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Real Estate Mathematics
Computing time
The two most common methods of measuring time for
interest computations are (1) 30-day month time and (2) exact time. Using
the 30-day method, the year is dived into 12 30-day method, and 360 days
in one year. Using the exact-time method, the year is represents by 365
days or, if a leap year, 366 days.
In computing interest, the time factor is represented
by a fraction. The numerator
represents the amount of days for which
interest is to be charged, the denominator the number of days in the year
? either 360 or 365. In calculating the number of days on which interest
is to be charged, the days are counted from the day following creation of
the obligation up to and including the last day. The rule is simplified by
remembering to skip the first day and include the last day.
Assuming interest is to be calculated for 67 days, the
fraction representing time would be 67/360 if the 360-day year is used,
and 67/365 if an exact time year is used. The 360-day year, however, is
the one used in real estate transactions, and it will be used in all
subsequent examples.
Where the period of time is expressed in seven years,
the time factor will not be shown as a fraction, but rather as a whole
number representing the actual years. Thus, in calculating the interest
for a loan in which the time is 11 years. Where the period of time is
expressed in even months, the time factor is shown as a fraction with a
denominator of 12. Thus, for a loan of 7 months, the time factor fraction
will be shown as 7/12. Where the period of time is to be expressed in
days, the time factor fraction will show the amount of days as the
numerator and 360 as the denominator. If the interest is to be calculated
for 79 days, the time factor fraction will be shown as 79/360.
Monthly Payment Formula:
Suppose you take out a 30 year mortgage for $100,000 at 7% interest, and want to know the monthly payments. To do that, you divide the interest rate by 12 to get (.07/12) = .00583; and multiply 30 x 12 = 360 to get the number of payments. Then the formula gives you:
payment = [$100,000(1 + .00583)360 x .00583] / [(1 + .00583)360 - 1]
= $665
Interest formula.
I = P x R x T
Interest = Principal x Rate x Time.
Question 1
What is the interest
on a loan of $18,000 for 3 years at 11%?
Interest = $18,000 x 0.11x 3
Interest = $5,940
The $18,000 is first multiplied by 0.11, giving an
answer of $1,980. The result is then multiplied by 3, giving an answer of
$5,940.
Question 2
What is the interest on a loan of $16,000 for 7 months
at 9%?
Interest = $16,000 x 0.09x 7/12 = $840
Question 3
What is the interest on a loan of $30,000 at 13 ? %
interest for 70 days?
Interest = $30,000 x 0.135 x 70/360
Interest = $787.50
Rate of interest may be found by use of the
formula.
R = I/ P x T
Example 1
If the loan amount is $120,000 the time is 3 years,
and the interest charges is $36,000, what is the rate?
Rate = $36,000 / ($120,000x3) = $36,000 / $360,000 = 1
/ 10 = 10%
Example 2
Amount of loan is $12,000, interest is $595, and time
is 7 months. What is the rate?
Rate = $595 / ($120,000 x 7/12) = $595 / $7,000 =
0.085 = 8 ?
%
Principal formula
Principal can be found by using the formula:
P= I / (R x T)
Principal = Interest / (Rate x Time)
Question 1
What is the principal amount of a loan on which the
interest is $35,625, the rate is 9 ? %, and the time is 5 years?
Principal = $35,625 / (0.095 x 5) = $35,625 / 0.475 =
$75,000
Question 2
If the interest is $350, the rate is 12%, and the time
is 5 months, what is the principal amount of the loan?
Principal = $350 / (12/100 x 5/12) = $7,000
(Rate is usually stated as decimal. However, in this
problem the 12% rate is not expressed as 0.12 but is shown as the fraction
12/100 because the time, 5 months, is better expressed as 5/12 than as its
decimal equivalent 0.41 2/3 )
Question 3
What is the principal amount of the loan if the
interest is $2,000, the rate is 8%, and the time is 3 months?
Principal = $2,000 / (0.08 x 0.25) = $2,000 / 0.02 =
$100,000
(In this problem, the time 3 months expressed
fractionally is 3/12, which reduces to1/4. The decimal equivalent of ? is
0.25, so it is more convenient to express both the rate and the time as
decimals)
Time Formula
Time can be found by using the formula
T= I / (P x R)
Time = Interest / (Principal x Rate)
Question 1
What is the time on a loan of $70,000 when the rate is
11 ? % and the interest is $28,175?
Time = $28,175 / ($70,000 x 0.115) = 3 ? years.
60-day, 6 % method
Numerous shortcut methods for finding interest are
available. One of the more common is the 60-day, 6 % method. The basis of
this method is that 6 % interest for 1 year equals 1 % interest for 2
months, or 60 days. The 1 % is easily found merely by moving the decimal
point in the amount of the principal.
To find interest at 6 % for 60 days, merely move the
decimal point in the principal amount 2 places to the left.
Example:
What is the interest on $2,550 at 6% for 60 days?
Moving the decimal point 2 places to the left in
$2,550 we have $25.50, which is the amount of interest. We can check this
by using formula I = P x R x T
Interest = $2,550 x 0.06 x 60/360 = $25.50
Example 2
What is the interest on $2,550 at 6% for 150 days?
$25.50 interest for 60 days
$25.50 interest for 60 days
$12.75 interest for 30 days (1/2 of the 60-day
amount)
$63.75 interest at 6% for 150 days.
Example 3
What is the interest at 2,550 at 8% for 150 days?
Calculation of the interest by the 60-day 6% method
gives us $63.75 as interest, but this is at 6%. To find the interest at
8%, we must realize that 8% is 1/3 more than 6%, so we take 1/3 of $63.75.
Therefore it equals:
$63.75 (at 6%)
$21.25 (additional 2%)
$85.00 (interest at 8%)
PERCENT
There are generally three elements: (1) base, (2)
rate, and (3) part (also referred to as percentage or portion)
The base is the principal amount and represents 100
percent
The rate is the relationship between the base and a
part of the base and is expressed as a percent.
The part of the base will vary in size, depending on
the rate used; any may be more or less than the base.
The following formulas are used:
Base = Part / Rate (B = P / R)
Rate = Part / Base (R = P / B)
Part = Rate x Base (P = R x B)
The following examples will illustrate the use of the
formulas.
Example 1
Lovran receives $3,000, representing an 8% return on an investment. What amount has he invested?
$3,000, rate is 8%, base is unknown
Using the formula: B = P/R
B = $3,000 / 0.08
B= $37,500
Example 2
McNaughton sells a building for $280,000 and receives
a commission of $19,600. What percent of the selling price is his
commission?
Base is $280,000, and part is $19,600
R = P/B
R = $19,600 / $280,000 = 0.07 = 7% rate
Example 3
John tells Smith that he can expect a 9% return on an
investment of $385,000. How much will Smith receives?
Base is $385,000 and rate is 9%
P = R x B
P = 0.09 x $385,000 = $34,650
COMMISSIONS
The following are examples of common types of
percentage problems found in real estate practice.
Example 1
Salesperson Thomas gets one half of the commission
that her firm receives of the sale of a building. If the building sells
for $850,000 and the commission rate is 5%, Thomas?s share is:
Commission = 0.05 x $850,000 = $42,500
$42,500 / 2 = $21,250
Example 2
Broker John sells a building for $286,500. He receives
a commission of 6% of the first $100,000 and 5% of the balance of the
selling price. How much does he get?
0.06 x 100,000 = $6,000
0.05 x 186,500 = $9,325
$6,000 + $9,325 = $15,325 John?s total commission
Example 3
Salesperson Wagner?s contract with broker Wong states
that if Wagner lists and sells a property he is to receive 60% of the
commission, less 10% for the advertising expense. On the sale of a house
at $128,500, what will Wagner receive? The commission rate is 6% of the
selling price.
0.06 x $128,500 = $7,710 commission on sale received
by office
0.60 x $7,710 = $4,626 Wagner?s share before
advertising deduction.
0.10 x $4,626 = $462.60 deduction for advertising
expense
$4,262 - $462.60 = $4,163.40 Wagner?s share
of commission
Profit and
Loss
Question 1
Jack purchases a lot for $5,000 and later sells it for
$8,000. What is the percent of profit on the original cost?
$8,000 selling price - $5,000 cost = $3,000 profit
Rate = Part ($3,000) divided by base ($5,000) = 0.60 =
60%
Question 2
Jack sells his residence for $360,000 and makes profit
of 25% based on the original cost. What was the cost?
In this type of problem, the selling price represents
the cost plus the profit. The cost is the base, which is always 100%, and
this is added 25% profit, so that the selling price of $360,000 represents
125%.
B = P / R
B = $360,000 / 1.25 = $288.000 original cost
Question 3
Lucia purchases 10 acres of land for $200,000. She
wants to sell the land and make profit of 35%. For how much must she sell
the property?
0.35 x $200,000 = $70,000 = profit of 35%
$70,000 desired profit + $200,000 cost = $270,000
selling price.
Net Listing
Stacey tells broker John the she wants to net $141,000
on the sale of this property. For how much must John sell the property in
order to make 6% commission in addition to the $141,000 for Stacey?
Since the selling price must include $141,000 plus 6%
commission, the reader should realize that the $141,000 must represent
94%. This, when added to the 6% commission, equals to the selling price or
100%.
Base (100%) = $141,000 / 0.94 = $150,000 selling
price.
The $150,000 selling price minus 6% commission of
$9,000 equals $141,000 net to the seller.
Capitalization
Capitalization of income is generally used to refer to
the annual expected rate of return on a property.
Example 1
Travers wants to purchase an income property that
shows a net yearly income of $120,000. If he wants an 8% return on his
total investment, how much should he pay for the building?
$120,000 (net income) / 0.08 (8% capitalization rate)
= $1,500,000 purchase price
Example 2
Luis purchases a building for $425,000. The building
shows a net yearly income of $25,500. What is the rate of return?
R = Net income / Present Value
R
= $25,000 / $425,000 = 0.06 = 6%
Loan ratio
Example 1
Broker Barry shows Mr. and Mrs. Duey a condominium
priced at $135,000. He things he can get a loan of 80% from ABC Savings
and Loan Company. If the down payment is to be cash over loan, what will
be the amount of the loan and the down payment?
0.80 x $135,000 = $108,000 loan
$135,000 - $108,000 = $27,000 down payment
Example 2
The selling price is $180,000. The loan is 75%. What
is the loan amount and down payment?
0.75 x 180,000 = $135,000 loan amount
$180,000 - $135,000 = $45,000 down payment
Discount
Example 1
A second note has been discounted 20%, and the amount of the discount is $1,200. What was the original amount of the note?
$1,200 / 20 = $6,000 original amount
Example
2
Conklin owns an old building, which he wants to
convert to cash in order to purchase some country acreage. The building is
free and clear.
Mistohosh offers Conklin $95,000 on the following
terms: Mistohosh will obtain a first loan of $60,000 from his bank, put
down $20,000 in cash, and Conklin is to carry back a second loan of
$15,000.
Conklin agrees, and after close of escrow, he sells
the note to John at 25% discount.
How much cash does Conklin now have for the purchase
of the country property?
Solution:
Coklin received $80,000 ($60,000 + $20,000) plus a
$15,000 note on the sale of his building. He discounts the note 25%:
$15,000 x 0.25 = $3,750 discount
$15,000 - $3,750 = $11,250
The $80,000 from the building plus the $11,250 from
the note equals $91,250 Conklin now has to invest.
Area Measurement
The licensee often must determine the square footage
contained in a parcel of land or in a building.
Example: The area of rectangular shape is found by
multiplying the width times the length.
Square feet in lot A:
50 x 100 = 5,000 square feet.