Chapter 18
IN THIS CHAPTER
Making the necessary conversions
Measuring buildings and land
Calculating percentages and commissions
Finding out about mortgage numbers
Seeing tax calculations in action
Wading through proration problems
Clarifying appreciation and depreciation
Assessing appraised value
Working in the real estate field is more than just selling real estate. It involves many skills and a thorough knowledge of many subjects, including math. As an agent you may have to measure a room or a house, or calculate the taxes on a piece of property. And most important of all, you need to know how to calculate your commission.
A quick note to all you math phobics (you know who you are!): Those of you who suffer from math-itis, or whatever you call it, should relax. Work through the material in this chapter, and do your best. Above all, though, don’t worry: The exam typically has few math problems. You’ll do fine without necessarily getting the math questions right. Later on, as a friend likes to tell all her mathematically challenged students, by the time they’re in the business a few months, they’ll be able to calculate their commissions down to the last penny, in their heads, while waiting at a stop light.
In this chapter, I review some common math concepts and calculations. Notice I say review. You’ve already seen most of what is in this chapter in middle school and high school. You simply may not have seen it in a real estate context. I show you how to apply some fairly simple math concepts to practical real estate problems. I’m sure as you go through this chapter that it all will come back to you.
Before you start doing most of the math problems in this chapter, you need to remember that test writers often throw problems at you that contain different units of measurements in the same problem. Before you can solve the problems, you need to convert those measurements to make sure you multiply feet by feet, yards by yards, and so on. Also, you may have to calculate the area of a room in square feet but later convert it to square yards. If any conversions are required to solve the problem, the exam writers won’t give you the conversion factors. They expect you to know them.
Because you’ll be selling land and buildings, knowing something about figuring out what size they are is important. For exam purposes, all you really need to know is how to calculate the area and volume of land and buildings after you have the measurements.
In this section, I take you through a series of examples that cover all the standard area and volume calculations that you may encounter on a state real estate exam.
A square is a four-sided figure on which each side is the same length. A rectangle is a four-sided figure where two sides opposite each other are equal. For example, a rectangle may have two sides opposite each other that are 150 feet long, and the two other sides opposite each other are 80 feet long. See Figures 18-1 and 18-2 for examples of a square and a rectangle.
The formula for calculating the area of a square and a rectangle is the same.
Length (L) × Width (W) = Area (A)
100 feet × 100 feet = 10,000 square feet
Or, say you have a rectangular piece of property that measures 80 feet by 150 feet. What is the area?
150 feet × 80 feet = 12,000 square feet
A triangle is a three-sided figure where all the sides join together. Look up at the end of a peaked roofed house and you’ll see a triangle. That triangle is called the gable end, and you may have to calculate the area to determine how much paint or siding is needed to cover it.
The formula for the area of a triangle is
½ Base (B) × Height (H) = Area (A)
Some of you may have learned this equation as
Base (B) × Height (H) ÷ 2 = Area (A)
Figure 18-3 shows a sample triangle.
0.5 × 30 feet × 12 feet = 180 square feet
A circle essentially is a single line that curves around and meets itself. Pick up that quarter in your pocket, and you’ve got a circle. The diameter is a straight line drawn through the center from one side of the circle to the other, and the radius is half of the diameter.
The formula to find the area of a circle is
Check out Figure 18-4 for an example of a circle.
Maybe you’re going to build a round patio. Or, if you live in farm country, you may have to rely on some of these calculations when determining the volume of a silo.
Unfortunately many houses aren’t single squares or rectangles but rather are combinations of several squares and rectangles. A rectangular house with a small breezeway and attached garage is a good example of what I’m talking about. And the key to calculating the area of a house like this is that you have to divide the figure into squares and rectangles and occasionally even triangles, calculate the area of each individual figure, and then add them together. You can also come pretty close by using this technique to calculate the area of an irregular piece of land.
The formula for calculating the volume of any six-sided, three dimensional figure involves calculating the area and multiplying it by the height. Why six sided? Most shapes that you may encounter in your real estate work will have six sides with opposite sides of equal dimensions. Think of a warehouse (the example that follows) or a nice concrete patio in your backyard. The first part of the formula looks like you’re finding the area of a flat surface. In the case of volume however you also want to account for the depth or height of the structure so the formula looks like this:
Length (L) × Width (W) × Height (H) = Volume (V)
200 feet × 100 feet × 15 feet = 300,000 cubic feet
In the case of a triangular-shaped object like an attic, in which the triangle’s height already is used to calculate the area, you multiply the area by the length of the attic. The whole formula is
½ Base (B) × Height (H) × Length (L) = Volume (V)
Finally, suppose you need to find the volume of a cylindrical object like a silo on a farm. Once again, you just have to multiply the area of the silo’s base by the silo’s height.
Remember, the area of a circle involves the radius, which is half of the diameter. (That’s why this example uses 10 feet instead of 20 feet.)
In case you’re wondering about other shape figures, you’ll probably not likely have to deal with the volume of an octahedron (eight sides) or a tetrahedron (ten sides) or some other unusual shape building. The preceding formula should get you through the exam, your real estate career, and that patio you may want to build in your backyard.
Using percentages is another type of real estate math problem. Commissions usually are figured on a percentage basis. Shared ownership of a property may be on a percentage basis. Vacancy rates usually are expressed as a percent. You’ll work with percents to figure out the selling price of a property. Property tax calculations may also involve percentages. So I think you can see the importance of understanding and being able to answer exam questions about percentages.
In this section, I provide the basics that you can apply to any problems asking you to use or calculate percentages. And by the way, I explain them to you as if you’re doing these calculations by hand. Some calculators have percentage keys that you can use. Feel free to use them, if you know how, or ignore them.
A percent — which is expressed as a number like 4 percent, 8 percent, 7.5 percent, or 0.8 percent — essentially describes what part of a hundred you’re talking about. The word “percent” comes from the Latin word centum, meaning a hundred. So 4 percent means four parts per hundred of whatever you’re taking about. If 4 percent of the eggs in a truckload are bad, then four out of every hundred eggs are bad. And if you’re allowed to build on 4 percent of your land area, it means you can cover 4 square feet out of every 100 square feet with a building. The calculations you do with percentages usually are most often multiplication and occasionally division, depending on the problem.
Say, for example, that you have to use 7 percent in a problem. If you divide 7 by 100, you get a number you can use in a calculator or by hand.
7 ÷ 100 = 0.07
That isn’t 0.07 percent. It’s simply 0.07, which is the same as 7 percent.
Another way to remember is that if a decimal point doesn’t actually appear in the number, then it’s assumed to be at the end. From there (the right side of a whole number), move the decimal to the left two spaces, even if you have to add one or more zeroes. Seven percent becomes 0.07. Check out a few more of these:
800% |
8 |
80% |
0.80 |
8% |
0.08 |
.8% |
0.008 |
.08% |
0.0008 |
8.5% |
0.085 |
Note: You probably won’t be working too much with small percentages, such as 0.08 percent or 0.8 percent, either on an exam or in real life.
Don’t forget the reverse of this concept so you can solve a problem like this.
$20,000 ÷ $400,000 = 0.05
Move the decimal point two places to the right to get the answer of 5 percent. Remember, to get from a decimal to a percentage, move the decimal two places to the right, or multiply by a hundred.
You probably were wondering when I’d finally get to this all-important section — the amount you get paid. Discussing percentages is foremost, because most commissions are based on a percentage of the sale price of a property. If you haven’t read “Percentages: Pinpointing What You Really Need to Know,” earlier in this chapter, take a few minutes to scan it before reading this section.
In this section, I take you through a few variations on commission problems, including splitting the commission with other brokers and agents. I also have you look at how to get to a selling price working backward from a commission. Because of antitrust laws, which say that all commissions are negotiable and that brokers cannot agree among themselves (price fixing) to set certain commission rates, I make up all the figures that I use for commissions and commission splits in the examples that follow. (For more about commissions and antitrust laws, check out Chapter 3.)
Here I start you out with a couple of basic commission problems and then move on to commission splits.
Remember to turn 5 percent into a number you can use (in this case, 0.05) before you proceed with the rest of the problem.
A variation on the commission problem has you figuring out your commission rate if you know the selling price of the property and the commission earned. I use the same numbers that I did in the last problem.
As a salesperson, your broker will pay you a share of the total commission earned for the sale. Remember that a salesperson works under the authority of a broker (see Chapter 3 for the full scoop).
All commissions and commission splits are negotiable between the salesperson and the broker. In a math problem, this split may be expressed different ways. For example, a problem may have a 60/40 split, which is 60 percent going to one party and 40 percent going to the other party. Whenever the commission shares are unequal, the problem will be clear as to what percentage each party gets. Examples of these types of problems follow.
Where two brokers cooperate on a sale, they’re often referred to as the listing side (the broker who originally got the listing agreement to represent the seller) and the buyer’s side (the broker who finds the buyer). In this particular question, you also have to know that “split” evenly means that each of the sides gets half of the total commission.
Establishing how much a house should sell for is an interesting calculation that has many uses, primarily because it enables you to work backward from a percentage to a number. But first I have a problem that I want you to try.
The number $298,920 obviously isn’t the $300,000 net the seller wants. If you got $318,000 for your answer, it’s because you added the 6 percent commission to the net that the seller wanted, but that isn’t how you calculate selling prices and commissions. You get a commission by taking it away from the selling price, not adding onto the net to the owner figure. If you did the same proof using the $319,149 and took 6 percent from that number, the owner gets the requested $300,000.
What it is |
Percentage |
Dollar amount |
---|---|---|
Selling price |
100 percent |
X (That’s what you’re looking for.) |
Commission |
6 percent |
You don’t know this and don’t need to know. |
Net to seller |
94 percent |
$300,000 |
See how the commission percentage and the seller’s net percentage always equal 100 percent, which is the selling price.
In other words, if $300,000 is 94 percent of some number that you want to find (the selling price, in this case), all you have to do is divide $300,000 by 94 percent (0.94). You get $319,149.
In this particular problem, you had to subtract 6 percent from 100 percent to get the 94 percent, which is the seller’s share. But the test writers may write the problem a little differently, trying to trick you.
Using the logic for this kind of problem, you can restate it like this: $18,000 is 6 percent of some number. In this case, that number is the selling price. And if you know the dollar amount and the percentage of the whole number it represents, all you have to do is divide by the matching percentage.
$18,000 (commission amount) ÷ 0.06 (commission rate) = $300,000 (selling price)
In this section, I do a few problems to review some of the math associated with mortgages. (Chapter 15 has the scoop on mortgages and additional math, such as problems involving loan to value ratios.) In these calculations, one important term to know is amortized loan, which means that each payment on the mortgage is a combination of principal and interest so that at the end of the mortgage term you have nothing left to pay.
A few standard problems that you may find on an exam deal with mortgage interest and principal payments. Here are the likely possibilities.
All interest on mortgage loans is expressed as an annual interest amount, so if your mortgage interest rate is 8 percent, that’s the annual rate. But most mortgages are paid on a monthly basis, so you sometimes need to calculate how much interest you actually paid in one month based on that annual rate.
First, look at an annual interest problem.
Remember that in a mortgage loan, the interest rate is always quoted annually and is always based on the loan’s unpaid balance.
Now here’s a monthly problem with different numbers:
You wind up with $12,000 for the first year’s interest. To figure out the first month’s interest, all you have to do is divide the first year’s interest by 12.
Note that this monthly interest calculation works this way only for the first month’s interest.
Test writers may go further and ask you to calculate the second month’s interest. To answer the question, you need to know what the total monthly payment is, and the test writers will tell you. I’ll continue using the numbers from the previous problem. In this case, the monthly payment, which includes principal and interest, is $1,432 (rounded), information they have to give you. The question is how much is the second month’s interest payment.
What you need to remember here is that in an amortized loan, you’re only reducing the amount you owe by the amount of principal you pay each month and not by the amount of the total payment, because each payment includes interest and principal.
A type of interest problem that seems to confuse people is the calculation of total interest. Total interest is the amount of interest you pay during the entire life of the loan, assuming that you pay off the loan by making the payments within the required time frame. In general, banks provide these numbers to people, but you need to be familiar with this calculation because it is fair game on an exam.
Unless you use a financial calculator, you’re going to calculate mortgage payments using a mortgage table. These tables, which are arranged according to the percentage of interest and years of the mortgage term, provide the monthly payment to amortize, or pay off interest and principal, for a $1,000 mortgage loan. After you get that monthly payoff number, which sometimes is called the payment factor, you multiply it by the number of thousands of dollars of the mortgage loan (which you get by dividing the loan amount by $1,000).
If you run into a problem like this on the exam, you’ll either get a sample of a mortgage table or be given the payment factor you need to solve the problem. You’ll have to remember the formula in the example. In the real world, that is, when you have your license and are working on your first million, printed mortgage tables, many online mortgage calculation sites, and financial calculators can make all this relatively simple.
If you want the lowdown on real estate taxes, check out Chapter 16, where you also find a bit of math on equalization rates. In this section, I solve a few sample problems that you may encounter about taxes. You need to know this information not only for the test, but also because every listing of a property for sale generally requires the agent to find out what the taxes are.
Many municipalities use assessment ratios to assess properties. An assessment ratio is the percentage relationship between the market value of a property, which is the amount the property will sell for in a normal market sale, and the assessed value, which is a value tax assessors use to calculate taxes and is usually related to market value. (I discuss the details of market value in Chapter 14 and assessment ratios in Chapter 16.) The use of an assessment ratio creates the possibility of three kinds of exam problems.
You can calculate taxes due using one of the following three methods, depending on how the municipality calculates taxes or how the exam question is asked.
You need to be familiar with all three methods for exam purposes.
Remember, when working with a millage, move the decimal three places to the left, because millages are tax rates expressed in mills, or tenths of a cent.
If it’s easier for you to see, you can also divide the assessed value by the unit in which the tax is stated, in this case, you could do the following:
If the tax rate is stated per hundred dollars, then divide by $100 instead of $1,000.
Proration is the allocation or dividing of certain money items at the closing. (I discuss proration and closings in more detail in Chapter 9.) An attorney, a real estate salesperson, a broker, or a representative of the title company does the proration calculations at the closing. In any case, most test writers expect you to know the basics of proration math. The key to remember about prorations is that the person who uses it needs to pay for it. Here’s an example that illustrates the point of proration.
The seller paid the full year’s taxes, but used the property only for the five months (January through May). Read the problem carefully. The buyer needs to pay the seller for the taxes already paid for the months the seller won’t own the property during the year. The rest is just the math of dividing up who paid and who pays.
This is the amount of the taxes that the seller used from January through May.
$250 per month × 7 months (the new buyer’s time in the house) = $1,750
This is the amount of the taxes the buyer will use, because he bought the house on June 1.
Because the seller had already paid the taxes for the whole year, the buyer owes the seller $1,750 for the taxes the buyer will be using but didn’t pay for. In the terminology of proration, the buyer gets a debit and the seller gets a credit.
Here’s another problem that has the buyer paying in arrears (or after the fact).
The seller used $1,600 of the paid taxes. Now to determine how much the seller didn’t use:
$200 per month × 4 months (the time the buyer owned the property) = $800
Because the buyer paid the full $2,400 for the previous year, during which he owned the house for only four months, the seller owes the buyer $1,600. The buyer gets a credit and the seller gets a debit.
The seller/owner owned the house for four full months (January through April) and 16 days in May. (Remember, the buyer is considered to own the house on the day of closing.)
The buyer owes the seller $2,240. The buyer gets a debit of $2,240, and the seller gets a credit of $2,240.
As if you weren’t confused enough already, you’ll find there’s only one kind of appreciation but there are two kinds of depreciation. Stay with me, though, because this stuff really is pretty easy.
Appreciation is an increase in a property’s value caused by factors like inflation, increasing demand, and improvements to the property. Depreciation is a decrease in the value of a property caused by lower demand, deflation in the economy, deterioration, or the influences of other undesirable factors like a new sewage treatment plant going in next door. Another type of depreciation, which I discuss in Chapter 17, is a tax benefit the government gives people on real estate investments.
Real estate salespeople and brokers, buyers, and sellers always are interested in how much a property’s value has increased or decreased. Investors, real estate brokers, and salespeople may want to know what tax benefits of a property are attributed to depreciation, although more often than not, an accountant needs to be doing these calculations for an investor. Test writers expect you to know the basics of how to calculate appreciation and depreciation.
People always are interested in how much more they can sell their property for than what they paid for it. They may not want to consider a loss to their property’s value, but that can happen, too. A real estate agent is expected to be able to track increases and decreases in a property’s value and to apply market increases to specific properties. So if the overall real estate market values in an area have increased by 10 percent during the last year according to government or other statistics, you need to be able to apply this increase in value to a specific property. By the way, not only market values can increase. Any of the types of values that I discuss in Chapter 14 can increase or decrease. I show you some math on how to handle these problems in the real world and on the exam.
The formula for this type of problem has two parts:
The formula for this type of problem also has two parts:
In either of these problems, you divide the change by the older value.
Now for a reverse problem:
The formula for this type of problem is
Current price (or value) ÷ percent of original price = price (or value) sought
Notice in this formula that I say “percent of original price,” which is always 100 percent plus the change, which is +30 percent. Now how did I know this, since the numbers were not given that way? Read the wording of the question carefully. It said that the house sold for 130 percent of the original price, not 130 percent more than the original price. Another way to look at this is to say, “$260,000 is 130 percent of some number” — in this case, the original price. If you can say it that way, you know to divide the dollar amount you’re given by the percentage number.
With numbers the problem looks like this:
Another problem similar to this reverses the question:
Once again, you can say, “$300,000 is 75 percent of some number.”
In these types of problems, you can usually get the number you’re looking for by dividing the number you’re given by the percent that it represents. You can find similar problems in the section on “Determining how much a house should sell for based on a commission rate” earlier in this chapter.
In Chapter 17, I explain the concept of depreciation that sometime is called cost recovery. Although in the real world accountants working for real estate investors usually do these calculations, test writers expect you to understand and be able to apply the basics. This type of depreciation has nothing to do with a property going down in value but rather is based on government regulations that are designed to help real estate investors reduce their tax burdens.
Try it both ways on your calculator to get a little practice in decimals and percentages.
If you want the lowdown on appraising property, check out Chapter 14. As a real estate agent, you won’t actually be doing appraisals, but you can use these formulas to roughly estimate property values, particularly if you deal in investment properties. You can read about real estate investing in Chapter 17, but in the meantime, state test writers expect you to understand and be able to apply these formulas. In this section, the only real math-driven formulas for appraising, at least at the level that you’re expected to know it, are formulas for finding value using the income approach. Two different methods are used in that approach to value. I give you formulas and examples for both.
The capitalization method uses one formula for finding a property’s value, and with a couple of variations you can find the property’s income and the capitalization rate. Capitalization is a technique for estimating a property’s value based on its income. You can read all about it in Chapter 14.
The formulas use the following symbols:
The formulas are as follows:
In the following three examples, I use the same numbers to illustrate how these formulas work.
The method for finding the value of a property using the gross rent is an easy matter of multiplication. The gross rent multiplier is used to estimate the value of small investment-type properties like small multifamily houses. You can find out more about it in Chapter 14. The formula is
Value = Gross monthly rent × gross rent multiplier (GRM)
The variations of this equation are
The following examples show you how to use these formulas with numbers.
You would rarely have to calculate the rent using the GRM formula, but just in case, here’s one last example.
Say you were looking at a building to invest in that was selling for $300,000 with a GRM of 100. What is the annual gross rent income?
It would be great to get 100 percent on the state exam, but if math really sends you into a spin, don’t worry. Not many math questions are on the salesperson’s or broker’s exam. In either case, if you really do well on the other questions, math won’t make or break you.
1. Find the volume of concrete needed to pour a patio slab 30 feet by 20 feet by 6 inches thick.
(A) 600 cubic feet
(B) 600 square feet
(C) 3,600 cubic feet
(D) 11 cubic yards
Correct answer: (D). Length × Width × Height = Volume
2. You’re selling a rectangular piece of property that measures 99 feet by 110 feet. What part of an acre is this property?
(A) .75
(B) .50
(C) .25
(D) .10
Correct answer: (C). Length × Width = Area
The answers are all in acres so you must convert by dividing the square feet by 43,560 square feet, which is the number of square feet in an acre. Watch out when one of the answer choices is the correct number of square feet but the question asks for acres.
3. A warehouse rents for $0.60 per cubic foot per year and measures 200 feet by 300 feet by 15 feet high. What is the monthly rent?
(A) $36,000
(B) $45,000
(C) $540,000
(D) $900,000
Correct answer: (B). Length × Width × Height = Volume
Note in this problem that $540,000 is a correct answer to a part of the problem. Be sure to read questions carefully or you may pick the right answer to the wrong question.
4. You sold 40 houses this year and 30 houses last year. What percent fewer houses did you sell last year?
(A) 133 percent
(B) 75 percent
(C) 66 percent
(D) 25 percent
Correct answer: (D). 40 (houses this year) – 30 (houses last year) = 10 fewer houses last year than this year
10 ÷ 40 = 0.25 or 25 percent
In this problem, you divide by 40 because you want to find out what percentage less than this year. If the question were how much more than last year, then you would have divided by 30.
5. How much is owed after the first month’s payment on an amortized mortgage loan of $150,000 at 7 percent interest for 25 years, if the monthly payment is $1,059?
(A) $148,941
(B) $149,816
(C) $149,125
(D) $139,500
Correct answer: (B). See the following math.
Take a look at “Making Mortgage Calculations without a Fancy Calculator,” earlier in the chapter to find out how to go to the second month for this type of problem. Don’t get thrown by what may appear to be an odd number in a question, like the 7 percent in this question (which I purposely put in) that at the writing of this edition of this book was much higher than market rates. Prices of homes vary around the country. Interest rates do fluctuate. And exams are sometimes out of date with respect to these kinds of things. Just use what the exam writers give you and don’t worry if the information seems a little unrealistic.
6. If a house has depreciated 30 percent since you bought it, and it’s worth $200,000 now, what did you buy it for?
(A) $285,714
(B) $240,000
(C) $230,000
(D) $160,000
Correct answer: (A). $200,000 is 70 percent of some number, the original purchase price of the house.
7. The seller pays $1,200 in taxes for six months in advance on April 1. She sells her house on May 1. What is the proration on the taxes?
(A) Seller gets a credit of $1,000; buyer gets a debit of $200.
(B) Seller gets a credit of $200; buyer gets debit of $1,000.
(C) Seller gets a credit of $1,000; buyer gets a debit of $1,000.
(D) Buyer gets a credit of $1,000; seller gets a debit of $1,000.
Correct answer: (C). The seller owned the property for one month of the period for which the taxes had been paid, so she used $200 worth of the taxes but had paid $1,200. She gets a credit of $1,000. The buyer will be in the house for five months of the tax period but paid none of the taxes and will use $1,000 worth. He gets a debit of $1,000. The buyer owes the seller $1,000 at the closing.
$1,200 (taxes) ÷ 6 months = $200 per month
8. A broker receives a commission check for $13,000 based on a 4 percent commission rate. How much did the house sell for?
(A) $52,000
(B) $300,000
(C) $325,000
(D) $336,960
Correct answer: (C). Commission earned ÷ commission rate = sale price
$13,000 ÷ 0.04 = $325,000
9. A room 14 feet by 18 feet is to be carpeted at a cost of $23 a square yard. What will be the cost to carpet the room?
(A) $214
(B) $579
(C) $644
(D) $5,796
Correct answer: (C). Length × Width = Area
10. What is the area in acres of a piece of property that is one mile square?
(A) 640
(B) 320
(C) 160
(D) 80
Correct answer: (A). One square mile is 640 acres. You can memorize this fact, because it’s also the size of a section in the Government or Rectangular Survey System. Or using the Length × Width = Area formula:
You may also want to remember that the phrase “one mile square” means a length of one mile on each side. Test writers also like to ask the area of a figure where the side is some portion of a mile, say ½ mile on each side. Remember that a mile is 5,280 feet.
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