Chapter 18

All in the Numbers: Real Estate Mathematics

IN THIS CHAPTER

Bullet Making the necessary conversions

Bullet Measuring buildings and land

Bullet Calculating percentages and commissions

Bullet Finding out about mortgage numbers

Bullet Seeing tax calculations in action

Bullet Wading through proration problems

Bullet Clarifying appreciation and depreciation

Bullet 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.

Warning Many students ask about rounding. Test writers are interested in you knowing basic formulas and techniques. They won’t give you questions where the difference between a right and wrong answer is a tenth of a number that you might get wrong due to rounding.

Remember Exam writers expect you to know the formulas involved in math calculations as well as how to actually do the calculations. You may run into questions that I call non-number math questions where you may be asked to identify the correct formula to solve a problem without actually being asked to doing the calculations. Be sure and memorize the formulas and get comfortable with your calculators.

Don’t Lose the Faith, but You May Have to Convert

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.

Remember Memorize the following conversion factors to ease your problem solving.

  • 1 foot = 12 inches
  • 1 yard = 3 feet
  • 1 mile = 5,280 feet
  • 1 square yard = 9 square feet (3 feet × 3 feet)
  • 1 acre = 43,560 square feet
  • A section = 1 square mile or 640 acres
  • A section = 5,280 feet (1 mile) on each side
  • 1 cubic yard = 27 cubic feet (3 feet × 3 feet × 3 feet)

Land and Buildings: Measuring Area and Volume

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.

Remember Area is the amount of space on a flat surface. Calculating the area means figuring out how large a flat space is. It may be determining the square footage of a house, the amount of carpeting or tile needed to cover the floor of a room, or the amount of wallpaper needed to cover a wall. (Remember a wall still is a flat surface.)

Remember Volume is the measurement of what it takes to fill up something. In real estate terms, calculating the volume of a warehouse, rather than just its floor area, is important when you’re informing a potential buyer how much stackable space the property has.

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.

Calculating the area of a square or a rectangle

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.

Diagram of a square measuring 100 feet on all four sides.

© John Wiley & Sons, Inc.

FIGURE 18-1: Square measuring 100 feet on all four sides.

Diagram of a rectangle having two sides opposite each other that are 150 feet long, and the two other sides opposite each other are 80 feet long.

FIGURE 18-2: Rectangle measuring 150 feet by 80 feet.

The formula for calculating the area of a square and a rectangle is the same.

Length (L) × Width (W) = Area (A)

Remember Like most other calculations in this chapter, when calculating area, the units of measure must be the same. You have to multiply feet by feet, yards by yards, and so on. Furthermore, the answer to any area problem is always in square units. If you’re multiplying feet by feet, your answer is in square feet.

Example You have a square piece of property that measures 100 feet by 100 feet. How big is the property?

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

Remember Notice in this second question where I put the longer figure for length. It really doesn’t matter in a problem like this what you call each number. On a piece of land, the distance across the front of the property generally is referred to as the width, while the distance going from the street back is the length or depth. Sometimes a problem may refer to a front foot or front footage, or frontage, which usually is the same as the width of the property along the street. If you get a problem that simply states two measurements of a lot, say 100 feet by 200 feet, assume that the first number is the measurement across the front of the lot, namely the frontage or width.

Figuring out the area of a triangle

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.

Diagram of a triangle having a base of 30 feet and a height of 12 feet.

© John Wiley & Sons, Inc.

FIGURE 18-3: Triangle with a base of 30 feet and a height of 12 feet.

Remember Similar to the way you calculate the area of a square or rectangle, when calculating a triangle’s area, don’t forget that all the units of measurement must be the same and the answer must be in square units. The height is a line that is perpendicular, or at a right (90-degree) angle, to the base.

Example Say you want to paint your roof’s gable end. The width of the house at that point is 30 feet. The height of the gable is 12 feet. How many square feet are you going to have to paint?

0.5 × 30 feet × 12 feet = 180 square feet

Calculating the area of a circle

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

  • π × radius squared (r2) (To square the radius, you multiply the radius by itself.)
  • π is the constant number 3.1416.

Check out Figure 18-4 for an example of a circle.

Diagram of a circle having a diameter of 16 feet and a radius of 8 feet, which is half of the diameter.

© John Wiley & Sons, Inc.

FIGURE 18-4: Circle with a diameter of 16 feet.

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.

Example You’re building a circular patio that’s 16 feet in diameter. How many square feet of patio block will you use?

  • 3.1416 × 8 (squared) = A (area)
  • 3.1416 × (8 × 8) = 201.06 square feet

Figuring out the area of an irregular shape

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.

Determining the volume of almost anything

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)

Example Say you want to calculate the volume of a rectangular warehouse that measures 100 feet by 200 feet by 15 feet high. The equation would look something like:

200 feet × 100 feet × 15 feet = 300,000 cubic feet

Remember Length × width is the formula for calculating the area, but by multiplying that answer (called the product) by the third dimension, the height, you get the volume. Just like when you calculate area, the measurements must always be in the same units. With volume, the answer is always in cubic units. So if all three measurements are in feet, the answer will be in 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.

Example If you had a silo 20 feet in diameter and 30 feet high, what is the volume?

  • π × radius squared (r2) × Height (H) = Volume (V)
  • 3.1416 × (10 feet × 10 feet) × 30 feet= 9,424.8 cubic feet

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.

Percentages: Pinpointing What You Really Need to Know

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.

Remember You have two choices for turning a percentage into a number you can work with. You can either divide the percent number by 100 or move the decimal point two places to the left.

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.

Example You make a $20,000 commission on the sale of a $400,000 house. What is your commission rate? Keep in mind that the word rate usually means percent. The calculation is

$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.

Remember One other thing you need to know how to do is to convert a fraction to a decimal and, in turn, a percentage. Don’t worry. If you don’t remember your sixth-grade math class, this step isn’t too difficult. Just divide the upper number (numerator) by the lower number (denominator). Doing so gives you a decimal. Now move the decimal point two places to the right (don’t forget to add zeroes as needed) and you have a percentage. Check out the following examples:

  • ½ = 1 ÷ 2 = 0.5 = 50 percent
  • 3⁄8 = 3 ÷ 8 = 0.375 = 37.5 percent
  • 9/16 = 9 ÷ 16 = 0.5625 = 56.25 percent

Commissions: Tracking Your Moolah

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.)

Figuring out how much you and everyone else earns

Here I start you out with a couple of basic commission problems and then move on to commission splits.

Figuring your commission

Example You’re a broker who sells a house. The owner has agreed to pay you a 5 percent commission. How much will you earn on the sale, if the house sells for $400,000?

  • Sale price × commission rate = commission earned
  • $400,000 × 0.05 = $20,000

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.

Figuring out your commission rate

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.

Example You sell a listed house for $400,000. You earn a $20,000 commission on the sale. What was your commission rate?

  • Commission earned ÷ sale price = commission rate
  • $20,000 ÷ $400,000 = .05 or 5 percent

Sharing the rewards: Commission splits

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.

Example Your firm receives a $20,000 commission that is to be split 60/40 between you and your broker. How much will you receive?

  • Commission amount × percentage share = commission amount share
  • $20,000 × 0.60 = $12,000

Tip Why did I use the 60 percent and not 40 percent? If you read the question carefully, the 60/40 split is in the same order as the “you and your broker.” If the problem doesn’t give you any more information, you have to interpret what test writers mean. Scan the following problem, which contains more than one split.

Example Suppose a $30,000 commission is earned on the sale of a house. The listing broker and the buyer’s broker agree to split the commission evenly. The listing salesperson receives 40 percent of the listing side. How much will the listing salesperson receive?

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.

  • $30,000 × 0.50 = $15,000 listing and buyer’s side because the commission is split evenly
  • $15,000 (listing side commission) × 0.40 (listing salesperson’s share) = $6,000 (listing salesperson’s commission)

Example A variation of this question uses the same information but asks you what the listing broker’s share was.

  • 100 percent (total percentage of listing side commission) – 40 percent (listing salesperson’s percentage) = 60 percent listing broker’s percentage
  • $15,000 (total listing side commission) × .60 (listing broker’s percentage share) = $9,000 (listing broker’s share)

Determining how much a house should sell for based on a commission rate

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.

Example Your seller wants to net (take away from the closing table) $300,000 after paying your 6 percent commission. How much does the house have to sell for to do this?

Warning This problem actually has two answers, $318,000 and $319,149 (rounded), but one of the answers, $318,000, happens to be wrong. Now before you start yelling or arguing or feeling bad that you got it wrong, allow me to explain. In virtually every math class that I teach, more than half the class comes up with the wrong answer. But by the time they do a few problems like the one that follows, nobody gets it wrong. Before I explain how you do this problem, I want to show you how to prove to yourself that the $318,000 is wrong. This proof may come in handy on an exam to check your answer.

  • Selling price × commission rate = commission paid
  • $318,000 × 0.06 = $19,080
  • Selling price – commission paid = net to seller
  • $318,000 – $19,080 = $298,920

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.

Remember Check out the following example if you want more clarification:

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.

Remember If you know a dollar amount and that is part of a larger dollar amount that you want to find, and you know the percentage of the total (100 percent) that it represents, you can always find the larger dollar amount by dividing the partial dollar amount by the percentage of the total that it represents. I’ll say that again with numbers:

  • $300,000 is the dollar amount that is part of a larger dollar amount that you want to find — that is, the selling price. $300,000 is 94 percent of the larger dollar amount, which is 100 percent. By dividing $300,000 by 94 percent, you get the larger amount, which is 100 percent of the total.
  • $300,000 ÷ 0.94 = $319,149

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.

Example You earn an $18,000 commission, which is 6 percent of the selling price. How much did the house sell for?

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)

Remember So what’s the moral of the story? When figuring commissions in relation to how much a house should sell for, memorize these two formulas:

  • Net to seller ÷ (100 percent – commission rate) = selling price
  • Commission amount in dollars ÷ commission rate = selling price

Making Mortgage Calculations without a Fancy Calculator

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.

Tip One of the questions I’m often asked is how necessary having a financial calculator is either for real estate work or for the state exam. A financial calculator is helpful and makes life easier if you know how to use it. As for the state exams, a simple inexpensive calculator does just fine. In fact, you can probably do most of the problems on your fingers and toes (if you have enough of them).

Calculating interest

A few standard problems that you may find on an exam deal with mortgage interest and principal payments. Here are the likely possibilities.

Annual and monthly interest

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.

Example You borrow $200,000 at 5 percent for 30 years in an amortized mortgage loan. How much interest will you pay the first year?

Remember that in a mortgage loan, the interest rate is always quoted annually and is always based on the loan’s unpaid balance.

  • Loan amount × interest rate = first year’s interest
  • $200,000 × 0.05 = $10,000

Warning The 30 years doesn’t matter. It’s extra information to confuse you.

Now here’s a monthly problem with different numbers:

Example You borrow $300,000 at 4 percent interest for 30 years in an amortized loan. What is the first month’s interest on the loan?

  • Loan amount × interest rate = first year’s interest
  • $300,000 × 0.04 = $12,000 annual interest

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.

  • First year’s interest ÷ 12 months = first month’s interest
  • $12,000 ÷ 12 = $1,000 first month’s interest

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.

  • $1,432 (rounded) (monthly payment) – $1,000 (first month’s interest) = $432 (principal payment)
  • $300,000 (loan amount) – $432 (first month’s principal payment) = $299,568 (loan balance after first month’s payment)
  • $299,568 (loan balance after first payment) × 0.04 (annual interest rate) = $11,982.72 (interest owed for the next 12 months)
  • $11,982.72 (rounded) (interest owed for the next 12 months) ÷ 12 months = $998,56 (interest paid for the first month of that next 12-month period, which is, in fact, the second month of the loan term of the loan)

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.

Total interest

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.

Example Say you borrow $300,000 at 5 percent for 30 years in an amortized mortgage loan. Your monthly payments are $1,610 (rounded) What is the total interest on the loan?

Warning Most people fool around with the 5 percent for a while, but you don’t need the percentage rate of the mortgage loan to work this problem. Watch this, because you’re not going to believe how easy it is.

  • $1,610 (monthly payment) × 12 months × 30 years = $579,600 total payments during the loan’s 30-year term.
  • $579,600 – $300,000 (original loan amount) = $279,600 interest paid during the course of the loan.

Remember Don’t forget that every amortized loan payment contains part principal and part interest. In this example in 30 years, you pay a total of $579,600 in principal and interest. So if you subtract the principal, or the amount you borrowed, what you have left is interest. It’s also a good demonstration of why you may want to pay that mortgage off as soon as possible.

Figuring out monthly payments

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).

Example The factor for a 20-year loan at 6 percent is $7.16. What is the monthly payment for a $150,000 loan?

  • $150,000 ÷ $1,000 = 150 (units of a $1,000)
  • 150 × $7.16 (factor to pay off $1,000) = $1,074 per month

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.

Oh, the Pain: Calculating Taxes

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.

Calculating the assessed value of a property

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.

Example A property has a market value of $200,000. The assessment ratio is 60 percent. What is the assessed value?

  • Market value × assessment ratio = assessed value
  • $200,000 × 0.60 = $120,000

Example What is the assessment ratio of a property whose market value is $200,000 and whose assessed value is $120,000?

  • Assessed value ÷ market value = assessment ratio
  • $120,000 ÷ $200,000 = 0.60 or 60 percent

Example What is the market value of a property assessed at a 60 percent assessment ratio whose assessed value is $120,000?

  • Assessed value ÷ assessment ratio = market value
  • $120,000 ÷ 0.60 = $200,000

Calculating taxes due

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.

  • Mills: This method bases the tax rate on so many tenths of a penny (or mills) in taxes for each dollar of assessed value.
  • Dollars per hundred: This method bases the tax rate on so many dollars of tax for each $100 of assessed value.
  • Dollars per thousand: This method bases the tax rate on so many dollars of tax per $1,000 of assessed value.

You need to be familiar with all three methods for exam purposes.

Mills

Example The tax rate in town is 24 mills. The assessed value of the property is $30,000. What are the taxes on the property?

  • Assessed value × millage = taxes owed
  • $30,000 × $0.024 = $720

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.

Taxes per hundred

Example The tax rate is $2.40 per $100 assessed value. What are the taxes on a property assessed at $30,000?

  • Assessed value × tax rate = taxes owed
  • $30,000 × ($2.40 ÷ $100) = taxes owed
  • $2.40 (tax rate) ÷ $100 (ratio of assessed value) = $0.024 tax rate (per dollar of assessed valuation)
  • $30,000 × $0.024 = $720

Taxes per thousand

Example The tax rate on a property assessed at $30,000 is $24 per thousand. What are the taxes?

  • Assessed value × tax rate = taxes owed
  • $30,000 × ($24 ÷ $1,000) = taxes owed
  • $24 (tax rate) ÷ $1,000 (ratio of assessed value) = $0.024 tax rate (per dollar of assessed valuation)
  • $30,000 × $0.024 = $720

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:

  • $30,000 ÷ $1,000 = 30 units of $1,000
  • 30 × $24 (tax rate per unit of $1,000) = $720.

If the tax rate is stated per hundred dollars, then divide by $100 instead of $1,000.

Putting Proration into Perspective

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.

Example The seller has paid taxes of $3,000 for the entire year ahead on January 1 and sells his property and closes title on June 1. Who owes what to whom?

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.

  • $3,000 ÷ 12 months = $250 per month
  • $250 per month × 5 months (the time the seller owned the property) = $1,250

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).

Example A buyer pays $2,400 in taxes in arrears for the entire year on December 31. He bought the house on September 1. Who owes what to whom?

  • $2,400 ÷ 12 months = $200 per month
  • $200 per month × 8 months (the time the seller owned the house) = $1,600

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.

State specific When you get out into the real world, find out what the local practice is for dividing up the year for prorations. Some areas and attorneys use the exact day, rather than the month. So in the previous examples, divide by 365 days to get the amount of taxes paid per day. Some people use the 12-month annual calculation and then divide by the exact number of days in a month when the closing date occurs midmonth. As far as I know, no state law governs this practice, but rather, it’s a matter of local practice. On the exam, test writers usually specify “actual days” if they want you to calculate it that way. If the time frame isn’t specified or you’re instructed otherwise on an exam, you need to use the third method, which is to divide a yearly cost or payment by 12 and the monthly number by 30, unless your real estate course or textbook specifies that one of the other methods is the only method used in your state. The other state-specific item that can affect proration is who owns the property on the day of closing. Unless your state says otherwise, assume that the buyer owns the house on the closing date. I’ll do a problem with a midmonth closing date to illustrate the 12-month/ 30-day method.

Example An owner sells a house, closes on May 17, but paid a full year’s taxes of $3,600 in advance on January 1. What is the proration of taxes?

  • $3,600 ÷ 12 months = $300 per month
  • $300 ÷ 30 days = $10 per day

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.)

  • 4 months × $300 per month = $1,200
  • 16 days × $10 per day = $160
  • $1,200 + $160 = $1,360 taxes used by the seller
  • $3,600 (total taxes for the year paid by the seller) – $1,360 (taxes used by the seller) = $2,240 (taxes used by the buyer)

The buyer owes the seller $2,240. The buyer gets a debit of $2,240, and the seller gets a credit of $2,240.

Appreciating Appreciation and Depreciation

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.

Getting more or less: Appreciation and depreciation of a property’s value

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.

Example You bought a house for $200,000 and five years later sold it for $250,000. What is the rate at which the house appreciated?

The formula for this type of problem has two parts:

  • New value – old value = change in value
  • Change in value ÷ old value = percent of change in value
  • $250,000 – $200,000 = $50,000
  • $50,000 ÷ $200,000 = 0.25 or 25 percent

Example A different type of problem deals with a decrease in value. You bought a house for $200,000 and five years later sold it for $150,000. By what percentage did it depreciate?

The formula for this type of problem also has two parts:

  • Old value – new value = change in value
  • Change in value ÷ old value = percent of change in value
  • $200,000 – $150,000 = $50,000
  • $50,000 ÷ $200,000 = 0.25 or 25 percent

In either of these problems, you divide the change by the older value.

Tip Keep an eye open for whether test writers ask you either of the preceding questions using the word “change” rather than “appreciate” or “depreciate.” For example: “By what percentage did the value of the house change?” In that case, your answers will have either a plus or a minus sign in front of them. The appreciation answer would be +25 percent, and the depreciation answer would be –25 percent.

Now for a reverse problem:

Example A house sold for $260,000, which is 130 percent of what you paid for it. How much did you pay or it?

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:

  • $260,000 ÷ 1.30 = $200,000
  • 1.30 is the decimal equivalent of 130 percent.

Another problem similar to this reverses the question:

Example You bought a house for $300,000, which is 75 percent of what you sell it for five years later. What price did you sell it for?

  • Original price (or value) ÷ the percent of selling price = price (or value) sought
  • $300,000 ÷ 0.75 = $400,000

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.

Depreciation: The government kind

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.

Example You have a building worth $250,000. You are allowed to depreciate it over a period of 27.5 years (also called the cost recovery period). What is the annual amount of depreciation?

  • Cost of property ÷ cost recovery period = annual amount of depreciation
  • $250,000 ÷ 27.5 = $9,090.91

Tip Now suppose you were asked what the annual rate of depreciation is. You don’t need the dollar amounts to figure that one out. You need only the cost recovery period.

Example You are allowed to depreciate a building over a period of 39 years. What is the annual rate of depreciation?

  • 1 ÷ 39 = .0256, which equals 2.56 percent per year
  • or you can also divide 100 by 39
  • 100 ÷ 39 = 2.56 percent per year

Try it both ways on your calculator to get a little practice in decimals and percentages.

Estimating Appraised Value

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

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:

  • I = Income or net operating income
  • R = Rate of return or capitalization rate
  • V = Value or sale price

Tip Don’t be confused by the possibility that any one of these letters can mean two different things (such as income versus net operating income). Memorize these terms and the problem will be clear as to what is being asked. Also remember that in these problems, all the numbers are based on annual figures. So if, for example, you’re given a monthly income number, you need to multiply by 12.

The formulas are as follows:

  • V = I ÷ R
  • I = V × R
  • R = I ÷ V

In the following three examples, I use the same numbers to illustrate how these formulas work.

Example Find the value of a building that has a net operating income of $30,000 where the capitalization rate is 10 percent.

  • V = $30,000 ÷ 0.10
  • V = $300,000

Example Find the income of a building that was sold $300,000 where the rate of return is 10 percent.

  • I = $300,000 × 0.10
  • I = $30,000

Example Find the capitalization rate of a building that sold for $300,000 and has a net operating income of $30,000.

  • R = $30,000 ÷ $300,000
  • R = 0.10 or 10 percent

The gross rent multiplier method

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

  • GRM = sale price (or value) ÷ rent
  • Rent = sales price (or value) ÷ GRM

The following examples show you how to use these formulas with numbers.

Example You’re appraising a building that generates a gross monthly rent of $3,000. You’ve calculated a GRM of 100. What is the value of the building?

  • Value = $3,000 × 100
  • Value = $300,000

Example You’re trying to calculate GRM using information on a building that recently sold for $300,000 with a gross monthly rent of $3,000. Find the GRM.

  • GRM = $300,000 ÷ $3,000
  • GRM = 100

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?

  • Rent = $300,000 ÷ 100
  • Rent = $3,000

Remember Don’t forget to note whether the numbers you’re given are based on monthly rent or annual rent and whether the question involves a Gross Rent Multiplier (GRM) or Gross Income Multiplier (GIM). GRMs use monthly rent numbers and GIMs use annual rent numbers. You may have to convert one to the other or use the number given. And just to make your life more interesting (and confusing) I have seen GRMs sometimes calculated on an annual basis and GIMs involve deduction of expenses. I doubt your course or the exam will get this deeply into the whole subject, but you should clarify and make sure you understand the terminology used for your exam.

Review Questions and Answers

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

  • 30 feet × 20 feet × 6 inches
  • 30 feet × 20 feet × (6 inches ÷ 12 inches — remember you have to convert inches to feet if the other numbers are in feet)
  • 30 feet × 20 feet × 0.5 feet = 300 cubic feet
  • 300 cubic feet ÷ 27 cubic feet = 11.11 cubic yards. (Make sure that you read the problem carefully to complete all the steps. In this case, you had to convert cubic feet to cubic yards.)

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

  • 110 feet × 99 feet = 10,890 square feet
  • 10,890 square feet ÷ 43,560 square feet = 0.25 acres

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

  • 300 feet × 200 feet × 15 feet = 900,000 cubic feet
  • 900,000 cubic feet × $0.60 per cubic foot = $540,000 annual rent
  • $540,000 ÷ 12 months = $45,000 monthly rent

Warning 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

Tip 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.

  • $150,000 × 0.07 = $10,500 (first year’s interest)
  • $10,500 ÷ 12 months = $875 (first month’s interest)
  • $1,059 (monthly payment of principal and interest) – $875 (interest) = $184 (principal paid off in first month)
  • $150,000 (original loan balance) – $184 (first month’s principal payment) = $149,816 (mortgage balance after first month’s payment)

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.

  • 100 percent – 30 percent = 70 percent
  • $200,000 ÷ 0.70 = $285,714

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

  • 18 feet × 14 feet = 252 square feet
  • 252 square feet ÷ 9 square feet = 28 square yards
  • 28 square yards × $23 per square yard = $644

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:

  • 5,280 feet × 5,280 feet = 27,878,400 square feet
  • 27,878,400 square feet ÷ 43,560 square feet per acre = 640 acres

Tip 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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