Basic Arithmetic: Fractions, Percentages, and Direct Capitalization

This chapter focuses on fundamental mathematical operations essential for real estate professionals in valuation and investment. It covers fractions, percentages, and direct capitalization. These concepts are practical tools used in real estate analysis and valuation.

The scientific importance of these operations lies in their ability to represent and analyze relationships between different values in the real estate market. Fractions are used to represent ownership shares or area ratios. Percentages are necessary for calculating price changes, evaluating investment returns, and determining real estate taxes. Direct capitalization, a core technique in real estate valuation, relies on understanding percentages and fractions to convert expected property income into an estimated value. Understanding these operations enables real estate analysts to make informed decisions, accurately assess risks, and provide sound investment recommendations.

1. Fractions

• A fraction represents a part of a whole, consisting of a numerator and a denominator. The numerator represents the part taken, and the denominator represents the whole. Example: 3/4 means three parts out of four equal parts.
• Types of fractions: proper (numerator < denominator, e.g., 1/2), improper (numerator ≥ denominator, e.g., 5/2), and mixed number (whole number + proper fraction, e.g., 2 1/4).
• Arithmetic operations: addition/subtraction (requires unified denominators), multiplication (numerator x numerator, denominator x denominator), division (invert the second fraction and multiply).
• Conversion: improper fraction to mixed number (divide numerator by denominator; the quotient is the whole number, the remainder is the new numerator), mixed number to improper fraction (multiply the whole number by the denominator, add to the numerator; keep the same denominator).
• Applications in real estate: calculating area of a land portion, profit distribution among partners, and determining occupancy rate.

2. Percentages

• A percentage expresses a number as a fraction of 100. The symbol "%" means "divided by 100". Example: 25% means 25/100 or 0.25.
• Conversion: percentage to decimal (divide by 100), decimal to percentage (multiply by 100), percentage to fraction (write as a fraction with denominator 100 and simplify), fraction to percentage (divide numerator by denominator, then multiply by 100).
• Arithmetic operations: calculating a percentage of a value (multiply the decimal form of the percentage by the value, calculating percentage change ((New Value - Original Value) / Original Value * 100), calculating the original value given a percentage and result (Result / Percentage in decimal form).
• Formula: Part = Percentage x Whole (A = B x C) which can be manipulated to find A, B, or C. The book has a percentage triangle.
• Applications in real estate: calculating occupancy rate, annual rent increase, and commission.
• Interest: Simple interest problems are similar to percentage and capitalization problems, but include time. The formula is Interest = Principal x Rate x Time (A = B x C x D). The formula can be expressed four ways to solve for each variable. An example is provided for calculating the interest earned on an investment over six months.

3. Direct Capitalization

• It is a method to estimate property value based on net operating income (NOI).
• Formula: Value = NOI / Capitalization Rate (V = I / R).
• Capitalization Rate: The ratio between NOI and value (R = I / V).
• Income Multiplier: The inverse of the capitalization rate (Multiplier = Value / Income).
• The capitalization rate and income multiplier are reciprocals of each other.
• Applications in real estate: estimating the value of a commercial property, determining an appropriate capitalization rate using recent sales data of comparable properties.
• The relationship between income (I), capitalization rate (R), and value (V) is presented as a triangle.
• Finding I: I = R x V
• Finding R: R = I / V
• Finding V: V = I / R

4. Additional Applications
* Calculating irregular areas: Divide the irregular land plot into simple geometric shapes (square, rectangle, triangle). Calculate the area of each shape separately and then sum the areas to obtain the total area. The area formulas are:
* Square: Area = Length x Width
* Rectangle: Area = Length x Width
* Triangle: Area = (Base x Height) / 2
* Calculating volumes: Volume = Length x Width x Height

5. Present and Future Value

• Present value is the value of money today.
• Future value is the value of money at a future date.
• Calculating the present value of a future amount is called discounting.
• The relationship between present and future value depends on the amount of interest that can be earned between the current date and the future date.

The chapter covers basic arithmetic operations necessary for real estate appraisal, focusing on fractions, percentages, and direct capitalization.

Fractions are used to calculate land and building areas, especially irregular shapes, by dividing them into simple geometric shapes, calculating individual areas, and summing them. The concept of reciprocals is explained to facilitate multiplication and division.

Percentages are a fundamental tool in appraisal used in techniques like allocation, direct capitalization, and sales comparison. The chapter explains converting percentages to decimals and vice versa, using the formula Part = Percentage × Whole, and a pyramid method to determine the unknown variable.

Direct capitalization is presented as a method for valuing properties based on expected income, explained by the formula Income = Rate × Value (IRV), which can be solved in three ways depending on the unknown variable. The capitalization rate and income multiplier are reciprocals.

Simple interest is covered, calculated using the formula Interest = Principal × Rate × Time. Consistent time units (years or months) are emphasized in interest calculations.

Present and future values are introduced, explaining that money today is worth more than in the future due to earning potential. "Discounting" is explained as calculating the present value of a future sum.

Basic arithmetic operations (fractions, percentages) are essential for calculating and appraising property areas. Percentages are widely used in valuation techniques. Direct capitalization is important for valuing properties based on income. Understanding the relationship between income, capitalization rate, and value is crucial. Time units must be standardized in interest calculations.

Mastering these operations enables appraisers to make accurate valuations, effectively analyze market data, make informed investment decisions, and understand financial tools used in real estate appraisal.