Lesson 1: Real Estate Math The Language of Math Topics This section focuses on the following topics: Key Terms Overview Fractions Decimals Percentages Learning Objectives At the conclusion of this section you will be able to: Name key terms used in real estate math problems Apply fractions, decimals, and percentages Translate fractions into decimal and percentage forms Key Terms Amortization: The repayment of a financial obligation over a specific period of time in a series of periodic installments. Appreciation: The increase in value of an object over time. Area: The two-dimensional surface space of an object. Decimals: A way of expressing fractions that is compatible with the use of calculators and computers. Depreciation: The decrease in value of an object over time. Page 1 of 103
Fractions: A method of expressing a portion of a whole and can be expressed in portions (fractions), decimals and percentages. Linear Measurement: A method of measurement used to determine the length of an object. Loan Discount: A one-time fee that a lender charges at closing to a borrower for granting the loan usually at a certain interest rate. Loss: The result when something is sold for less than what was originally paid for it. Percentage Leases: A lease that typically charges a minimum base rent, plus some percentage of the income generated from the tenant s business at the property. Percentages: An expression of a fraction that states a portion of the whole where the whole is equal to 100% and means a portion per hundred. Profit: The result when something is sold for more money than was originally paid for it. Prorate: To divide proportionately. Rate: The cost per unit. Overview This topic will explore how practical mathematics applies to real estate practice including measurements in land and buildings, financial calculations, income property performance, valuation of real estate, and other areas in real estate practice. Many decisions to buy and sell real estate are based on many numerical and financial factors that the real estate agent must be prepared to address and explain. Having an understanding of basic mathematical calculations is critical for the real estate practitioner. Page 2 of 103
It is also worth noting that most consumers are going to demand explanations on figures and numbers and the real estate professional will be need to be able to explain them to the consumer. Most buyers and sellers are most concerned with the numbers and math in a transaction and is another reason that the agent needs to be skilled in mathematical calculations. A calculator will be needed to work on the mathematical problems in this lesson. A simple calculator that can add, subtract, multiply and divide is all that is really needed. A calculator with a % key may be more helpful but most importantly is that you know how to use it. Calculators differ, so choose one that is comfortable for you, and you are familiar with its use (up to 5 decimal points is best). It is recommended you use the same calculator for the state licensing exam. Specialty calculators, such as a mortgage or financial calculator, are not necessary for this lesson or the license exam but may be handy in your practice. Although calculators are permitted during the licensing exam, the calculator must be silent, non-programmable, non-printing and solar or battery operated. Do not use any other electronic devices such as an app on a tablet or phone as these devices are not permitted into the licensing testing area. Fractions Fractions, decimals, and percentages are all different ways of expressing a portion of a whole. For example, one half of a whole can be represented mathematically as 1/2,.50 and 50%. Fractions have two parts called the denominator and the numerator. The denominator shows the number of equal parts in the total or whole. The numerator shows the number of parts of the total or whole that is being considered. In the example below, the total or whole has been divided into ten equal parts, and you have eight of these equal parts: 8/10 (Read: eight-tenths ) 8 is the Numerator 10 is the Denominator Page 3 of 103
Proper Fractions This could be a way of expressing eight pennies as being eight-tenths of a dime. 8/10 is an example of a proper fraction. A proper fraction is a portion less than the whole or total. The whole or total can be expressed as 1. In the previous example, the dime is the total or whole, as in 1 dime equals 10 cents. Improper Fractions 12/10 is an example of an improper fraction or a number larger than 1. It is improper because 12 is greater than 10. 12/10 can also be expressed as a mixed number: 1 and 2/10. A mixed number is a whole number plus a fraction. In the previous example, this would be the same as saying 1 dime and 2 pennies. Some fractions can be reduced to make it easier to solve the problem. To reduce a fraction, determine the largest number by which both the numerator and the denominator can be evenly divided, then divide each by that number. 50/100 is equal to 1/2 because both 50 and 100 are divisible by 50. Calculations with Fractions Reducing Fractions To reduce a fraction, divide both the numerator and the denominator by the same largest possible common even divisor (i.e., the largest common denominator). Example: 3/12 3 is the largest common denominator: 3 divided by 3 is 1 and 12 divided by 3 is 4 = (3 3) / (12 3) = 1/4 When the numerator and the denominator cannot again be divided evenly by a common number, the fraction is in its "simplest form." Page 4 of 103
Whatever you multiply or divide a numerator with, you must do exactly the same to the denominator, so it doesn't change the value of the fraction. This principle DOES NOT apply to adding or subtracting identical values from numerators and denominators, which, unfortunately, DOES change the value of the fraction. Prove it to yourself. Common Denominators Whenever denominators are not the same, fractions cannot be added or subtracted until a common denominator is found. The lowest number that can be divided evenly by all of the denominators is called the lowest common denominator. What is the lowest common denominator of the following fractions? 3/4, 1/2, and 5/8 The largest denominator is 8 and the other two denominators (4 and 2) will divide evenly into 8. Thus, 8 becomes the lowest common denominator. Example: 8 divided by 4 = 2 and 8 divided by 2 = 4 and 8 divided by 8 = 1 Adding Fractions Example: 3/4 + 1/2 + 5/8 =? Step 1: When adding fractions together, first determine which of the denominators will divide evenly into the other denominator(s). Use the largest one. As stated on the previously, the largest common denominator in this example is 8. The other denominators, 4 and 2, will divide evenly into 8. Step 2: Convert the fractions so that they all have the same common denominator. Do this by dividing the lower denominators into the largest common denominator to figure out what to multiply both the denominator and numerator by. Page 5 of 103
So to change the original fractions of 3/4 and 1/2 to new fractions in terms of the common denominator of 8, do the following: 3/4 = (3 x 2) / (4 x 2) = 6/8 If the common denominator is twice as large as the original (i.e., you have to multiply the original denominator by 2 to get it from fourths to eighths), the new numerator must likewise become twice as large - so you multiply the original numerator by 2 as well. 1/2 = (1 x 4) / (2 x 4) = 4/8 In this case, 2 goes into 8 four times so multiply both the numerator and denominator by 4 to get the answer. Now the fractions are restated in terms of the lowest common denominator and the numerators are ready to be added. 3/4 + 1/2 + 5/8.. is the same as... 6/8 + 4/8 + 5/8 Step 3: Add up all the new numerators to get the "sum." Then, if the answer is an improper fraction, divide the numerator by the denominator to get the final sum. 6/8 + 4/8 + 5/8 = 15/8 = 1 and 7/8 Adding and Subtracting Fractions Problem: 3/4 + 5/8-5/6-1/24 =? Restated with common denominator: 18/24 + 15/24-20/24-1/24 = 12/24 Answer reduced: 12/24 = 1/2 In the example, each of the original ("old") denominators, 4, 8, 6, and 24 is evenly divisible into 24, so 24 becomes the lowest common denominator. To convert 3/4 into??/24, figure out how many times the old denominator, 4, divides into 24. That is 6. In effect, the new denominator (the common denominator) is the old denominator multiplied by 6. Thus, to Page 6 of 103
get the new numerator, the old numerator has to be multiplied by 6 as well so the fraction keeps its same value. So 3/4 thereby becomes 18/24. Another Example: Problem: 7/8 + 11/16 5/4 + 13/32 =? Restated with common denominator: 14/32 + 22/32 40/32 + 13/32 = 9/32 Here the largest common denominator where every other denominator is divisible into is 32. Another Example: Problem: 5/7 + 8/9 + 2/3 = 143/63 or 2 and 17/63 Here to find the lowest common denominator, the 7 had to be multiplied by 9 which is 63. Since the numerator 3 was divisible into 9 the two largest denominators are then multiplied together (7 x 9) to get the 63. Therefore, since the largest denominator was 9, which then had to be multiplied by 7, the numerators also had to be multiplied by 7 giving the figures 45/63 + 56/63 + 42/63 = 143/63. Multiplying Fractions To multiply fractions, multiply the numerators, then multiply the denominators, place the product of the numerators over the product of the denominators, and reduce the fraction to its simplest form (if necessary). Example: 1/4 x 5/6 = (1 x 5) / (4 x 6) = 5/24 Dividing Fractions To divide fractions, simply invert the fraction you are dividing BY (turn the divisor upside down) and proceed as you would in the multiplication of fractions. Page 7 of 103
Example: 3/4 divided by 1/2... is exactly the same as... 3/4 multiplied by 2/1 3/4 x 2/1 Or: = (3 x 2) / (4 x 1) = 6/4 = 3/2 = 1 and 1/2 To divide a whole number by a fraction, invert the fraction that you are dividing by, and complete the problem as you would for the multiplication of whole numbers by fractions. Example: 4 divided by 3/5 = 4/1 x 5/3 = 20/3 = 6 and 2/3 To divide a fraction by a whole number, place the whole number (which in this problem is the divisor) over 1 and invert it, and complete the problem as in the multiplication of fractions. Example: 3/5 divided by 4 = (3/5) (4/1) = 3/5 x 1/4 = 3/20 Decimals Decimals are a way of expressing fractions in a form that is compatible with the use of calculators and computers. This form is derived by the use of a decimal point. The numbers to the right of the decimal point indicate what portion of the whole is being considered. The numbered spaces consecutively to the right of the decimal point represent tenths, hundredths, thousandths, and so on. Fractions to Decimals Fractions can be converted to equivalent decimals by dividing the numerator by the denominator. Thus, if using a calculator, fractions are represented in decimal form as follows: 8/10 = 8 divided by 10 =.8 12/10 = 12 divided by 10 = 1.2 Page 8 of 103
Adding or Subtracting Decimals Example: 1.2 +.05 = 1.25 To add or subtract decimals, vertically line up the decimal points under each other, insert zeros for missing digits and add or subtract as you would with whole numbers. Example: 1.20 + 0.05 = 1.25 Multiplication of Decimals To multiply decimals, simply multiply the numbers as if they did not contain decimal points, then count the number of decimal places in each number and move the decimal point that total number of decimal places to the left in the answer. 0.4 x 0.65 (1) 4 x 65 = 260 (2) Now move the decimal three places to the left (because the.4,.6, and.5 mean there are three decimal places) and the answer is.260 or.26. Division of Decimals To divide decimals, consider the following diagram and terms: The dividend is the number that is being divided. The divisor is the number the dividend is being divided by. The quotient is the answer. If the divisor does not contain a decimal but the dividend does, then the decimal can be carried straight up into the quotient as shown in the following example, where.64 is divided by 8: Page 9 of 103
In the next example when we divide.64 by.008, there is now a decimal in the divisor as well as the dividend. In this case, we eliminate the decimals in both the divisor and dividend by shifting the decimals three places to the right. Thus, when we move the decimal point up into the quotient, we find we are effectively dividing 640 by 8, which yields a result of 80 as shown below. Changing a Decimal to a Fraction To change a decimal to a fraction, write the decimal as a whole number and place it in a fraction over "1" followed by as many zeros as there were decimal places to the right of the decimal point in the original number. That's it. If the resulting fraction can be reduced, do so. Examples: 0.25 = 25/100 = 1/4 0.123 = 123/10000.0004 = 4/10000 = 1/2500 2.625 = 2 + 625/1000 = 2 and 5/8 (Note: 2 +625/1000 = 2625/1000) Page 10 of 103
When changing a decimal to a fraction, with the decimal point removed, the number in question becomes the numerator and the place value of the last digit to the right becomes the denominator. The resulting fraction may or may not be in its simplest form. If it is not, it should be reduced. Example: 0.25 = 25/100 = 1/4 Example: 0.123 = 123/1000 Percentages A percentage represents a portion of a whole. A whole is expressed as 100%. Percent means per hundred. Most real estate calculations are based on the calculation of percentages. For example, 40% means 40 parts out of the 100 parts. Percentages greater than 100% contain more than one whole unit. Thus, 175% is one whole plus 75 parts of another whole. Conversion of a Percentage to a Decimal To solve problems involving percentages, first convert them to either a decimal or a fraction. To convert a percentage to a decimal, you divide the percentage by 100, which is the same as moving the decimal point two places to the left and removing the percent sign. 50% = 0.5 2% = 0.02 175% = 1.75 Conversion of a Percentage to a Fraction To change a percentage to a fraction, place the percentage over 100. Then, if possible, reduce the fraction. 50% = 50/100 = 1/2 2% = 2/100 = 1/50 Page 11 of 103
175%= 175/100 = 7/4 Changing a Percent to a Fraction or a Decimal To change a percent to a fraction, the percent becomes a fraction when placed over its denominator of 100 (the whole to which it is related). If the percent is a whole number, drop the percent sign, place the number as the numerator over 100, and reduce the resulting fraction, whenever possible. Examples: 12% = 12/100 = 3/25 150% = 150/100 = 3/2 = 1 and 1/2 Changing a Fraction or a Decimal to a Percent To change a fraction to a percent, first divide the numerator by the denominator. Make the result a decimal. Then, move the decimal point two places to the right and add the percent sign. Examples: 1/4 = 0.25 = 25% 2 and 3/4 = 2 + 0.75 = 2.75 = 275% Percentage Problems The majority of math problems that you will encounter involve percentages. There are three factors in any percentage Problem: 1. Amount paid is the amount invested. 2. Rate (%) is the percentage. 3. Amount made is the amount earned. In percentage problems, one of three factors is missing. There are three rules for finding the missing factor: 1. To find the amount paid divide made by the rate. Page 12 of 103
2. To find the amount made multiply paid by rate. 3. To find the rate divide made by paid. Keep these other factor terms in mind: Made = part, return, profit, commission, net income, or interest Paid = total, investment, cost, price, value, or principal Rate = rate of return, profit, commission, capitalization, or interest In rate problems, think of the word "is" as equals (=), the word "of" as multiplication (x), and the word "per" as division. For example, 5% of $125,000 is $6250. The typical percent problem will supply you with two of the variables. You may determine the third through the use of the proper equation: Amount paid? = made rate Amount made? = paid x rate The rate? = made paid The underlying formula for determining these amounts is the formula IRV or Income = Rate x Value. This is the universal formula in real estate for determining the values of properties and promissory notes. Normally, a math problem with this formula will involve knowing two of the three values and obtaining the third. Therefore, the Rate of Return will equal the Income divided by the Value. The Value will then be equal to the annual income by the rate of return. You will have to convert the percentage rate into a decimal before completing the operation. This is done by moving the decimal point two spaces to the left and dropping the percent sign. This changes a percent into its equivalent decimal form. 8.5% =.085 50% =.5 Page 13 of 103
110% = 1.1 To convert a decimal number into a percentage, you move the decimal point two spaces to the right, adding zeros if needed..095 = 9.5% 1.2 = 120%.009 =.9%.07 = 7% Example: What is 65% of $175,000? Solution: 65% = 0.65 Therefore: $175,000 x.65 = $113,750 Example: $50,000 is 31% of the value of a property. What is the value? Solution: $50,000 divided 31% (.31) = $161,290.32 Example: A property was worth $125,000 when the investor purchased it and has appreciated in value by 17% since. What is the current value of the property? Solution: Current Value = 117% of $125,000 Therefore: $125,000 x 117% (1.17) = $146,250 T-Method Percentage problems follow the T-Method and solve for part, total, or rate. (Just remember that the Part and the Total are never in the same part of the T) Part T-top I Page 14 of 103
Total Rate T-bottom I To determine a specific percentage of a whole, multiply the percentage by the total. This is shown in the following formula: Total x Rate = Part 100 x 20% = 20 For Example: If a broker is to receive a 5% commission on the sale of a $100,000 ($100,000 is the total sale price) house, what will the broker s commission be? Total Sale Price x Commission rate = Broker s commission $100,000 x 0.05 = $5,000 Rule 1 If we are given two numbers at the bottom of the T (the total and the rate; or as in the previous example, sale price and commission rate), then the mathematical operation used to solve for the result (the part; or the broker s commission) is multiplication. $100,000 X 0.05 = $5,000 This is the formula used in calculating mortgage loan interest, broker s commissions, loan origination fees, discount points, earnest money deposits, and income on capital investments. Rule 2 When given the part (at the top of the T) and the rate (at the bottom), the mathematical function to find the whole is solved with division: divide the part by the rate. Visualize this rule by remembering that because the part is on top of the T, it is always on top of the division sign. So, given the rate and the part, put the part on top and divide by the rate. Example: If a seller nets $100,000 on the sale of his or her house after paying a 5% commission to the broker, what was the sale price of the property? Page 15 of 103
Here, it is possible to get confused. Is $100,000 a total or a part? Since the seller receives (nets) only a percentage of the sale price, rather than all of it, it is a part. So we put $100,000 on the top of the T. We know that it is equal to 95% of the selling price (100% minus 5% commission paid, or 95%), so 95% goes in the lower right of the T. $100,000 / 95% = $105,263.16 Rule 3 When given the part (at the top of the T) and the total (at the bottom), the mathematical function to find the rate is, as above, division. Let s continue the previous example, but let s assume that we know that the broker received $5,000 in commission and that the selling price was $100,000. What was the commission rate of the broker? Because $5,000 is the part, it belongs at the top of the T. $100,000 is the sale price (or total) and goes in the bottom left. Thus, 5,000/100,000 =.05 or 5%. Collectively, the three rules of the T-method allow us to find any one of the three variables (part, total, and rate) when the other two are known. Example: The realty company received a $2,000 commission for the sale of a house. The broker s commission was 2% of the total sale price. What was the total sale price of the house? $2,000 is the commission portion of the entire sale price of the house. It is a part, so it belongs on top. 2% is the commission or rate, so it goes to the bottom right. 2,000 / 0.02 = $100,000 Sale Price vs. Net Price A variation of this technique is used to compute the sale price of a property if the owner wants to net a certain amount, say $100,000 after expenses. The first step is to estimate the sale expenses and express them as a percentage of the sale price. For example, if Page 16 of 103
the sale commission is 5.5% and the seller s other costs amount to.5%, the total expense is (5.5 +.5) or 6%. Further, if the sale price is 100% with expenses estimated at 6%, this means that the seller s net of $100,000 is 94% (100 6) of the sale price, or $100,000 = 0.94 x Sale Price To solve for the sale price, divide both sides by 0.94: $100,000 /.94 = (.94 x Sale Price) /.94 Notice that.94 /.94 = 1. Therefore, these can be canceled, which gives us the following: $100,000 / 0.94 = Sale price or $106,382.98 To check your work, take 6% of this amount using the T-method with the two bottom elements, total and rate: $106,382.98 X 0.06 = $6,382.98 When subtracted from the sale price of $106,382.98, this leaves the seller s desired net price of $100,000. Page 17 of 103
Measurement of Dimensions Topics This section focuses on the following topics: Introduction Linear Measurement Area Volume Legal Descriptions Rectangular Survey System Lot and Block System Learning Objective At the conclusion of this section you will be able to: Determine the area and volume of a given object or parcel. Key Terms Cubic Measurement: When a shape encloses a space, the shape has volume. Cubic measurement is normally used to calculate the volume of a building, such as a warehouse. The volume is measured in feet, inches or yards as a cube. It is a threedimensional measurement. Height x Width x Length = Volume. Front Foot Measurement: Used when you are dealing with the frontage of a lot. The frontage is normally the street frontage, but it could be a water frontage if the lot is on a lake or stream or ocean edge. When a lot measurement is given, such as 75' x 150', the first figure (75') refers to the front feet. A front foot measurement can be by the foot or yard or other measurement, as long as it is the linear measurement of the distance involved. Page 18 of 103
Linear Measurement: A running foot, the same as a progressive distance such as putting one foot in front of another. It is a one-dimensional measurement. Example: The distance around the perimeter of the property can be measured in linear feet. Square Measurement: An area such as length multiplied by width. This measurement is normally used to calculate floor space of a building or the area of a lot. It is a twodimensional measurement. Introduction This section covers dimensional mathematics including linear measurements, area, and volume. Linear measurements are used to express the length, width, and height of objects in terms of inches, feet, yards, and miles. This section shows how to convert from one unit of measurement to another. Area is the two-dimensional surface space of an object. It is expressed in square units such as square inches, square feet, and square yards. Volume is the three-dimensional measurement of an enclosed space. Volume is expressed in cubic units: i.e., cubic inches, cubic feet, and cubic yards. An understanding of linear measures and the ability to manipulate area and volume are among the tools essential to real estate professionals. Units of Measurement 12 inches = 1 linear foot 12" x 12" x 12"= 1 cubic foot 3 feet = 1 lineal yard 3 feet x 3 feet x 3 feet = 1 cubic yard 1 mile = 5,280 lineal feet 1 rod = 16 1/2 lineal feet 1 mile = 320 rods 1 mill = 0.10 of 1 cent 1 hectare = 2.471 acres 1 square foot = 144 square inches 1 square yard = 9 square feet Page 19 of 103
1 township = 36 sections 1 section = 1 square mile 1 square mile = 640 acres 1 acre = 43,560 square feet 1 acre = 10 square chains 360 degrees = full circle 90 degrees = 1/4 circle 1 degree = 60 minutes 1 minute = 60 seconds Linear Measurement Linear measurement is used to determine the length of an object. The object may be in the form of a straight, curved or crooked line. The measurements are expressed as inches, feet, yards, meters or miles. In real estate, the terms per foot, per linear foot, per running foot, and per front foot refer to the total length of an object. Per foot means for each foot, so a lot might be priced $10 per foot. Per front foot refers to the frontage of a lot. It could refer to street frontage or water frontage and is important to the value of the lot, depending on the amount of usable frontage. When lot dimensions are given, the first dimension represents frontage, unless otherwise specified. At times it may be necessary to convert linear measurements of one form to another. The following are standard conversions used in real estate: 12 inches = 1 foot Inches divided by 12 = feet Feet multiplied by 12 = inches 36 inches = 1 yard Inches divided by 36 = yards Yards multiplied by 36 = inches Page 20 of 103
3 feet = 1 yard Feet divided by 3 = yards Yards multiplied by 3 = feet 5,280 feet = 1 mile Feet divided by 5,280 = miles Miles multiplied by 5,280 = feet Example: Determine the cost of a fence to enclose a lot at $5 per linear (running) foot of fence. Because fences are priced per linear (running) foot, it is necessary to first determine the total amount of linear feet of the lot. Next, you should multiply this amount by the cost of fencing each linear foot. For example, if the lot were 75 feet by 125 feet the problem would be solved as follows: 125 75 75 + 75 + 125 + 125 = 400 linear/running feet 400 linear/running feet x $5.00 (cost of fence per linear/running foot) = $2,000 In general, the formula for determining the perimeter (the length around) of a rectangle is: 2 x (Length + Width) Page 21 of 103
We could have solved by using this formula; where length = 75 and width = 125. This achieves the same result: Perimeter = 2 x (75 + 125 ) = 2 x (200 ) = 400 (You should use the method with which you are the most comfortable.) Example: A tract of land is for sale for $3,000 per front foot. How much will it cost to purchase the property if the tract is 200 by 175 feet? 200 feet is the frontage because it is the first measure given. 200 front feet x $3,000 = $600,000 (Cost to purchase) Area Area refers to the two-dimensional surface space of an object. It is measured in square units such as square inches, square feet, square yards or acres. Areas may be any shape, but typically in real estate, area involves rectangles, squares, and triangles. Conversion of Area At times it may be necessary to convert area measurements from one form to another. The following are standard area conversions used in real estate: 144 square inches = 1 square foot Square inches divided by 144 = square feet Square feet multiplied by 144 = square inches 1,296 square inches = 1 square yard Square inches divided by 1,296 = square yards Square yards x 1,296 = square inches 9 square feet = 1 square yard Square feet divided by 9 = square yards Square yards multiplied by 9 = square feet Page 22 of 103
43,560 square feet = 1 acre Square feet divided by 43,560 = acres Acres multiplied by 43,560 = square feet Area of Squares and Rectangles The area of a square or a rectangle is the length multiplied by the width. Length x Width = Area of a square or rectangle Example: If the length of a lot is 100 feet and the width is 100 feet, what is the area of the lot? 100 x 100 = 10,000 square feet Example: How many acres are there in a tract of land that measures 225 feet x 400 feet? 225 x 400 = 90,000 square feet divided by 43,560 = 2.066 acres Example: If hardwood floors cost $15.00 per square foot to install, how much would it cost to install hardwood floors in a room that is 18 feet by 23 feet? 18 x 23 = 414 square feet multiplied by $15.00 = $6,210 At times, the area to be measured is not a perfect square or rectangle. For example, the area may be in an L shape. In this instance, measure the two sections of the shape, calculate them individually, and add them together. Don t include the area of overlap of the two sections. Example: What is the area of the following L-shaped floor plan? Page 23 of 103
23ft 2 10ft 7ft 18ft First, measure the vertical section of 23 feet. This is the length of the first rectangle. To find its width, take the horizontal section of 18 feet minus 10 feet. So the width is 8 feet. Multiply length times width to find the area 23 x 8 = 184 square feet. Second, find the area of the second rectangle. Its length is 7 feet and its width is 10 feet, so multiply the length and width: 10 x 7 = 70 square feet. Finally, add the two results: 184 + 70 = 254 square feet, the total area of the floor plan. Area of a Triangle Triangles are encountered in real estate because of tapering property lines and roofs. The area of a triangle can be found by multiplying the base by the height and dividing the answer by two. The base is the bottom of the triangle. The height is the imaginary straight line extending from the base to the point (corner) opposite. This line creates a 90 degree angle with the base (in other words, height is perpendicular to the base). (Base x Height) / 2 = Area of a triangle Example: How many square feet are contained in a triangular lot that is 350 feet at the base and is 200 feet high? Page 24 of 103
200 height 350 base 350 feet x 200 feet = 70,000 square feet divided by 2 = 35,000 square feet Area of a Circle To calculate the area of a circle we use the formula: 3.142 x Radius squared The linear distance around a circle, the circumference, is given by the formula: 2TT x radius. 3.142 is an approximation of the number π (read: pi), which is the ratio of a circle s circumference (the length around the circle) to its diameter (the length of any line segment that passes through the circle s origin, or center, and whose endpoints lie on the circle). The radius of the circle is one half of the diameter, i.e., a line segment starting at the origin of the circle and ending at any point along the circle. Example: Calculate the area of a circle with a radius of 10 feet. Page 25 of 103
10 3.142 x 10 squared (10 x 10 ) 3.142 x 100 square feet = approximately 314.2 square feet Volume Volume is the cubic capacity of an enclosed space. Volume is used to describe the amount of space in any three-dimensional area and is expressed in cubic units. The formula for computing cubic or rectangular volume is: Volume = Length x Width x Height Example: The living room of a condo is 15 feet long, 6 feet wide and has a ceiling height of 10 feet. What is the volume of the living room? 15ft 10ft 6ft Volume = 15 x 6 x 10 Volume = 900 cubic feet Page 26 of 103
There is another way to calculate volume if the solid is a triangular prism (think roofs). Volume = 1/2 (Base x Height x Width) Example: Calculate the volume of the house shown below. 10 50 30 14 Here we first divide the house into two solids, a rectangular prism (the box part of the house) and a triangular one (the roof part). The triangular prism at the top of the house has a height of 10 feet, a base of 30 feet and a width of 50 feet. Thus, we know: Volume = 1/2 (Base x Height x Width) Volume = 1/2 (10 x 30 x 50 ) Volume = 7,500 cubic feet The next step is to calculate the volume of the rectangular prism. It shares two dimensions with the top: the triangle s base is the width of the rectangle and the triangular prism s width is the length of the rectangle. The rectangle is 14 feet high, so: Volume = 30 x 50 x 14 Volume = 21,000 cubic feet The total volume of the two solids is their sum: 7,500 + 21,000 = 28,500 cubic feet of airspace Finally, the formula for computing the volume of a cylinder is expressed below. Volume = π x Radius squared x Depth Page 27 of 103
Example: What is the approximate volume of a straight tube that is 30 feet long and 10 feet wide? Remember to use 3.142 to represent π. First, determine the radius of the tube, which is 1/2 the diameter. The diameter is 10 feet, so the radius is 5 feet. We know the tube is 30 feet long, so using the formula: Volume = 3.142 x 5 feet squared x 30 feet Volume = 3.142 x 25 x 30 = 2,356.5 cubic feet Cubic measurements of volume are used to compute the construction costs per cubic foot of a building or the heating and cooling requirements for a building. Keep in mind that when calculating the area or the volume, all dimensions used must be given in the same unit of measure. For example, the student should not multiply feet by inches, but instead feet by feet and inches by inches. Sample Problems - Measurements Example: The excavation for a basement of a new home measured 75 feet by 45 feet by 17 feet deep. At a cost of $17.50 per cubic yard, how much did the excavation cost? Solution: 75 feet x 45 feet x 17 = 57,375 cubic feet. There are 27 cubic feet in a cubic yard; therefore 57,375/27 = 2125 cubic yards. $17.50 x 2125 = $37,187.50. Example: A parcel containing 120 yards by 260 yards would contain how many acres? Page 28 of 103
Solution: 1 yard = 3 feet Therefore, the dimensions of the parcel would be 360 by 780. Therefore, the parcel s size in square footage would be 280,800 square feet (360 x 780). One acre = 43,560 square feet Therefore, the parcel would contain 6.47 acres (280,800 divided by 43,560 square feet per acre). Example: How many square feet in a triangular parcel of ground measuring 225 feet at the base and 275 feet high? Remember: Area of Triangle = 1/2 x Base x Height Therefore: 225 x 275 = 61,875 divided by 2 (or x.5, 1/2 = 0.5) = 30,937.5 square feet. How many acres are in this parcel? Solution: One Acre = 43,560 square feet 30,937.5 square feet divided by 43,560 square feet per acre = 0.71 acres. Example: At $5700 per acre, what would be the selling price of a parcel of ground measuring 711 feet by 729 feet? Solution: Area = 711 x 729 = 518,319 square feet 518,319 square feet divided by 43,560 square feet per acre = 11.90 acres. Price = 11.90 acres x $5,700 per acre + $67,830 Page 29 of 103
Exercises Integrating Multiple Mathematical Problems Example: A lot measuring 300 feet by 726 feet is purchased by a builder for $167,000. Twenty percent of the lot is used for street and improvements at a total cost of $44,000, and the rest subdivided into half-acre lots. How much must each lot sell for in order for the builder to realize a 20% return on the total investment? Solution: $167,000 + $44,000 = $211,000 builder investment Builder s desired return = 20% or $211,000 x.2 = $42,200 Total price needed = $211,000 + $42,200 = $253,200 Total square footage of lot = 300 x 726 = 217,800 square feet. However, 20% of the land is used for the street and improvements. Therefore, 217,800 x.2 = 43,560 square feet. The total available square footage for subdividing is 217,800 43,560 = 174,240 square feet. One acre = 43,560 square feet; 1/2 acre = 21,780 square feet Number of 1/2 acre lots = 174,240 divided by 21,780 square feet = eight 1/2 acre lots; Therefore: $253,200 divided by 8 = $31,650 per lot. Example: A home was worth $150,000 and the lot has a present value of $7,700 and the home is 16 years old and has depreciated by 11/2 % per year. What is the present value of the property? Solution: If the property has depreciated by 1.5 % each year, the property has since depreciated by a total of 24% (16 years x 1.5% per year). Therefore, the property is now worth 76% of what it was worth 16 years ago. The property by itself would be worth $114,000 ($150,000 x.76). However, the lot is worth $7,700 Page 30 of 103
today, so if added to the current value of the improvements would equal $121,700 for the property. Example: How many 65 by 85 lots can you fit into an acre of land? Solution: One acre = 43,560 square feet: A lot here would equal 5,525 square feet. (65 x 85 ) 43,560 square feet divided by 5,525 square feet would equal 7.88 lots Example: A rectangular piece of property contains 17,000 square feet, has 50 of frontage and sells at $225 per front foot. What is the depth of the property? What is the price per square foot? Solution: 17,000 square feet = 50 width x?depth: Therefore: Depth = 17,000 divided by 50 or 340 feet 50 x $225 per front foot = $11,250 total price Therefore: $11,250 divided by 17,000 square feet = 66 cents per square foot. Exercise: A high rise office building has a quarterly gross income of $357,000. The expenses amount to 31% of the income. What is the annual net income? Solution: Annual income = $357,000 x 4 = $1,428,000 Expenses: $1,428,000 x 31% (.31) = $442,680 Net Income = $1,428,000 - $442,680 = $985,320 Page 31 of 103
Legal Descriptions A legal description is defined as information that a qualified surveyor can use to find the property and mark each corner. There are four accepted methods of describing real estate: 1. Metes and bounds 2. Rectangular survey system 3. Recorded plat of subdivision 4. Monuments and markings Metes and Bounds Real estate professionals are required to be able to read and interpret the metes and bounds of a legal description. metes = measurement bounds = direction In plotting metes and bounds, there must be a point of beginning (POB). The direction from the POB is determined from a compass bearing to the next point in the plotting of the property. First, we need to know that the directions stated in a metes and bounds description are the magnetic compass directions for all boundary lines. Recite the north or south direction first, followed by the direction of declination from north or south (i.e., N45 E means looking north, decline 45 to the east; S90 W means looking south, incline 90 to the west). In preparing a metes and bounds description, we must add the distances to be measured from point to point (i.e., N45 E for 2,500 feet). Page 32 of 103
Using a compass, determine the direction to travel by the degrees measured on the compass. A compass is graduated into degrees. A degree is a unit of measurement and a circle is equally divided into 360 degrees. Looking down on the compass, you will find that the needle always points to magnetic north. In order to know which direction is magnetic north, turn the compass so that the needle is pointing at "0" or north. Imagine that you are standing at the POB. The first direction begins with either north or south, whichever is closest (except that due easterly and westerly directions are N90 E or N90 W), and gives an angle of declination or deflection to the east or west, as in N35 W or N60 E. This is followed by a distance. "POB N1 E 250 feet" fully defines the first Page 33 of 103
segment of a property boundary (from POB proceed one degree east of north for 250 feet). Example: If the first boundary segment is described as N45 E 2,640 feet, the angle from POB is as shown below and the distance to the next point is 2,640 feet. The complete boundary description of this property would be: POB N45 E 2,640', N90 E 1,867', S180 5,507', N90 W 4,640' POB. An actual description would have the degrees with minutes and seconds, and the distances to the nearest hundredth of a foot. Page 34 of 103
From the previous example you should be able to determine the boundaries from the following metes and bounds description: Beginning at the brass marker in the center of the intersection of New Days Road and Old Folks Avenue, proceed N45 W 40' to the point of beginning. From POB proceed N89 W 1,000', then N1 E 1,000', then S89 E 1,000', then S1 W 1,000' to POB. In actual use, you almost never find straight N, S, E or W. It is more often something like N 89Â 59' 37" E. Rectangular Survey System This means of describing land, also known as the government survey method, is a system of numbered squares, based on two sets of intersecting lines (principal meridians and baselines). Rectangular survey systems are found in 30 of the states. Most of them are in the western United States where all states use this method except Texas. However rectangular survey systems are used in some southern and Midwestern states as well where the government has established meridians and base lines. Page 35 of 103
Principal meridians run north and south. Base lines run east and west. Both are located by reference to degrees of longitude and latitude. In order to locate a position on a map, the latitude and longitude of the position must be known and intersected to pinpoint the position. On maps with north at the top, lines of latitude are horizontal lines, and lines of longitude (including meridians) are vertical lines. The intersection of a specific meridian (vertical) and base line (horizontal) provides a unique point of reference for determining the location of nearby properties. Any point on the earth's surface can serve as such an intersection and the meridian (and sometimes the base line) are often named for that point. The intersection of the Fox and the Maronga rivers might, for example, yield a coordinate point named the "Fox and Maronga River Intersection Principal Meridian (and Base Line)." Page 36 of 103
A grid is established with range lines running parallel to the principal meridian at 6-mile intervals on each side of it, and township or tier lines running parallel to the base line at 6-mile intervals north and south of it. Ranges are bounded by the vertical range lines, the principal meridian being the first. Range 1W is the first six-mile-wide strip of land to the west of the principal meridian. R2W would be a strip of land between six and twelve miles west of the principal meridian. The lines running parallel with the base line, 6 miles apart, are called township lines. In between the township lines is a strip of land called a township tier. Township tiers are designated by consecutive numbers north or south of the base line. Page 37 of 103
Example: The strip of land between 6 and 12 miles north of a base line is Township 2 North (T2N). Townships are the basic unit of reference in the rectangular survey system. Any two adjacent range lines and any two adjacent tier or township lines bound a square called a township. Each township is 6 miles square and contains 36 square miles or sections. Each township has a legal description based on its principal meridian (and sometimes base line). Example: "T3N R4W Fox and Maronga River Intersection Principal Meridian (and Base Line)" refers to a 6-mile by 6-mile (36 square mile) township lying between 12 and 18 miles north and 18 and 24 miles west of the river intersection. Page 38 of 103
This township is designated Township 2 North, Range 3 East of the principal meridian. This township is the second tier north of the base line. The township is also located in the third range strip east of the principal meridian. Finally, reference is made to the principal meridian because the land being described is within the boundary of land surveyed from that meridian. The description is abbreviated as T2N, R3E of the principal meridian. This township is designated Township 3 North, Range 1 West of the principal meridian. This township is the third tier north of the base line. The township is also located in the first range strip west of the principal meridian. Finally, reference is made to the principal meridian because the land being described is within the boundary of land surveyed from that meridian. The description is abbreviated as T3N, R1W of the 5th principal meridian. Townships are squares, 6 miles on each side. Page 39 of 103
There are 36 square miles in one township. Each township is divided into 36 sections. The sections are numbered from right to left, beginning with mile 1 starting at the (NE) top right hand corner and going through 6. Section 7 lies directly below section 6 and goes back, this time from left to right, to section 12. Directly below section 12 lies 13, and goes, right to left again, to section 18. This continues until all 36 square miles form one township. Every section is a square, 1 mile on each side. Therefore, a section equals one square mile. It is also 640 acres. It can be divided into halves, and further subdivided again and again into quarters and smaller tracts until the acreage required is defined. 1/2 section = 320 acres 1/4 section = 160 acres 1/8 section = 80 acres 1/16 section = 40 acres Page 40 of 103
To determine the location and size of a property described in the rectangular survey method, start at the end of the description, read from right to left, and work backwards towards the beginning. Example: Locate the south 1/2 of the NW 1/4 of the SE 1/4 of section 11. Starting with the section, read backwards to first locate the southeast quadrant (in yellow). Then find the northwest corner of that area (noted in green), and then divide that area in half to find the property in question in black. Page 41 of 103
Determining Acreage Math Shortcut: To calculate acres in a rectangular survey system description, multiply all the denominators and divide that number into 640 acres. For instance, the size of the SE1/4 of SE1/4 of SE1/4 of Section 1 = 4 x 4 x 4 = 64 640 64 = 10 acres Page 42 of 103
Example: How many acres are there in the E 1/2 of the NE 1/4 of the SE 1/4 of a section? 2 x 4 x 4 = 32; 640/32 = 20 acres How many square feet are in the W 1/2 of the NW 1/4 of the NW 1/4 of the SW 1/4 of a section? 2 x 4 x 4 x 4 = 128; 640/128 = 5 acres x 43,560 square feet per acre = 217,800 square feet How many acres are there in Sections 16 & 15 and the N 1/2 of Section 21 and the NW 1/4 of the NW 1/4 of Section 22? Since a section is 640 acres, Sections 16 & 15 will add up to 1280 acres. One-half of Section 21 will add another 320 acres now totaling 1600 acres. Now for Section 22 we take 4 x 4 = 16; 640/16 = 40 acres. Now 1280 + 320 + 40 = 1640 acres. Example: How many acres would be contained in the following legal descriptions? NW 1/4 SW 1/4 N 1/2 SW 1/4 and the SW 1/4 SE 1/4 S 1/2 NW 1/4 of Section 21 Hint: Always work backwards when tabulating acreage in a Township Section. Section = 640 acres: Therefore: SW 1/4 is 160 acres; N 1/2 of SW 1/4 is 80 acres; SW 1/4 of the N 1/2 of the SW 1/4 is 20 acres and the NW 1/4 of the SW 1/4 of the N 1/2 of the SW 1/4 of the Section would be 5 acres. Plus: The NW 1/4 of Section 21 would be 160 acres: the S 1/2 of that would be 80 acres; the SE 1/4 of that would be 20 acres and the SW 1/4 of that would be 5 acres. Therefore: 5 acres + 5 acres would be 10 acres. Page 43 of 103
Lot and Block System A lot and block survey is performed in two steps. First, a large parcel of land is described either by metes and bounds or by rectangular survey. Once this large parcel is surveyed, it is broken into smaller parcels. As a result, a lot and block legal description always refers to a prior metes and bounds or rectangular survey description. Example: Lot 2, San Juan Estates, located in a portion of the southeast quarter of Section 20, Township 4 North, Range 2 East of the San Juan Principal Meridian in the county of San Juan, state of New York. Monuments and Markings Datum: A datum, or reference datum, is a point, line or surface from which distances or elevations are measured. Bench mark: Bench marks are permanent reference datum points that have been established throughout the United States. They are usually embossed brass markers set into solid concrete or asphalt bases. The location and elevation of such a marker is accurately known and recorded by the U.S. Geological Survey. While used to some degree for surface measurements, their principal reference use is for marking datums. One of the most important figures in real estate can either be price per acre or price per square foot. For Example: A tract of land in a recreational area is measured at 456,890 square feet. A corner of the tract, 31 feet wide and 123 feet deep is owned by the county. The privately owned land sold for $7,118 per acre. What was the total amount realized from the sale? Solution: The area of the rectangular parcel is 31 feet x 123 feet = 3,813 square feet. The privately owned land then would be 456,890 square feet 3813 square feet = 453,077 square feet. One acre is = 43,560 square feet. Therefore 453,077/43,560 = 10.40 acres. 10.40 acres x $7,118 per acre = $74,027.20 Page 44 of 103
Financial Math Topics This section focuses on the following topics: Key Terms Introduction Interest and Percentage Calculations Amortization Rate Prorations Learning Objectives At the conclusion of this section you will be able to: Identify the principles involved in rate calculations. Use equations to calculate interest. Explain amortization. Apply the principles of prorating. Key Terms Amortization: The process of structuring loan payments so that a series of level payments, including principal and interest, will retire the debt in full at the end of the loan term. Conventional loan: Any loan that is not directly guaranteed by the federal government. Discount points: Added loan fee charged by a lender to make the yield on a lower-thanmarket-interest VA or FHA loan competitive with higher-interest conventional loans. One discount point is equal to 1 percent of the loan amount. Discount points may not be added to the loan amount. Page 45 of 103
Fixed rate mortgage: A mortgage wherein the interest rate remains constant throughout the life of the loan. Installment note with a balloon payment (partially amortized note): A promissory note with periodic payments of principal and interest and a large payment at the end (maturity date or due date). LTV (Loan to Value): The loan-to-value ratio is the percentage of a property's market value that a lender is willing to loan, with the property offered as collateral. Mortgage Insurance: Insurance that indemnifies the lender in the event of default by a borrower. It is required on all FHA loans and any conventional loan whose loan to value ratio exceeds 80%. Negative Amortization: Amortization where the loan balance actually increases rather than decreases as a result of unpaid, deferred interest on the loan. Promissory Note: The basic instrument used to evidence the obligation or debt. It is an unconditional promise, in writing, by one person to another, promising to pay on demand, or over a fixed determinable time, a certain sum of money. Straight Note: A promissory note in which a borrower repays the principal in one lump sum, at maturity, while interest is paid, in installments or at maturity. Introduction This section covers several aspects of financial math. These include the topics of interest, amortization, rate, and proration. Interest is the cost incurred for the use of money over a period. Amortization is the repayment of a financial debt over a period of time in a series of payments. Rate is a cost per unit that is charged to an assessed value. Prorations involve the debits and credits charged to buyers and sellers based on their proportionate ownership of the property at the time of closing on real estate transactions. Page 46 of 103
Interest and Percentage Calculations Interest is the cost incurred for borrowing money. The amount of interest paid is determined by the annual interest rate, the amount of money borrowed (the principal), and the term of the loan. The formulas for percentage calculations also are used for interest computations. Principal x Annual interest rate (a percentage) x Time (in years) = Interest payment For example, presuming that the principal doesn t change, what is the annual interest on a $10,000 loan, which has an interest rate of 10%? $10,000 x 0.10 x 1 year = $1,000 interest Annual interest is the interest for an entire year. If we want the interest for just one month we have to divide the annual interest by 12 (number of months in a year). $1,000 / 12 = $83.33 monthly All the above calculations are based on simple interest. Now compare that with compound interest. The difference between the two is that compound interest has each period s interest added to the principal before the next interest payment is figured, so compound interest will be a higher amount (which is great if the money is in a retirement fund, but not so great if it s a mortgage payment). Example: If $100,000 is placed in an account bearing 10% simple interest, what would the account balance be after three years? $100,000 X.10 = $10,000 yearly interest $10,000 X 3 years = $30,000 total simple interest $30,000 (total simple interest) + $100,000 (principal) = $130,000 account balance If this amount were calculated with compound interest (yearly compounding), instead of simple interest, we would have to make sure to compound the interest of each year s principal: $100,000 X 0.10 = 10,000 first year s interest Page 47 of 103
$100,000 (1 st year s principal) + $10,000 (interest) = $110,000 (new principal) 110,000 X 0.10 = $11,000 second year s compound interest $110,000 (2 nd year s principal) + $11,000 (interest) = $121,000 (new principal) 121,000 X 0.10 = $12,100 third year s compound interest $121,000 (3 rd year s principal) + $12,100 (compound interest) = $133,100 account balance As you can see, compound interest yields $3,100 more than simple interest. Amortization The repayment of a financial obligation over a specific period of time in a series of periodic installments is known as amortization. Specifically, it is the payback of the principal owed to the lender. Each payment should cover the interest accrued over the time period for which payment is made and a portion of the principal. The interest portion is tax deductible, whereas the amortization portion is not. The payment is first applied to the accrued interest with the remaining balance applied to the principal. Example: What would the balance be on a 30-year loan for $150,000 with a 10% annual interest rate after the first two monthly payments of $1,500 each? To solve this problem, begin by calculating balances from the first to the second payment. Remember that the rate of interest for a month will be one-twelfth the annual rate. So solve for this in the following way: Monthly interest rate =.10/12 = 1/120 = 0.008333 = 10/12 % Page 48 of 103
Thus, the interest due the first month is the monthly interest rate times the loan principal for that month: $150,000 loan principal X 10/12 % monthly interest rate = $1, 250 The payment on the principal for the first month is the monthly payment less the first month s interest: $1500 monthly payment $1250 interest = $250 payment on loan principal So the loan balance after the first monthly payment is shown below. $150,000 $250 = $149,750 Our second month figures are as follows: Monthly interest is $149,750 loan principal x 10/12 % monthly interest rate = $1,247.92 interest Payment on loan principal is $1500 monthly payment $1,247.92 = $252.08 principal payment New balance = $149,750 (previous balance) $252.08 (monthly payment on principal) = $149,497.92 The following chart exhibits interest rate factors in determining the principle and interest monthly payments on a loan. The factors are the principal and interest monthly payment for a $1,000 loan for the corresponding rate and term. For instance, if a borrower borrows $1000 for 30 years at 5.5%, his principle and interest payment (P & I) would be $5.68. Of course borrowers borrow much more than $1,000 and usually sizable increments such as $200,000. By logic, if a $1,000 loan at 5.5% would have a principle and interest monthly payment of $5.68 over a 30-year term, how much would a $200,000 loan have? To calculate this simply take the $200,000 loan amount and divide by $1,000 to get the number of $1,000 Page 49 of 103
increments in the loan and then multiply that number by the interest factor from the chart. $200,000 divided by $1,000 multiplied by 5.68 will provide a payment of $1136.00 (P & I). Of course in most cases, the other monthly recurring costs such as property taxes, property insurance, and possibly mortgage and flood insurance (if applicable) would also be added for the total monthly payment. Example: A borrower borrows $268,000 for 30 years at 5.75% interest. What would be the monthly P & I payment on the loan? Solution: Interest rate factor for 5.75% for 30 years: 5.84; $268,000 divided by $1000 = 268 $1000 increments. 268 x 5.84 = $1565.12 (P & I). Page 50 of 103