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Lesson 1- Basic
Percentage Formula and Process
Basic Percentage Formula:  Percentage, Percent, Base
"Percentage is a Percent of the Base"  is the basic percentage relationship expressed in words.
In symbols, P = R·B
R is identifiable since it is a number with the percent symbol, %.
P and B are numbers which can be mis-identified by students.  However, there are "keys" to correctly identifying each:
  • The word "is" should be adjacent to P, the percentage.
  • The word "of" precedes the base, B.
  • In this course, P is the smaller number, B is the larger number.  This is not universally true, but it is true in this course.
When the P=R·B relationship is applied, or used, in a specific context, appropriate descriptive terms are used rather than percentage, percent and base. For example, Sales Tax Amount is Sales Tax Rate  times Purchase Amount.
The P=R·B  relationship is used throughout this course in all topics and often times goes unnoticed  due to the verbage being different from percentage, percent and base.
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Basic Percentage Formula:  Variation
From the basic percentage formula, P=R·B , using elementary algebra, the following variations are both evident and useful.
  •     
R equals P divided by B
  •  
 B equals P divided by R
Examples of the proper use of the basic percentage formula and variations are given in the text and on the sample exam.
Process:  Increase or Decrease
It is extremely helpful to the student to understand that in a large number of applications, percents are used to describe a process whereby the base, B, is either increased or decreased. It is helpful to adopt appropriate notation (different from the text) to accurately describe the process.  The following diagram should be helpful.

Graphical representation of Increase/Decrease Process
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Remark: Note problems 13-24 on page 57 to see the use of the words increase or decrease to describe the process. Note also in problems 25-47 that the same words increase/decrease, or synonyms, are used in most problems.  If the problem, or situation is not specific in using increase or decrease, one must ascertain the process from the context.

Also, note problems 1-12 on page 57.  The words increase/decrease are not used, but one can easily determine the case from the given information. Finally note the set-up of these problems emphasizes the process: from Base to either increased base or decreased base.

From the variation of the basic percentage formula, R equals P divided by B, it follows for the two process cases:

Rate of Increase = RI = P divided by B = Amount of Increase divided by B, since P = Amount of Increase

Rate of Decrease = RD =P divided by B = Amount of Decrease Divided by B, since P= Amount of Decrease.

Thus, the proper mathematical formulas representing the result for each process is:

BI = B + RI · B, since  RI · B = P = Amount of Increase and the increased base, BI, would be the Sum of the base plus the increase amount.

BD = B - RI · B, since RD · B = P = Amount of Decrease and the decreased base, BD, would be the difference between the base and the decrease amount.


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Process:  Accumulation Factor and Complement

From the two relationships previously developed, using factorization from elementary algebra, it follows that:

BI = B(1+RI) and BD = B(1-RD)
(Process is accomplished by multiplication by either 1+ RI or 1-RD)
1 + RI is called the accumulation factor.
1 - RD is called the complement of RD, or complement.
These formulas are not given in the text, but are extremely important and useful for the course.

 
Example  9: 
  
Find RI when 400 is increased to 500.
RI =  Amount of Increase divided by B =

 
Example 10: Find BI  when 400 is increased by 25%.
Two methods are shown using two Basic formulas.

BI = B + RI · B

 BI= B(1+R)
RI = 25% = .25
B = 400
RI times B =  (.25)(400) = 100
BI =400 + 100 = 500		 RI = 25% = .25
1+ RI = 1.25
B = 400
BI =400(1.25) = 500
Example 11: 500 represents an increase of 25% from some base B. Find B.
Process is increase.
BI =500
RI = .25% = .25
BI= B(1+RI)
Solve for B

B = 400		 (Increase process is "reversed" by dividing!)
Note: The formula is extremely important for all the concepts in the course. This is a fact that often confuses students.  The tendency is to want to subtract since a smaller number is desired.  But, one must recognize that the process is an increase - accomplished mathematically by multiplication by the accumulation factor - and must be reversed mathematically by division.
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Formula Summary

P=R· B

 R equals P divided by B

B equals P divided by R

RI =  Amount of Increase divided by B

RD =

BI = B+RI times B = B(1+RI)

BD = B-RD times B = B(1-RD)

B = 

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Copyright  ©  Jim Pack, 2000. All Rights Reserved. Last modified 11/20/07