math

Units of Measure:

  • 1 acre = 43,560 square feet

  • 1 square yard = 9 square feet

  • 1 inch = 0.08 feet (divide inches by 12 to convert inches to feet)

  • 1 discount point = 1% of loan amount

Finding Area:
The area of a square or rectangle is:
Length X Width = Area, or
L \times W = A

The area of a triangle is:
Base X Height divided by 2 = Area, or
B \times H = \frac{A}{2}

The formula for the area of a triangle is half the formula for a square or rectangle because, as you can see in the diagram, a triangle is half of a square or rectangle. The height of a triangle is a perpendicular line to the base. Please note the examples shown. Area is expressed in square feet.

Finding Volume:
Volume equals Length X Width X Height = V, or
L \times W \times H = V

To convert inches to feet, divide by 12:
6 inches = 6 divided by 12 = 0.5 feet

For commission problems, use the “T” formula:

  • Small #

  • %

  • Big #

The horizontal line in the formula means "divide," and the vertical line means "multiply."

In other words:

  • The Small # divided by the % = the Big #, or

  • The Small # divided by the Big # = the %, or

  • The % X the Big # = the Small #.

You can use the T formula for mortgage problems, also. However, remember that interest rates are based on a year, so interest payments must be on a yearly basis before putting them in the formula. For mortgage problems, adhere to this structure to ensure accuracy in calculations.

Example 1:

Mr. and Mrs. Patterson are borrowing $75,000. If their first month's interest payment is $625, what is the interest rate on the loan?

  • Remember to first multiply $625 by 12 months to get the yearly interest paid:
    625 imes 12 = 7,500

  • Set up the formula:
    \frac{7,500}{75,000} = \text{interest rate}

  • Calculation:
    \frac{7,500}{75,000} = 0.10

  • Therefore, the interest rate = 10%.

Example 2:

A house appraised for $110,000. The house sells for 95% of the appraised value. The first month's interest payment on an 80% loan is $627. What is the loan amount? What is the interest rate on the loan?

  • Selling price:
    110,000 \times 0.95 = 104,500

  • Loan amount:
    \frac{627 \times 12}{0.80} = 7,524

  • Set up the formula:
    \frac{7,524}{83,600} = \text{interest rate}

  • Calculation:
    \frac{7,524}{83,600} \approx 0.09

  • Therefore, the interest rate = 9%.

Example 3:

A mortgage company makes a loan for 80% of the appraised value of a house. The first month's interest payment is $700, and the interest rate is 8%. What is the appraised value of the house?

  • Using the T formula to find the loan amount:
    700 \times 12 = 8,400

  • Set up the formula:
    \frac{8,400}{0.08} = 105,000\text{ (loan amount)}

  • The loan amount is 80% of the appraised value:
    \frac{105,000}{0.80} = 131,250\text{ (appraised value)}

  • Therefore, the appraised value of the house = $131,250.

When buying or selling a house, at the closing, the taxes are prorated so that the buyer and seller each pay their share for the part of the year that they own the house. Proration may be based either on a 360-day year (30 days in every month), or a 365-day year. When using a 360-day year, you can just find the cost per month then multiply times the total number of months. When using a 365-day year, you must find the cost per day, then figure out the total number of days and multiply the daily cost by the number of total days. To find the amount owed by the buyer or seller, divide the amount for taxes by twelve to get a monthly cost, then multiply that number by the appropriate number of months or days to find the amount owed.

Example 1:

Mr. Stone buys a house from Mr. Wood. The closing is on April 15th. Mr. Wood paid $550 for the property taxes on January 1st. How much does Mr. Stone owe Mr. Wood for the taxes? Use a 360-day year, seller pays for day of closing.

  • The monthly cost of the taxes is:
    \frac{550}{12} = 45.83

  • Mr. Stone will owe from April 16th to December 31st = a total of 8 ½ months.
    45.83 \times 8.5 = 389.56
    (Mr. Stone owes Mr. Wood)

Example 2:

Taxes on a property are $730 paid in arrears. The property is sold, and the closing is on July 20th. The tax year is from April 1st to March 31st. Using a 365-day year, at the closing who will owe the taxes and how much will they owe? The seller pays for day of closing.

  • Since the taxes are paid in arrears (at the end of the tax year), at the closing the seller will owe the buyer for his portion of the taxes. First, find the cost per day for taxes:
    \frac{730}{365} = 2.00/day

  • Then, calculate the number of days that the seller will owe for (April 1st to July 20th):
    30 + 31 + 30 + 20 = 111 days

  • Then multiply the number of days times the cost per day:
    111 days \times 2.00/day = 222 (seller owes buyer)

The tax rate is usually expressed in mills and may be called the millage rate. A mill is one thousandth of a dollar, that is, one penny equals 10 mills.

Example 1:

Houses in Dulsville are assessed at 50% of market value. If a house sold for $125,000, and the tax rate is 25 mills, what is the annual tax?
First find the assessed value:

  • 125,000 \times 50\% = 62,500 (assessed value)

Then, multiply the number of thousands of assessed value by the mills:

  • 62.5 \times 25 = 1,562.50

Tax = $1,562.50

If the tax rate is expressed as $ per 100 of assessed value, use the formula:

Tax = Tax Rate ($) \times # \text{ of } 100's \text{ of assessed value}

Example 2:

The tax rate in Newtown is $5 per $100 of assessed value. If the tax on a house is $2,500, what is the assessed value of the house?

  • 2,500 = 5 \times # \text{ of } 100's \text{ of assessed value}

  • 2,500 \div 5 = # \text{ of } 100's \text{ of assessed value}

  • = 500 \text{ (of 100's)}; 500 \times 100 = 50,000; \text{ assessed value is } 50,000.

The IRV formula is used to calculate capitalization rates and value for investment property. The IRV formula works exactly like the T formula:

I = \text{net yearly income}
R = \text{capitalization rate, or rate of return}
V = \text{value}

Income divided by Rate = Value
Income divided by Value = Rate
Rate times Value = Income

The net income is determined by subtracting the yearly expenses for the property from the gross yearly rents. Monthly debt service (mortgage payment) is NOT considered an expense in this type of problem.

Example 1:

An apartment building has 12 apartments which rent for $400 per month. Annual expenses include a 10% management fee, $150 per month for pest control, $200 per month for maintenance, and $300 per month for taxes and insurance. Given a capitalization rate of 10%, what is the value of the building?

Gross yearly rents are
400 \times 12 \text{ apartments} \times 12 \text{ months} = 57,600

Expenses are:

  • Management fee =
    57,600 imes 10 ext{ extperthousand} = 5,760

  • Pest control =
    150 imes 12 ext{ months} = 1,800

  • Maintenance =
    200 imes 12 ext{ months} = 2,400

  • Taxes & insurance =
    300 imes 12 ext{ months} = 3,600

  • Total expenses =
    5,760 + 1,800 + 2,400 + 3,600 = 13,560

Net yearly income =
57,600 - 13,560 = 44,040

Capitalization Rate Calculation:

  • 44,040 = 10 ext{ extperthousand} imes ext{Value}

  • ext{Value} = rac{44,040}{0.10} = 440,400

Example 2:

An apartment building has a capitalization rate of 10%. The value is $500,000. Annual expenses are $20,000. What is the gross income?
To obtain a capitalization rate of 12%, what would the gross rents have to be?

  • Net Income Calculation at 10%:
    I = 10 ext{ extperthousand} imes 500,000
    = 10 ext{ extperthousand} imes 500,000 = 50,000 ext{ (net income)}
    50,000 + 20,000 = 70,000 ext{ (gross income)}

  • Net Income Calculation at 12%:
    I = 12 ext{ extperthousand} imes 500,000
    = 12 ext{ extperthousand} imes 500,000 = 60,000 ext{ (net income)}
    60,000 + 20,000 = 80,000 ext{ (new gross income)}

The Gross Rent Multiplier, also called gross income multiplier, is a number which, when multiplied by the rent for a property, will give the value of the property. Gross rent multipliers are found by dividing the market value by the rent for several properties in a given area and taking the average of the values found. This number may then be applied to other similar properties to determine value.

NOTE: Gross monthly rent is used for residential properties. Gross annual income is used for commercial and industrial properties.

Formulas:

  • GRM = Market Value / Rent, or

  • GRM X Rent = Market Value

Example 1:

The gross rent multiplier for a given area is 125. If a house rents for $750 per month, what is its value?
125 imes 750 = Market Value
= 93,750

Example 2:

Given the following comparables, what is the GRM for this area?

Comps

Sale Price

Rent

Comp A

$80,000

$650

Comp B

$83,000

$675

Comp C

$89,200

$725

Finding the GRM's:

  1. For a property selling at $80,000 with a rent of $650:
    80,000 \, \div \, 650 = 123.08

  2. For a property selling at $83,000 with a rent of $675:
    83,000 \, \div \, 675 = 122.97

  3. For a property selling at $89,200 with a rent of $725:
    89,200 \, \div \, 725 = 123.04

Finding the average:

Adding the values:
123.08 + 122.97 + 123.04 = 369.09
Dividing by 3 for the average:
369.09 \div 3 = 123.03, \, or \, 123 \, (GRM)$$