Chapter 20

Changing mixed numbers to improper fraction. Multiply the whole number by the denominator of the fraction, add the result to the numerator and place the new total over the original denominator.

B. Finding a Common Denominator for Fractions. When adding or subtracting fractions, each fractions must have a common denominator. The common denominator is a number into which the denominator of all given fractions will divide evenly.

To change both fractions to have common denominators:

1) Divide the new denominator by the denominator of the original fraction.

2) Multiply the numerator and the denominator of the original fraction by that number.

Example:

C. Adding Fractions. Change all fractions so that they have common denominators. Add the numerators and place the total over the common denominator. Change improper fractions to mixed numbers in the final answer.

D. Multiplying Fractions. Multiply the numerators and multiply the denominator.

E. Dividing Fractions. Invert the fraction you are dividing by and multiply.

Note: In doing fraction problems, reduce all answers to a simples fraction or a mixed number.

II. DECIMALS. A decimal number represents a part or portion of a whole number. It is written with a decimal point to its left. It is the same as a fraction but the denominator is 10, 100, 1,000, 10,000, etc.

A. Adding and Subtracting Decimals. Keep the decimal points in line.

B. Multiplying Decimals. Count off the same number of decimal places in your answer as there are in the two numbers being multiplied.

a) 2.3 x 5 = 11. b) 1.5 x .5 = .75 c) 25 x .05 = 1.25

C. Dividing Decimals. In order to explain the process it is best to give names to the various numbers in a decimal division problem:

1) Change the divisor to a whole number by moving the decimal point to the right.

2) Move the decimal point in the dividend to the right an equal number of places.

3) Place the decimal point in the answer above the point in the dividend.

III. PERCENTAGES. A percent is a part of a proportion of a whole number expressed in hundredths. A percent is a fraction with the denominator always 100. In stating percents, the denominator of 100 is replaced by the percent (%) sign.

A. Converting Percent to Decimal. In solving percent problems, the percent must be changed to its decimal form. Move the decimal point two places to the left and drop the percent (%) sign.

B. Change Fraction Percents to Decimal Form. Divide the numerator by the denominator.

C. Change Mixed Number Percents to Decimal Form. Change the mixed number to an improper fraction. Proceed as in (B) above.

D. Changing Decimals to Percent Form. Move the decimal point two places to the right.

PART 2

INTRODUCTION TO SOLVING WORD PROBLEMS

The majority of the problems found in real estate math involve percentages. The areas involved are as follows:

Commission problems

Loan interest problems

Depreciation problems

Profit and loss problems

Capitalization problems

Return on investment problems

There are variables involved in all percentage problems:

RATE: (percent)

PRINCIPAL: (a whole amount or 100%)

EARNINGS: (The result or multiplying Rate by Principal)

In solving word problems, you will be given two of the variables (or you will have to arrive at them in prior steps) and be asked to determine the missing variable.

Example: Broker sells a home for $20,000 and receives a commission of 6% of the sales price.

What is the Broker's commission?

Solution: 6% of $20,000 = what?

.06 x $20,000 = $1,200

In the above example, we multiplied rate times the sales price to find the missing variable. however, if the problem had given us the commission rate and the commission earned, and we were asked to determine the sales price, we would have to solve it by dividing commission rate into earnings.

Thus, in solving word problems, we must first label the variables, determine which one is missing and solve for it by either multiplying or dividing. To help you know what to do in each case, we have developed the Box Formula.

THE BOX FORMULA

The Three-Variables- Rate, Principal and Earnings - can expressed visually in the following formula:

To use the formula, first, determine the two known variables and place them under their proper headings. The sign above the unknown or missing variable will tell you what to do. As the arrow indicates, compute your answer by working from the left to right.

Example:

a) Rate is 6%, sales price $40,000. What is the total commission?

b) Rate is 5%, broker earns $1,500 commission. What is the sales price?

c) Price is $30,000, commission earned is $1,800. What is the commission rate?

The most difficult part is solving word problems is to properly label the variables. The examples given in this review will help you to make the proper determinations, and with a little practice, you should have no problem.

The chart below indicates the labels given to the different variable in each type of percentage problem and shows you how they fit into the formula.

STRATEGY FOR SOLVING WORD PROBLEM

1. Read the problem carefully and thoroughly.

2. Determine what the question is. What is the missing variable?

3. Pick out the known variables and label them.

4. Place the known variables in the formula and do the operation indicated.

Example (a) A house is sold for $50,000. How much does the broker receive if the rate of commission is 6% of the sales price?

The formula indicates that you multiply.

.06 x $50,000 = $3,000

Example (b) A house is sold for $60,000. The broker receives a commission of $4,200. What is the rate of commission?

The formula indicates that you divide.

(Note: The above solution could be expressive as follows: $4,200 ÷ $60,000 = .70, which is the way it would be done on a calculator.)

Example (c) A buyer takes a loan for one year at 12% interest. At the end of the year he pays back the loan plus $6,000 interest. How much was the loan?

The formula indicates that you divide.

PART 3

COMMISSION PROBLEMS

1. The VARIABLES in Commission problems are as follows:

RATE: Rate of Commission

PRINCIPAL: Sales Price

EARNINGS: Commission Earned

Example (1) A broker's commission for selling a home for $90,000 is 6% of the sales price. How much was the broker's commission?

Example (2) A house is sold for $70,000. The broker earned a commission of $4,900. What was the commission rate?

Example (3) A house is sold for $80,000. The commission is 6% of the sales price. The salesperson who made the sale gets 40% of the commission, 10% goes to cover expenses and the broker gets the balance.

Question: How much is the total commission?

.06 x $80,000 = total commission = $4,800

Question: How much did the salesperson get?

.4 x $4,800 = $1,920

Question: How much did the broker get?

Example (4) An owner want to net $112,800 from the sale of his home after paying a broker a commission of 6% of the sales price. How much should the broker sell the house for?

Solution: The owner will receive 94% of the sales price, so

94% of the sales price will equal $112,800

.94 x sales price = $112,800

$112,800 ÷ .94 = $120,000

PART 4

MEASUREMENT PROBLEMS

There are three types of measurement problems which you must deal with. They are:

I. Basic Rules.

A. All measurements must be in like terms. Problems cannot be solved by multiplying feet by yards, etc.

Example: 6 ft., 6 in. should be converted to 6.5 ft. rather than 78 inches.

B. To find AREA (square feet), multiply width by depth.

C. To find VOLUME (cubic feet), multiply width by depth by height.

D. The PERIMETER (distance around) or LINEAR measure is determined by adding the measure of each side..

II. Tables and Measures

III. Areas of Rectangles, Trapezoids, and Triangles.

A. A RECTANGLE is a closed figure with four sides at right angles to one another.

To find the area of a rectangle, multiply the length by the width. In measuring lots, the width is always the distance along the road or right of way.

b) What is the area of the above figure in square yards?

Since there are 9 sq. ft. per sq. yd., divide 1,800 by 9 = 200 sq. yds.

B. Using LIKE MEASURES. As stated above, all measures must be computed in the same kind of units.

Example: a) How many square feet in a lot 60 ft. x 30 yds.?

Change 30 yards to feet by multiplying 30 x 3 = 90 ft.

Square footage would be 60 x 90 = 5,400 sq. ft.

C. AREA OF TRAPEZOID. A trapezoid is a four-sided figure in which only two of the sides are parallel. To find the area of a trapezoid, multiply 1/2 the sum of the two parallel sides by the height or distance between them.

D. AREA OF TRIANGLES. A triangle has three sides that meet in three angles. To find AREA OF A TRIANGLE, multiply 1/2 the base times the height. The height is always measured on a vertical line which forms as 90° angle with the base.

PRACTICE TRIANGLE PROBLEM.

Mr. Jones sells the lot shown at the right for $6 per square foot. He pays a broker a commission of 6% of the sale price. What was his net?

IV. Area of Odd Shaped Lota. To find the area of any straight-sided lot, break up the area into familiar shapes (rectangles or triangles). Compute the area of the smaller units and add them together for the total area.

V. Volume. Volume is the amount of space contained within a cube. Its is measured in terms of cubic inches, feet, or yards. To determine the cubic area of a cube, multiply the area of the base by the height.

Cubic Measure of a Triangular Shaped Object. 1/2 the area of the multiplied by the height.

Practice Volume Problem. A) Compute the volume of the object in the diagram below:

Solution: 1) Compute volume of rectangular shape cube: 20 x 80 x 40 = 64,000 cu. ft.

2) Compute volume of roof: 40' x 8' = 25,600 ÷ 2 = 12,800 cu. ft.

3) Add the two figures: 64,000 + 12,800 = 76,800 cu. ft.

VI. Linear Measure (Distance around figures). PERIMETER (distance around) of a straight sided figure is found by adding up the measure of each side.

PART 5

PROFIT OR LOSS PROBLEMS

I. The VARIABLES in Profit or Loss problems are as follows:

Ex: a) Owner bought his home for $30,000 and later sold it for $33,000. What percent of profit did he make?

Steps in solving:

1. What is the question? Ans: What is the RATE of P/L.

2. Find the two known variables. PRINCIPAL is the investment or cost, or original purchase price. EARNINGS is his actual cash profit, which in this case is $3,000.

3. Make a sentence which states the problem: "$3,000 is what percent of $30,000." Switch the word around to fit into the Box Formula.

Ex: b) A property was bought for $45,000 and sold as a 20% profit. What was the selling price?

Make a sentence: "What is 20% of $45,000"?

Switch the wording to match the Formula: "20% of $45,000 is what?"

Profit = .2 x 45,000 = $9,000

Selling price = $45,000 (cost) + $9,000 (cost) = $54,000

Ex: (c) A building was sold for $23,000 at a profit of 15%. what was the original purchase price?

The sales price of $23,000 consists of the original purchase price plus the profit (cash). If you sold a house for the same amount that you paid for it, you would have sold it for 100% of your purchase price. If you sold it for a 15% profit, then you would have sold it for 115% of what you paid for it originally. Thus, the $23,000 represents 115% of the original purchase price. Forming our sentence, we get: "$23,000 is 115 percent of the original purchase price?"

Note: If the EARNINGS is greater than the PRINCIPAL, the RATE is greater than 100%.

II. WHEN PROPERTY IS SOLD AT A LOSS, the RATE will be less than 100%.

Ex: a) Owner bought a home for $20,000 and sold it at a 5% loss. What was the selling price?

This problem can be solved in two ways:

Note: In doing LOSS problems, the RATE is always less than 100%.

PART 6

INTEREST

Interest is a charge which you pay a lender when you borrow money. It is rent which is paid for the use of someone else's money.

In computing interest problems you will be dealing with three familiar variables, plus a fourth-TIME.

RATE: The percentage charged for the use of money on the basis of one year.

PRINCIPAL: The amount of money borrowed or due at any one time.

EARNINGS: The charge for the use of the money borrowed.

TIME: Used to compute interest charge when money is borrowed for either more or less than a year.

TYPES OF INTEREST

1. Simples Interest: Paid only on the unpaid balance owing.

2. Compound Interest: paid on the balance plus the accrued interest.

3. Add-On Interest: The interest is determined for the life of the loan, and then added to the principal. The borrower signs a note for the principal and interest combined and pays it back in equal monthly installments. (This is commonly used for installment sales and results in almost double interest, since the borrower is paying for the use of money which he already paid back.)

4. Discount Interest: Borrower signs a note for the principal and interest combined, but only receives the principal. He then pays interest on the full amount of the note.

Note: In this section we will be dealing with simple interest only.

THE FORMULA is the same with addition of time when required.

Ex: a) What is the interest on a loan of $20,000 at 8% for one year?

Ex: b) In the above example, what would be the interest for 2 years and 6 months?

RULES IN CALCULATING INTEREST

1. Interest may be calculated annually, semiannually (1/2 of annual) quarterly (1/4 of annual), monthly (1/12 of annual) or daily (1/360 of annual).

2. In calculating interest on a daily basis, we use a 360 day year with each month having 30 days regarding of the month.

Ex: a) On a loan of $6,000 at 6%, what is the interest due for 3 months and 5 days?

Interest for 1 year = .06 x $6,000 = $360.00

Interest for 1 month = $360 ÷ 12 = $30.00

Interest per day = $30 ÷ 30 = %1.00

Interest for 3 months and 5 days = $90 plus $5 = $95.00

Shortcut: 6% yearly interest is 3% per half year, 1.5% per quarter, or 1/2% per month.

INTEREST ON REAT ESTATE LOANS may be figured in one of two ways.

1. Level Principal Payment. An equal amount on principal is paid each month plus interest which diminishes as the principal goes down.

Ex: a) What is the interest due for the first and second months on a mortgage of $12,000 for 20 years with 6% interest per annum if the principal is paid in equal monthly installments?

Principal payment = $12,000 ÷ 240 (months) = $50/month

Annual interest on $12,000 @ 6% = .06 x $12,000 = $720.

Interest for one month = $720 ÷ 12 = $60.

First month's interest = $60

Principal, after first month's payment has been reduced to $11,950.

Annual interest on $11,950 = .06 x $11,950 = $717.

Interest for one month = $717 ÷ 12 = $59.75

First month's interest is $60. Second month's interest is $59.75.

2. Direct Reduction Loan. The total monthly payment, which includes principal and interest, remains the same during the life of the mortgage. The interest portion of the payment decreases as the principal payments increase. Special mortgage loan payment charts are used to determine the monthly payments and the monthly amortization of the principal.

Example: Brown has a $40,000 mortgage on his home. The interest rate is 12% per annum. The monthly mortgage payment on P. & I. is $421.29. What is the balance due on the mortgage after the first payment?

Total annual interest: .12 x $40,000 = $4800/year

Monthly interest: $4,800 ÷ 12 = $400

Principal payment for first month: $421.29 less $400 = $21.29

Balance Due: $40,000 less $21.29 = $39,978.71

PART 7

DISCOUNTS - POINTS, ORIGINATION FEES

Mortgage lenders often charge discounts to increase their "effective yield" when making loans at competitive interest rates or attractive repayment terms. The discount is a charge made at the inception of the loan and is measured in points, with each point representing a charge of 1% of the loan. Federal regulations allow either party in the sale to pay discount points on conventional and F.H.A. mortgages. Discount points may not be charged to buyers in V.A. loan transactions.

Example: Bank approves a mortgage for $60,000 for a customer. The bank will write the mortgage at a fixed rate of interest of 13% per year, or at a variable interest rate of 11% for the first year. The bank requires 4 points for the variable rate mortgage. How much is the discount charge to be paid at the closing?

Solution: .04 x $60,000 = $2,400.

Service charge or Original Fee: This is a one time charge also measured in points made by the bank when originating a loan. It is usually paid by the buyer and in the case of a V.A. loan the buyer may pay no more than 1%

Practice Problem. Sale price is $90,000 = $72,000 x .03 = $2,160

PART 8

CAPITALIZATION AND RETURN ON INVESTMENT

CAPITALIZATION is a term used to describe a method of appraising income producing property. It is the process of estimating the value of property through the use of a given rate. Value is ascertained by dividing the annual net income (gross income minus expenses) by the investment rate of return for that property.

Ex: a) An apartment building has an annual net income of $18,000. If the capitalization rate for this type of property as 9%, what is the estimate of value?

Ex: b) What is the expected net annual income on a property valued at $180,000 with a capitalization rate of 12%?

Ex c) A building valued at $80,000 produced an annual net income of $6,400. What is its capitalization rate?

NET ANNUAL INCOME is arrived at after deducting normal operating expenses. principal and interest payments on a mortgage are not considered as an expenses. The capitalization rate for the property is determined by an appraiser and reflects what an investor would expect as a rate of return for a particular type of property.

RATE OF RETURN ON INVESTMENT is the actual rate of return produced by an investment based upon its annual net income.

[RATE OF RETURN = CAPITALIZATION]

Rate of Return is actually the same as Capitalization rate and the terms are used inter-changeably.

PRACTICE PROBLEMS

1) An apartment building valued at $90,000, to earn 15% on the total investment annually, should return a net monthly income in what amount?

2) Owner's building produced a semi-annual gross income of $8,000. His expenses were 60% of his gross income. If the capitalization rate is 8%, what is the building worth?

RATE OF RETURN ON CASH INVESTED. When using capitalization rate, the property is treated as being free and clear with no mortgage. The return on investment is not the actual cash received in hand, since the mortgage payments are not taken into consideration. Most investors want to know what they can expect as a return on their cash invested after deducting all expenses including debt service (mortgage payments).

Ex: a) An investor buys a building for $120,000 and puts down $30,000 in cash. If net income is $9,600 and mortgage payments are $6,000 annually, what is his rate of return on cash invested?

Net cash return = $9,600 less $6,000 = $3,600

GROSS INCOME MULTIPLIER. This is a method of evaluating income property by multiplying the gross annual income by a number or factor determined from comparable properties. The GIM is not reliable or acceptable approach in appraising.

Example: Income properties in the area are selling for 8 times the gross annual income. if you are considering buying a comparable building with gross annual income of $30,000, how much would you pay for the building?

Answer: 8 x $30,000 = $240,000

GROSS RENT MULTIPLIER is a yardstick used in evaluating one-family residences or duplexes which are owned for rental income. Value is determined by multiplying the monthly rent by a number or factor developed from comparable sales. For example, a single family home with a monthly rent of $400 sells for $60,000 or 150 times the monthly gross rent. If you were considering buying a comparable property with a $500 monthly rent, applying the GRM, it would be worth $75,000.

PART 9

DEPRECIATION AND APPRECIATION

DEPRECIATION is defined as the loss in value to a property due to one or all of three factors:

Only the building or improvements depreciate. No depreciation is taken for land.

STRAIGHT LINE DEPRECIATION. The value of the improvement depreciated by an equal amount or percentage each year during its estimated economic life. Straight line depreciation will be used exclusively in these problems.

Ex: a) A building worth $100,000 depreciates at a rate of 2% per year. Its annual amount of depreciation is $2,000 per year.

DEPRECIATION RATE is determined by dividing the economic life (in years) into 100%.

Example: A building with an economic life of 50 years will have a depreciation rate of 2%.

(100% ÷ 50 =2%)

USING THE FORMULA TO SOLVE DEPRECIATION PROBLEMS

1) To determine annual depreciation, the varies are follows:

RATE = Annual depreciation rate %

PRINCIPAL = Book value, original cost or reproduction cost. (All referred to as COST in these examples.)

EARNINGS =Annual depreciation (expressed in dollars.)

Ex: b) A building which cost $180,000 to construct depreciates at 4% annually. What is the yearly depreciation in dollars?

2) To determine total amount of depreciation, the variables are:

RATE = Accrued depreciation rate. (Annual rate x Age)

PRINCIPAL = Cost

EARNINGS = Total amount of depreciation.

Ex: c) A building cost $80,000 to reproduce. It is now 8 years old and has depreciated at a rate of 5% annually. How much has it depreciated to date?

3) To determine present value (market value, or remaining value), use the formula as follows:

RATE = Present value rate (100% less accrued depreciation rate).

PRINCIPAL = Cost.

EARNINGS = Present value.

Ex: d) A building cost $80,000 new. It is now 6 years old and has depreciated at a rate of 4% annually. What is its current value?

PART 10

REAL ESTATE TAX PROBLEMS

Real estate taxes are charges levied against real estate according to its value in order to help pay for the municipality's budget. The amount tax is determined by multiplying the assessed value by the tax rate.

TAX RATE. A rate established by a municipality by dividing the total assessed valuation into the annual budget.

Ex: Big town has an annual budget of $336,000 and the total assessed value of the property is $8,000,000. What is the tax rate?

Tax rates can be expressed as so many dollars per $100 or per $1000 of assessed valuation.

Ex. $42 per $1000 = $4.20 per $100 = .42 per $10

MILLS. Tax rate are also expressed as so many MILLS. Mill means dollars per $1,000.

Ex: 42 Mills is the same as $42 per $1,000

ASSESSED VALUATION. This is the value established by a city or town for tax purposes. It is usually based upon a percentage of market value.

APPLYING THE FORMULA FOR SOLVING TAX PROBLEMS

RATE = Tax rate

PRINCIPAL = Assessed valuation

EARNINGS = Annual real estate tax

Ex: a) A property is assessed at 60% of market value of $40,000. What is the tax if the rate is $42 per thousand?

Solution: 60% of $40,000 = .6 x $40,000 = $24,000 assessed valuation.

Multiply assessed valuation by tax rate and divide by $1,000.

b) The taxes on a home are $600. If the tax rate is $30 per thousand and the property is assessed at 50% of market value, what is the market value?

Step One: Determine assessed value. $600 ÷ .03 = $20,000

Step Two: Determine market value. $20,000 ÷ .5 = $40,000

c) Market value is $60,000. Property as assessed at 80% of market value. Taxes for the year are $1,536. What is the rate?

Step One: Determine assessed value. .8 x $60,000 = $48,000

Step Two: Determine tax rate. $1,536 ÷ $48,000 = .032 x 1000 = 32

DEED TAX STAMPS OR TRANSFER TAX

Most states required a deed tax or transfer tax. The amount and who pays the tax varies according to state law. There is no federal deed tax. Some states tax only the equity (price less take-over encumbrances) while others tax the full purchase price. Check the law of your state.

PART 11

PRORATION

PRORATION is the dividing of the expenses of a property between buyer and seller in proportion to the time they owned the property. The most common expenses are:

Real Estate Taxes.

Insurance premiums (assigned policy).

Mortgage Interest (take-over mortgage).

Fuel, etc.

Rents.

RULES

1) Prorations are made to, and including, day of closing. Seller is usually responsible for the day of closing.

2) Calculate time involved. (Either seller's time or buyer's time depending whether item is owed or prepaid.

3) In prorating, a year has 360 days, there are 12 months to a year and each month has 30 days regardless of the month.

4) Carry your figures three places to the right of the decimal point until your final answer, then round off to 2 decimal places. Round up 1¢ if the third decimal is five or larger.

5) To determine who owes what, ask the following questions?

a) HAS THE EXPENSES BEEN PAID?

b) WHO PAID THE EXPENSES?

c) WHO HAS, OR WILL RECEIVE THE BENEFIT?

d) DETERMINE THE TIME INVOLVED AND CHANGE TO DOLLAR AND CENTS.

Ex: a) Real estate taxes for the calendar year are $360 and have been paid in full. Settlement and closing takes place on September 1. Who gets credit for tax and for how much?

In the above example the expenses has been PREPAID by the seller.

Annual taxes = $360

Monthly taxes = $360 ÷ 12 = $30

4 months @ $30 = $120 credit seller

Ex: b) Closing takes place on July 1 and taxes for the calendar year are not due until November 1. Taxes are $1,200 for the year. Who gets credit and for how much?

Annual taxes = $1,200

Monthly tax = $1,200 ÷ $100

6 months x $100 = $600 credit buyer

TYPES OF EXPENSES TO BE PRORATED

I. ACCRUED. Accrued by seller, but not paid. CREDIT to BUYER.

II. PREPAID. Prepaid by seller, but not fully used at closing. CREDIT to SELLER.

PRORATING TAXES

1. If the taxes are owed but not yet due, calculate the time that seller has owned the property and credit this time to buyer.

Ex: Closing date is July 20. Yearly taxes are $1,440 and have not been paid.

2. If taxes have been PREPAID for a period beyond the closing, seller will get a CREDIT for the time he will not own the property during the tax year.

Ex: Closing is November 15. Taxes of $720 for the calendar year have been paid in full.

NOTE: Generally seller is responsible for the day of the closing. However, in some cases, when the closing is on the first of the month, seller is only held responsible up to the last day of the preceding month.

PRORATING FIRE INSURANCE PREMIUMS (HAZARD INSURANCE)

Fire insurance policies may be written for 1, 3 or 5 years and are usually paid in advance to take advantage of reduced yearly rates. When property is sold and the policy is transferred (assigned) the buyer will owe the seller for the unearned premium (paid for but not used).

Ex: Closing date is September 20, 1975. A 3-year fire insurance policy written August 12, 1974 will be transferred to buyer. Paid up 3-year premium is $300. How much credit is due seller.

To determine time, set up the dates as follows:

Subtract by using the "borrowing method"

PRORATION OF MORTGAGE INTEREST

When a mortgage is taken over (or assumed) at the closing, the accrued interest due will have to be prorated. Interest on the mortgage is usually paid in arrears for the period of time preceding day of payment. it is important to determine at the closing whether interest is due up to, and including, the day of the payment or up to, but not including the day of the payment.

Example of interest paid up to and including day of payment.

Ex: Buyer will assume a mortgage with a balance due of $12,000 @ 6% per annum. Interest is paid monthly, with the last payment being paid on March 1, including interest for that day. Closing is March 22. How much interest will be due buyer?

Example of interest paid up to, but not including day of payment.

Ex: Balance due on mortgage is $12,000 @ 6% per annum and is paid quarterly. The last mortgage payment was made July 1 including interest for the month of June. Closing date is August 15. How much will we credit buyer?