days =                         $36.99


(1/2 of 6% amount)


Interest @ 9% for 60 days =                       $110.97


We still need 9 days more interest; so


6  days = 1/10 of 60 days 6 days =$11.09 


(1/10 of $110.97)


3  days = 1/2 of 6 days                                     3 days$5.55 


(1/2 of $11.09)


Interest @ 9% for                                           69 days=$127.61






In three-variable formulas, one variable is equal to the other two. Place one variable in the top half of an imaginary circle. Place each of the other two variables in adjacent quarters in the bottom half of the circle. Divide the circle horizontally by a diameter line. This is a division line (/).Divide the bottom quarters of the circle vertically by a radius line. This is a multiplication line ( X ). To find the unknown variable, cover with finger the unknown variable and complete the calculation.



Other three-variable formulas:


Area = Length X Width (A = L X W)


Tax = Assessed Value X Rate (T = A X R)


Commission = Sales Price X Rate (C = S X R)






Common Linear Measurements


One foot = 12 inches


One yard = 3 feet or 36 inches. One rod = 16 1/2 feet, 5 1/2yards


One furlong = 40 rods


100 feet = 6.6 rods


One mile = 5,280 feet, 1,760 yards, 320 rods, 80 chains


Surveyors' Measurements


1 link = 7.92 inches;1 rod = 25 links;1 chain = 4 rods, or 66feet.


(These are the old surveyors' measurements. Modern surveyors use a steel tape or what is called an engineer's chain which is 100 feet long with links of one foot. Thus, a mile measured by a modern steel tape chain is 52.8 chains.)


Spatial or Area Measurements (Length X Width)


1 square foot = 144 square inches


1 square yard = 9 square feet


1 square rod = 30 1/4 square yards


1 acre = 10 square chains, 160 square rods, 4,840 square yards,43,560 square feet


(An acre is an odd and inconsistent measurement. It is supposed to have been the amount of land that a farmer could plow in a day with oxen and the old wooden plow. As a square, it is approximately 208.71 feet on a side.)


A section = 1 square mile, 640 acres


A township = 36 square miles


A quarter section = 160 acres


Area of a square or rectangle = length X width in unit of linear measurement used


Area of a triangle = base X 1/2 altitude


Cubic Measurements (Length X Width X Height)


1 cubic foot = 1,728 cubic inches


1 cubic yard = 27 cubic feet






one foot = 0.3048 meter


one yard = 0.9144 meter


one mile = 1.6093 kilometers or 1609 meters


one meter = 39 inches one kilometer = 3281 feet or .62 miles or 1000 meters 




one square foot = 0.0929 Sq. meter


one square yard = 0.836 sq. meter


one acre = 4068.8 sq. meters


one square mile = 259 hectares or 2.59 sq. km.


one square meter = 10.76 sq. feet


one hectare = 2.47 acres or 10,000 sq. meters


Principal -Interest - Rate ?Time


Principal is the amount of money on which interest is paid usually the loan amount. In a sense, interest is the "rent" paid by a borrower of money to a lender of money for the use of the money. It is the cost of borrowing money. Interest is term ed "simple or "compound.” Simple interest is interest paid only on the principal owed; compound interest is interest paid on accrued interest as well as on the principal owed.


The basic simple interest formula used in finance has four components: interest, principal, rate and time.


I = P X R x T or I = PRT


1.Interest (cost of borrowing expressed in dollars money paid for the use of money).


2.         Principal (amount of the loan in dollars on which the interest is paid).


3.         Rate (cost of borrowing expressed as a percentage of the principal paid in interest for one year).


4.         Time (length of time of the loan usually expressed in years).


One must know what three of these four components are in order to compute the fourth (and unknown) component.


1.         Interest Unknown - Interest = Principal x Rate x Time or I = PRT.


Example: Find the interest on $2500 for 7 years at 13%.




I = $2500 x . 13 x 7


I = $325 X 7


I = $2275


2.         Principal Unknown - Principal = Interest divided by Rate x Time


            or P = I/RxT.


    Example: How much money must be loaned to receive $2275 interest at 13% if the money is loaned for 7 years?


a. P=I/RT                             c.P = $2275/.91


b. P=$2275/.13x7                  d.P = $2500


3.Rate Unknown Rate = Interest divided by Principal X Time


or R=I/P X T.


Example: In seven years $2500 earns $2275 interest. What is the rate of interest?


a. R=I/PT                           c. R=$2275/$17,500


b. R=$2275/$2500x7d. R=.13 or 13%


4. Time Unknown - Time = Interest divided by Rate x Principal or




      Example: How long will it take $2500 to return $2275 interest at an annual rate of 13%?


a. T=I/PR                             c. T=$2275/$325


b. T=$2275/$2500x.13      d. T=7 years


Compound Interest


Compound interest is more common in advanced real estate subjects, such as appraisal and annuities. Compound interest tables are readily available but the principle is discussed here.


As previously stated, compound interest is interest on the total of the principal and its accrued interest. For each time period, called the conversion period, interest is added to the principal to form a new principal amount and each succeeding time period has an increased principal amount on which to compute interest. Conversion periods may be monthly, quarterly, semi-annual or annual.


The compound interest rate is usually an annual rate and must be changed to "interest rate per conversion period" or "periodic interest rate. " To find compound interest (interest on interest) the formula for simple interest of I = PRT is used but the symbol "i" is used instead of RT, "i" being the annual interest rate divided by the conversion periods per year; so I = Pi.


Example: A $5000 investment at 12% interest compounded annually for 3 years earns how much interest at maturity? Solution : Using I = Pi


I = $5,000 X (.12/1)


      First year's I = $5,000 X . 12 or $600.Add to $5000 principal.


      Second year's I = $5,600 x .12 or $672.Add to $5,600 principal.


      Third year's I = $6,272 X . 12 or $752.64.Add to $6,272 principal.


At maturity borrower will owe $7,024.64.Loan has earned interest of $2,024.64 in 3 years.


Example: How much interest will a $1,000 investment earn over 2 years at 16% interest compounded semi-annually? Solution: Since the conversion period is semi-annual, the interest rate is computed every 6 months. The periodic interest rate is 16% divided by two conversion periods: i = 8% for the calculation: I = Pi


1.Original principal amount$1,000.00


2.Interest for 1st period ($1,000X.08) 80.00


3.Balance beginning 2nd period1,080.00


4.Interest earned for 2nd period ($1,080 X .08) 86.40


5.Balance beginning 3rd period1,166.40


6. Interest for 3rd period ($1,166.40 X .08)93.31


7. Balance beginning last period1,259.71


8. Interest for last period ($1,259.71 x .08)100.78


9.Compound principal balance1,360.49


      I for 2 years = $1,360.49 - $1,000 or $360.49.


The same problem using annual simple interest results in $40.49 less interest for the lender ($1,000 X .16 X 2) = $320 simple interest.


Obviously no one in actual practice is going to go through the tedious process outlined above to calculate compound interest. Instead standardized compound interest tables will be used to give the answer quickly.


The exact formula for computing compound interest is:


      S = P (1 + i)n


      S = the compounded amount, principal and interest, or accumulated value-the Sum


      P = the original principal amount or present value of S


      i = the rate of interest per period


      n = the number of conversion periods in the term or periods in which the principal is compounded.


To illustrate this formula, suppose you deposit $100 in a savings account paying 6% interest compounded quarterly. In this case,


      S=compounded sum being sought




      i = 1 1/2% or .0150 (the interest per period, the rate being 6% per year divide 6% by 4 to get the quarterly rate)


      n = 4 (the number of conversion periods in the problem)


      S = P (1 + i)n


      S = 100 (1 + . 0150)4


      S = 100 (1.0150) (1.0150) (1.0150) (1.0150)


      S = 100 (1.0613634)


      S = $106.14


The total interest is $6.14 ($106.14-$100.00).By solving for simple interest, the interest on $100 at 6% per annum is only $6.00.


Licensees should become well versed in using these symbols, formulas and compound interest tables to be able to calculate annuity payment values (mortgage payments, rents), sinking fund needs (accumulation of money in equal periodic deposits at compound interest to meet obligations, such as replacing depreciated assets or debt retirement) and compute amortizations.


Effective Rate of Interest


he "nominal" or "named" interest rate is that rate of interest stated in the loan documents, while the effective interest rate is the rate the borrower is actually paying, commonly called the annual percentage rate (APR).In other words the loan papers may say one thing but the result is another, depending upon how many times a year the actual earnings rate is compounded. The effective rate of interest = the annual rate which will produce the same interest in a year as the nominal rate converted a certain number of times. For example, 6% converted semi-annually produces $6.09 per $100; therefore, 6% is the nominal rate and 6.09% is the effective rate. A rate of 6% converted semi-annually yields the same interest as a rate of 6.09% on an annual basis. 


Discount Rate


When the loan proceeds disbursed by the lender are less than face value (the original principal sum stated in the promissory note), the lender has deducted "up front" a loan fee and perhaps discount points as compensation or remuneration for making the loan on the agreed terms. The borrower thus receives less than must be repaid under the contract. This lending practice is called "discounting. " Sometimes under a bank's commercial loan terms the total interest is deducted in advance from the principal amount of the loan before the loan proceeds are given to the borrower. This method is usually associated with short-term bank loans, but real estate lenders also discount loans by collecting "points" in advance to increase yield.


When more accurate yield and interest tables are unavailable, it is possible to approximate effective interest cost to borrowers or a yield rate to a lender by using the following formulas where discounted loans are involved:


i = approximate effective interest rate (expressed as a decimal)


r = contract interest rate (expressed as a decimal)


d = discount rate, or points deducted (expressed as a decimal)


P =        principal of loan (expressed as the whole number 1 for all dollar amounts)


n = term (years, periods, or fraction thereof)


The formula for approximating the effective rate of interest on a discounted real estate loan may be written:


i = r + (d/n)/(P-d)




What is the estimated approximate effective interest on a $60,000 mortgage loan, with a 20 year term, contract rate of interest being 12% per annum, discounted 3%, so that only $58,200 is disbursed to the borrower?


             i = .12 + (.03/20)/(1-.03) = .12 + (.0015)/.97 = .121546 or 12.15%


Where published yield tables are available, both the tables and the above formula should be compared, especially when tables do not have fractional interest rates or uncommon terms. By interpolation, errors in approximating the result can be minimized.


Principal-Plus-Interest. Another manner of figuring interest is the Principal-Plus-Interest or the "interest extra" approach. Here the borrower makes a fixed payment on the principal each time plus paying the interest on the unpaid balance. Since the interest is different each time, the total payment is different each time. This method is not widely used in California in the case of first mortgages or trust deeds. It is occasionally found in the case of secondary financing.


Straight Loan. This is an arrangement where the principal of the loan is paid off in one lump sum rather than periodic payments. This lump sum payment is usually made when the loan becomes finally due. It is normal, however, that the interest payments be made at intervals during the term of the loan.


Extended Term Amortization. Often in commercial and industrial real estate loans with a very stable and secure tenant the amortization payments are based on a payment schedule that is longer than the actual term of the loan.


For example, the actual term of the loan may be 25 years, but the monthly amortization payments may be based on a term of 30 years. Obviously when the term of the loan ends there will be a balance to be paid off in a lump sum or "balloon payment."


Computers Tables Formulas. The mathematics of financing is generally done with appropriate mortgage tables and formulas and with the computations completed by hand-held high-speed calculators.


Courses in instruction in the use of calculators, computers and the sophisticated tables and formulas in general use are available in most communities.




There are some expenses connected with owning real estate that are commonly paid for in arrears or in advance. For example, fire insurance premiums are normally paid for in advance. Landlords usually collect rents in advance, too. On the other hand, mortgage interest accrues in arrears. Taxes, however, may be paid in advance or in arrears, depending on the time of year of sale. Real property tax adjustment (proration) depends on a number of things, such as: date current installation of taxes is due? What portion of year's taxes has seller paid, is the escrow closing at a time of year when current tax bill has not yet been issued by the tax collector, and, if so, should the parties consider the possibility that the tax bill will substantially increase over the preceding year and agree to a further adjustment between themselves when the new tax bill is available?


When a property is sold, a portion of these expenses has usually been used up by the seller and are rightfully an expense of the seller. However, often a portion of the expenses of ownership has not been used up and when the title to the property is passed on to the buyer, the benefit of these advances will accrue to the buyer. It is only fair that the buyer, therefore, reimburse the seller for the unused portions of these expenses of homeownership.


These adjustments are normally made by the process of proration. This simply means apportioning the expenses (or benefits) fairly to each party.


For example, if the seller sells the property six months after paying the annual fire insurance premium, 1/2 has not been used up and will accrue to the benefit of the buyer. In this case, the buyer should pay 1/2 of the insurance premium to the seller. This example is over-simplified, of course. In practice, prorations are figured down to the day with the so called Banker's rule of 30 days to a month and 360 days to a year usually applying.


Taxes. The real property tax year runs from July 1 of one year to June 30 of the following year (fiscal year).Taxes may be paid in one or two installments. The first installment is due November 1. The second is due February 1.The entire bill may be paid at the time the first installment is due. Suppose an escrow for the sale of a small residence closed January 16.Taxes for the full year in the amount of $600 are to be prorated as of close of escrow between buyer and seller. How much should the buyer be charged in escrow to reimburse the seller for advance paid taxes?


Taxes are paid through June 30th.Buyer will owe seller for 5 1/2 months tax adjustment.$600/12 = $50/month.5 1/2 x 50 = $275 due seller.


Insurance. Fire insurance rates are usually quoted at so much per $100 of coverage per year. Assume the rate is 95 cents per $100. For a $47,000 insurance policy paid in advance to March 31, what will be the buyer's reimbursement to seller in the above escrow if the insurance premium is an annual premium and adjustment is as of close of escrow?


Premium = $. 95 x 470 = $446. 50


Remaining days under policy = 75 days


$446.50/360 days = $1.24/day


$1.24 x 75 = $93 due seller.


Rents. Suppose the seller of a single-family dwelling rented it for $450 a month. Rent was to be adjusted in escrow as of the closing date of May 1.Rent is paid on the 25th of each month, and the tenant is current in paying rent. Who is to be charged and credited in the escrow on the rent adjustment? What is the amount of the adjustment?


Rents are paid in advance. Seller has collected rents to May 25. Escrow will debit seller and credit buyer as follows:


$450/.30 = $15/day


24 (days of the adjustment) x $15 = $360 credit to buyer.


Mortgage Interest. The buyer is purchasing a home subject to two assumable trust deeds of record. Interest is to be prorated between buyer and seller as of close of escrow. Both loans are current. Escrow closes on the 21st day of the month. The first trust deed has an unpaid principal balance of $59,200, monthly installment payments are $711. 54 including interest at 11. 5%. Payments are due the first of each month. The second trust deed unpaid principal is $11,700, principal and interest monthly payments are $200 per month, interest is 13% per annum, monthly installments are due the 24th day of each month. Whose account is credited for interest in the escrow?


First trust deed                                         Second trust deed




21 days                                                    21 days (closing


(closing date)                                                 date)


-1 day                                                    -24 days


(interest due date)                                   (interest due date


20 days of                                                27 days of


interest owed                                                 interest owed


by seller                                                         by seller


$59,200 x .115/360                                 $11,700 x .13/360xx 20 = $378.22,                                                          27 = $114.08,


credit buyer and                                       credit buyer and


debit seller.                                              debit seller.


Taxes and Assessments. As of June 1978 a limit on real property taxation was set by constitutional amendment. This limitation is one percent of "full cash value" (market value) plus a maximum authorized two percent annual inflationary factor and an additional sum (averaging 1/5 to 1/4 of one percent) to pay for any existing indebtedness on affected property, approved by voters prior to the passage of the amendment.


In computing the tax to be paid on California real property for the 1978-79 and subsequent Assessment Roll years, the following is to be considered:


(1) the "full cash value" of the property established as of (a) the lien date of the 1975 base year, or (b) the date a property is purchased, newly constructed, or has a change of ownership, after the 1975 lien date; (2) the maximum two percent inflationary factor applied to the base value, which inflationary (cost of living) factor is to be added to the base year value for each lien date after the lien date on which "full cash value" is determined; (3) an additional amount may be added to pay for any bonded indebtedness previously approved by the voters, which amount will vary from county to county and will not be applicable to all parcels.


For example, a county tax bill may be computed as follows: assume an appraised value of $57,000; deduct homeowner's exemption of $7,000, leaving $50,000 as amount taxable; multiply that amount by the tax rate of .012 (1% tax limit plus two-tenths of one percent allowed for bonded indebtedness); the resulting tax bill is $600. 00.


Because real property is reassessed as a result of incidents of transfer (unless the transfer is subject to a statutory exemption), a supplemental tax assessment is usually imposed by the county assessor. This supplemental assessment increases the taxes due for the remainder of the fiscal year within which the transfer occurred.


(End of the CalBRE Reference Book excerpt)