Traditional Mathematical Elements of Percent

Ted Mitchell

The key to understanding percentages

• Is to remember that the metric used to measure the input is the same as the metric used to measure the output

• Dollars are being transformed by a process into more or less dollars

• Pounds of weight are being transformed by a process into more or less pounds of weight

• Number of customer is being transformed into a different number of customers

• Number of transactions is being transformed into a different number of transactions

Elements of Problems using Percent, Rate, and Ratios

• 1) The Base, I• is the initial state, the size of the input or the amount of the

principal and represents the value of a 100 percent.• 2) The Output, O • is a portion of the base, the value of the final state, or the

size of the output, amount remaining from the base • 3) The Percent, % • is a decimal which represents the rate or the ratio of the

final state to the initial state, the size of the output to the size of the input, or the rate of transformation that was visited on the amount used as input.

The Base, I

• is the size of the initial state, the amount of the input or the principal and represents the value of a 100 percent.

• The size of the base is often represented by the letter I

The Output, O, • is a portion of the base, the amount of the final state, or the size

of the output • In mathematics, it is often referred to as the Percentage of the

Input or Percent of Initial state• e.g., What is the Percentage of Profit to Sales Revenue when

there is a 90% Return on Sales?• In business, it is easier to understand the question when the

term percentage is replaced with terms such as the amount of production, size of output, value of the final state or the portion of the input that is left after the conversion or transformation

• e.g., What is the Amount of the dollar Profit generated from the dollar Sales Revenue when there is a 90% Return on Sales

A Percent, %

• Is defined as a ratio of one part in every hundred

• Is a rate which is often called a “per centum”• Is a “per cent” rate or ratio in which the “cent”

means a 100.• Percent, %i, is a context free rate compared to

explicit rates such as miles per gallon, mpg, or dollars per pound,

Percents are

• Abstract because they are context free• Statements involving a 90% rate of efficiency

or 90% reduction rate have no context• Normal conversion rates sound concrete

because they have a context• Statements involving 90 Miles per gallon, 6

sales per call, 200 cups per server, or dollars per hour imply a specific output being converted from a specific input

A Percent is context or value free

• When the units of input and output are the same, i.e., dollars, pounds, hours, miles, then the conversion rate is a simple percent.

• Percent, %, is a value-free rate or context-free ratio because the units of measurement cancel each other out

• The percent symbol, %, reminds us to treat the percent as a rate and NOT as a Whole Number

Using Percent• Output= (Conversion rate or efficiency)x Input• O = (O/I) x I• O = %i x I• The symbol %i will indicate that the number

being represented is to be treated as a ratio with the Input or size of the Initial state being the denominator of the Ratio, %i = O/I

• Remember rates and ratios can be NOT treated like whole numbers

Equations that describe

• The relationship between the Base, B, the Output, O, and the Percent is a mathematical identity

• It is true by definition!• Percents are used regularly in business plans

and marketing analysis

In a business conversation• It is invariably awkward to express the Decimal representing the

rate, ratio, or fraction• For convenience it is common to convert a decimal into a

numerical percentage in order to make the number with a decimal less awkward to express in a conversation

• A decimal is converted into a numerical percentage by multiplying the decimal by 100

• ¾ = 0.75 and 0.75 x 100 = 75%• A number followed by a percentage symbol is a decimal that has

been multiplied by 100 for convenience of the English language• A percentage should always be converted back to a decimal before

doing any calculations• A percentage is converted back into a decimal by dividing the

percentage by 100• 75% ÷ 100 = 0.75

Two Difficulties with Using %

• The first difficulty working with percentages is the abstract manner in which they are presented in Middle School Math Class

• The second difficulty is the way we use percents to describe things in our day to day lives

The First difficulty

• In learning to work with inputs, percents, outputs and final states is the abstract manner in which traditional questions are asked in mathematics.What is 12% of 300?

• The question should be written in a context • What is the number that results when taking 12% of the number

300?• What is the resulting output when taking 12% of an initial input

of 300?• What is the value of the final state of a process that takes 12% of

a state’s initial value of 300?• Output, F = 12% x 300 Input

Math Problems are Abstract Questions

• 75% of 52 is what number?• Business problems should be written in the

concrete context of a Two-Factor Model• Output = 75% x an Input valued at 52• What is the final weight of the 52 pounds of

input when the 52 pounds is reduce by 75%?• The input was $52 and the conversion process

reduced the value by 75%. What was the dollar value of the output?

Math Problems are easier in a context

• What percent of 9 is 4?• Business problems should always be in a context• Output of 4 = conversion percent x Input of 9• The screening process reduced the 9 original

applicants to 4 finalists. What percent of the original candidates became finalists?

• The input was 9 pounds and the conversion process reduced the weight to 4 pounds. What was the percent of the conversion process?

Make Abstract Problems more Concrete with a Context

• 60% of what number is 12?• Business problems always have at least an implied

context to be made explicit• Output of 12 = (60% conversion) x (an Input)• The screening process of 60% reduced the number

of original applicants to 12 finalists. What was the number of the original applicants?

• A drying process reduced the original weight of the product by 60% down to 12 pounds. What was original weight of the product?

Two Difficulties with Using %

• The first difficulty is the abstract math• The second difficulty is the way we use and

think about percents on a day to day basis• We have limited number of scenarios in which

we use percentages• In what situations have you used a percentage

lately?• Price discounts, interest rates, your grade

calculations, tax rates, mortgage rates,

What is it that makes the discussion of growth and reduction rates

• So awkward and difficult?• Common usage!• 1) How often do you hear: Hey guys! I got a

great deal on this new car. My price was only 80% of the list price.

• The more common statement is: Hey guys! I got a great deal on this new car. I got 20% off the list price.

• I got a 20% discount

In Common Usage

• We don’t distinguish between a Percent, %i, and a percentage change, %∆i

• 2) How often do you hear: Hey guys! The bank grew my investment by 105%.%i = 105%

• The more common statement is: Hey guys!• My account grew by 5%. • %∆I = 5%• Or we say, “The bank has an interest rate of 5%.”

• Which is (105-100)/100 = ∆5/100 or a %5 change

To become proficient

• With percents, percentage changes, etc.• 1) You must learn to put them in a context!• 2) You must learn that there are two types of

percentagespercents, %i and percent change %∆i