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Formulas, Symbols, Math

Review, and Sample Problems

Mathematics and Analytical Skills Review ................................................................. 1

Summary of Basic Formulas ....................................................................... .............. 11 Direct Capitalization ........................................................................ .................... 11 Yield Capitalization ........................................................................ ..................... 13

Present Value of Increasing/Decreasing Annuity ................................................... 14

Mortgage-Equity Analysis ........................................................................ ............. 15 Investment Analysis ........................................................................ .................... 19 Symbols ................................................................ ............................................ 21 Standard Subscripts ........................................................................ ................... 24

Capitalization Selection Tree .......................................................................

............ 25 Sample Problems with Suggested Solution Keystrokes

for the HP-10B, HP-12C, HP-17B, and HP-19B .......................................................... 27

SMATHREV-E

For Educational Purposes Only

The opinions and statements set forth herein reflect the viewpoint of the Appraisal Institute at the time of publication bu

do not necessarily reflect the viewpoint of each individual. While a great deal of care has been taken to provide accurate

and current information, neither the Appraisal Institute nor it s editors and staff assume responsibility for the accuracy o f

the data contained herein. Further, the general principles and conclusions presented in this text are subject to local, state,

and federal laws and regulations, court cases, and any revisions of the same. This publication is sold for educational

purposes with the understanding that the publisher and its instructors are not engaged in rendering legal, accounting, o

r any other professional service.

Nondiscrimination Policy

The Appraisal Institute advocates equal opportunity and nondiscriminatio n in the appraisal profession and conducts its activities in accordance with applicable federal, state, and local laws.

part of this publication may be reproduced or incorporated into any information retrieval system without written permission

from the publisher. Appraisal Institute Mathematics and Analytical Skills Review 1

Mathematics and Analytical Skills Review

I. Order Of Operations

Background - A universal agreement exists regarding the order in which addition, subtraction, multiplication, and division should be performed.

1) Powers and roots should be performed first.

2) Multiplication and division are performed next from left to right in the order

that they appear.

3) Additions and subtractions are performed last from left to right in the order

that they appear.

Example 1: 3 + 4 × 5=

3 + 20 = 23

Example 2: 7 × 3

7 × 9 = 63

Note. If grouping symbols such as parentheses "( )," brackets "[ ]," and braces "{ }," are present, the operations are simplified by first starting with the innermost grouping symbols and then working outward.

Example 3: {[12 - 2 × (7 2 × 2)] ÷ 3}

2 {[12 - 2 × (7 4)] ÷ 3} 2 {[12 - 2 × 3] ÷ 3} 2 {[12 - 6] ÷ 3} 2 {6 ÷ 3} 2 2 2 =4

2 Appraisal Institute Mathematics and Analytical Skills Review

II. Subtracting/Adding Negative Numbers

Background - Every negative number has its positive counterpart, which is sometimes called its additive inverse. The additive inverse of a number is that number which when added to it produces 0. Thus, the additive inverse of 5 is +5 because (5) + (+5) = 0. Subtracting a negative number is the same as adding its positive counterpart. Adding a negative number is the same as subtracting its positive counterpart.

Example 1: 17 - (-3) =

17 + (3) = 20

Example 2: 12 + (-8) - (-10) =

12 + (-8) + (10) =

12 - (8) + (10) = 14

III. Multiplication/Division With Negative Numbers Background - When numbers of opposite signs are multiplied or divided, the result is negative. When numbers of the same sign are multiplied or divided, the result is always positive. When dividing or multiplying, the two negative signs cancel out.

Example 1: 6 × (-7) ÷ 3 =

(-42) ÷ 3 = (-14)

Example 2: (-32) ÷ (-4) = 8

Appraisal Institute Mathematics and Analytical Skills Review 3

IV. Addition/Subtraction of Fractions

Background - Simplifying fractions by addition or subtraction requires the use of the lowest common denominator. The denominator on both fractions must be the same before performing an operation. Just as when adding dollars and yen, the yen must be converted to dollars before addition.

Example 1: $120 + ¥13,000 =

$120 + ¥13,000

130 dollars

Yen $120 + $100 = $220

Example 2:

32
43
33 42
34 43
98
12 12 17 12

V. Multiplication/Division of Fractions

Background - Multiplication with fractions is very straightforward, just multiply numerator by numerator and denominator by denominator. When dividing with a fraction, the number being divided (dividend) is multiplied by the reciprocal of the divisor. Frequently this has been stated "invert and multiply."

Example 1:

24 2 4 8

35 3 15

Example 2:

341233 4433

4 Appraisal Institute Mathematics and Analytical Skills Review

VI. Compound Fractions

Background - Frequently a mathematical expression appears as a fraction with one or more fractions in the numerator and/or the denominator. To simplify the expression multiply the top and bottom of the fraction by the reciprocal of the denominator.

Example:

2 5 4 12 45144
2 20 1 2 20 1 10 Note. When multiplying or dividing the numerator and denominator of the fraction by the same number the value of the fraction does not change. In essence, the fraction is being multiplied/divided by 1.

VII. Exponents

Background - Exponents were invented to make it easier to write certain expressions involving repetitive multiplication: K × K × K × K × K × K × K × K × K = K 9 . Note that the exponent (9) specifies the number of times the base (K) is used as a factor rather than the number of times multiplication is performed.

Example: 6

4 = 6 × 6 × 6 × 6 = 1,296 Appraisal Institute Mathematics and Analytical Skills Review 5

VIII. Fractional Exponents

The definition of a fractional exponent is as follows:

M / N MN

XX This equality converts an expression with a radical sign into an exponent so that the y x key found on most financial calculators can be used.

Example 1:

4545
12 12 = 7.3009

Example 2:

10.255 10 10 10

= 1.5849

IX. Subscripts

Background - Concepts or variables that are used in several equations generally use subscripts to differentiate the values. Example: Capitalization rates are expressed as a capital "R." Since there are a number of different capitalization rates used by appraisers, a subscript is used to specify which capitalization rate is intended. An equity capitalization rate, therefore, is written as R E

6 Appraisal Institute Mathematics and Analytical Skills Review

X. Percentage Change

Background - Calculating percentage change or delta "" is required in several of the capitalization techniques. The formula for " " is: final value starting value starting value

Example 1:

What percentage of change occurs if a property

purchased for $90,000 sells for $72,000?

Answer: $72,000 $90,000

$90,000 $18,0000.20$90,000 =20%

Example 2:

What percentage of change occurs if a property

purchased for $75,000 sells for $165,000?

Answer: $165,000 $75,000

$75,000 $90,000 $75,000 = 120% Appraisal Institute Mathematics and Analytical Skills Review 7

XI. Cancellation of Units

Background - Many appraisal applications involve the multiplication and/or division of numbers with "units" associated with them, e.g., $/sf, sf, ft, yr., etc. The proper handling of these units is necessary to describe the mathematical result correctly. According to the identity principle, any number/variable divided by itself is equal to 1 and can thus be removed from the equation. 2 2 57157
Xxy X xy

Example: What value would be indicated for a

12,000 sf building if it is worth $55/sf?

Answer:

$5512,000 sfsf = $660,000 sfsf

8 Appraisal Institute Mathematics and Analytical Skills Review

XII. Solving Equations

Background - In many instances, an equation or formula exists in a form that is not convenient for the problem at hand, e.g., with value as the goal and the available equation is: I = R × V. Using equation solving techniques, the formula can be rewritten to solve for value with V = I ÷ R as the result. The rules of equation solving are quite simple and are as follows:

1) Adding or subtracting the same number/variable to both sides of the

equation will not change the solution.

2) Multiplying or dividing both sides of the equation by the same

number/variable will not change the solution.

3) Raising both sides of the equation by the same power or taking the

same root will not change the solution.

Example: Income = Capitalization Rate × Value

I = R × V

Divide both sides by R.

IRV RR IVRquotesdbs_dbs4.pdfusesText_7

[PDF] [PDF] Formulas, Symbols, Math Review, and Sample - Appraisal Institute

Formulas, Symbols, Math

Review, and Sample Problems

SMATHREV-E

For Educational Purposes Only

Nondiscrimination Policy

Mathematics and Analytical Skills Review

I. Order Of Operations

1) Powers and roots should be performed first.

2) Multiplication and division are performed next from left to right in the order

3) Additions and subtractions are performed last from left to right in the order

Example 1: 3 + 4 × 5=

3 + 20 = 23

Example 2: 7 × 3

7 × 9 = 63

Example 3: {[12 - 2 × (7 2 × 2)] ÷ 3}

2 Appraisal Institute Mathematics and Analytical Skills Review

II. Subtracting/Adding Negative Numbers

Example 1: 17 - (-3) =

17 + (3) = 20

Example 2: 12 + (-8) - (-10) =

12 + (-8) + (10) =

12 - (8) + (10) = 14

Example 1: 6 × (-7) ÷ 3 =

Example 2: (-32) ÷ (-4) = 8

IV. Addition/Subtraction of Fractions

Example 1: $120 + ¥13,000 =

130 dollars

Example 2:

V. Multiplication/Division of Fractions

Example 1:

24 2 4 8

35 3 15

Example 2:

341233 4433

4 Appraisal Institute Mathematics and Analytical Skills Review

VI. Compound Fractions

Example:

VII. Exponents

Example: 6

VIII. Fractional Exponents

M / N MN

Example 1:

Example 2:

10.255

10 10 10

IX. Subscripts

6 Appraisal Institute Mathematics and Analytical Skills Review

X. Percentage Change

Example 1:

What percentage of change occurs if a property

Answer: $72,000 $90,000

Example 2:

What percentage of change occurs if a property

Answer: $165,000 $75,000

XI. Cancellation of Units

Example: What value would be indicated for a

12,000 sf building if it is worth $55/sf?

Answer:

8 Appraisal Institute Mathematics and Analytical Skills Review

XII. Solving Equations

1) Adding or subtracting the same number/variable to both sides of the

2) Multiplying or dividing both sides of the equation by the same

3) Raising both sides of the equation by the same power or taking the

Example: Income = Capitalization Rate × Value

Divide both sides by R.