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Lesson 1 - Percents, Decimals, Fractions, Rounding, Calculator Usage
 
Objective: To develop an understanding of the basic concepts and a proficiency with the basic mathematical procedures for this course.  Additionally, the basic mathematical formulas with appropriate symbols will be introduced.
Background: All of the topics covered in this course utilize percent in some way. Percent is essential in the description and definition of all the business concepts.  The mathematical basis for all the concepts utilizes percent in an appropriate manner in the mathematical formulas. Proper mathematical procedures are essential in order to ensure consistency with the one standard in each area of business, mathematics, and academia.

Understanding what information percents provide and understanding what percents "do" is essential in both business and mathematics.
The emphasis in this material will be:
  • Appropriate mathematical methods relative to percents, decimals, rounding, answer format, calculator usage, symbols, formulas and mathematical manipulation of formulas.
  • Development of basic mathematical formulas with appropriate symbols.
  • Development of basic concepts and how mathematical formulas accurately reflect these concepts.
Reading Assignments: As you look at the text, please note there is some difference in the symbols used herein verses those used in the text. Also, some formulas and terms are developed which are not in the text. The purpose is for clarity and enhancement of the learning process.
Definitions: Terms and symbols used in this material are defined:
R = Percent, or, rate BI = Increased Base
B = Base BD = Decreased Base
P = Percentage 1 + RI = Accumulation Factor
RI =  Percent (or, rate) of Increase 1-RD = Complement of Rate of Decrease (Complement, for short)
RD = Percent (or, rate) of Decrease  
Mathematical Methods: Percents,  Decimals, Fractions, Rounding, Calculator Usage
A. The standard method in using percents is to change the percent to decimal form.  Sometimes percents are not given but are determined from given information.  The mathematics used will produce a decimal representation for the percent which then is changed to percent form. A "model" useful in the process of changing percents to decimals and decimals to percents is:

Decimal  

arrow pointing to the right under an arrow pointing to the left

Percent
(% is the symbol for percent)
(D is to the left of P; or P is to the right of D in the alphabet.) The arrow indicates the direction the decimal point should be changed.  The change encompasses two decimal places since  "percent" means per hundred (two "places").
Example 1: 7.65% = 0.0765 (Darrow pointing to the left P, Move the decimal two places to the left, drop  "%" symbol)
Example 2: 0.0854 = 8.54% (Darrow pointing to the rightP, move the decimal two places to the right, use the "%" symbol)
B. The standard in business is to express percents with two decimal places. Evidence of this is prominent in financial data from all media sources. Therefore, the standard for this course is to give percents with two decimal places.  Note that this requires decimal representations with five decimal places to accommodate  rounding.
Example 3: 0.08543 = 8.54%
Example 4: 0.08547 = 8.55%
Example 5: 0.08500 = 8.50% (explicitly); or 8.5%

 

C. On some occasions, although becoming more and more rare in actual practice, percents are given with fractions rather than decimals. The standard method is to change the fraction to decimal form and then change the percent to decimal form. However, this can be accomplished only if the decimal form for the fraction terminates.  If repeating, another approach is used.  Examples help clarify this consideration.
Example 6: Eight and one-half percent equals eight and five tenths percent equals decimal point 085 (as decimal)
one-half(fraction)= 0.50 (decimal) 

 
Example 7: seven and one eighth percent equals 7 decimal point 125 percent equals decimal point 07125 (as decimal)
one eighth (fraction) = 0.125  (decimal)

 
Example 8: eight and one third percent equals eight and bar three percent equals eighty three thousandths with a bar over the three. (the "bar" is used to indicate  that the digit(s) is (are) repeated.


The tendency for students is to terminate (stop) this repeating decimal representation at some chosen point.  Often this leads to errors and is not consistent with standard methodology.

 

Note 1:  Standard methodology on numerical data is to round answers only.  Input, or given, data and numerical values obtained in the process of arriving at answers (intermediate values) are not rounded.  Procedures used and calculator techniques should adhere to this standard. 


Note 2: The standard procedure for handling percents with fractional parts producing repeating decimal representations is to convert to fractional form. This can be accomplished by using the table on page 48, or by using the definition of percent (per 100, or, divide by 100).


Note 3: This situation is not a recurring one for this course nor in real-world considerations. There are a few problems in the course with this consideration. In the real world, the overwhelming number of occurrences is percents in decimal form rather that fractional form.


Example 8 (again): Using page 48, eight and one third percent equals one twelfth
     Also, using page 48, sixty-six and two thirds percent equals two thirds    and   sixteen and two thirds percent equals one sixth

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