1. Putting the Mathematics of Percentage Rates, and Percentage Rates of Change, into the Context of a Marketing Machine
Ted Mitchell

2. Learning Objectives
At the end of this lecture students should be able to
1) Redefine the traditional mathematical context of a Percent with its base and proportion in terms of the Three Elements in a Two-Factor Marketing Model
2) Understand why a Rate and a Percent should not be used as a Whole Number
3) Understand that some rates are value-free decimals and need to be stated as a percent to ensure that the rate is not confused with a whole number
4) Transform traditional context-free “Math Questions” about percent into problems in a business context

3. Two-Factor Models
Are the best context in which to learn that a value-free rate or ratio is reported as a percent, that outputs are considered the final states of a transformation process, and bases are discussed as the inputs and initial states.

4. Simple Two-Factor Model
Is to visualize basic marketing operations as simple machines
Marketing Machines have three elements
1) Output
2) Input
3) Conversion Rate
The Marketing Machine is Output = Conversion Rate x Input

5. They are called Two-Factor Models
Because the Amount of the Output is determined by Two Factors
Factor1) The amount of Input
Factor 2) The rate or efficiency of the conversion
Output = Factor 2 x Factor 1
Output = (Conversion Rate, r) x Input
Conversion Rate is defined as the ratio of Output to Input
Conversion Rate, r = (Output / Input)
and is written as the rate of the Output per Input

6. Simple Marketing Machine
Input = one of the 4 P’s
Conversion Process Efficiency = Output/Input
Crank Handle
Output $ $ $ $ $

7. Common Concrete Rates of Conversion No Need For Percents or Percentage Changes!
Output = Conversion Rate x Input
Customer Visits = Customers per Hour x Hours Open
Quantity sold = Sales per Salesperson x Number of Salespeople
Quantity sold = Sales per Ad x Number of Ads
Sales Revenue = Price per Unit x Number of Units Sold
Customer Called Upon = Calls per Day x Number of Days Worked

8. When Rates of Conversion
Have the same unit measures in their Input and their Outputs. then the metrics cancel each other out and the conversion rate is a value free rate
There is room for confusion
$ Sales Revenue = (conversion from advertising) x $ advertising

9. Simple Marketing Machine$
Input is Advertising Dollars $ $ $
Conversion Process Efficiency = Output/Input = $R/$A =conversion percent
Crank Handle
Output $ $
Output is Dollars of Sales Revenue $ $ $

10. There is room for confusion
$ Sales Revenue = (conversion from advertising) x $ advertising
Conversion rate = Dollars of Sales Revenue / Dollars of Advertising
Observe the advertising machine $500 of Revenue = (conversion rate) x $200 of Advertising
Conversion rate = $500/$200 = 2.5
$500 of Revenue = 2.5 x $200 of Advertising
To Prevent the decimal from being treated as a Whole Number we convert it to a percent
$500 of Revenue = 250% x $200 of Advertising
Some percentage rates of return or efficiency are very common and have acquired labels to help prevent confusion
250% “Sales Revenues Returned on Advertising”
250% “Return on Advertising” remains ambiguous

11. Ambiguity Abounds Due to Two Similar Machines
1) With an Output measured as Revenue
Sales Revenue, $R = (% conversion from advertising) x $ advertising, $A
Revenue, R = (R/A) x Advertising, A
Revenue, R = %A x Advertising, A
2) With an Output measured as Profit from Advertising Advertising Profit = (Sales Revenue-Advertising) = R-A
Notation: (R-A) = ∆A means the size of the difference from $A to $R
Advertising Profit, (R-A) = ((R-A)/A) x Advertising, A
Advertising Profit, ∆A = (∆A/A) x Advertising, A
Advertising Profit = (%∆A) x Advertising, A

12. Simple Marketing Machine$
Input is Advertising Dollars, A $ $ $
Conversion Process Efficiency = Output/Input = (R–A)/A = size of conversion percent
Crank Handle
Output
Output is Dollars of Profit From Advertising
Profit = Revenue – Advertising, Profit = (R-A) $ $ $ $ $

13. Ambiguity Abounds Due to Two Similar machines
1) with a conversion rate of advertising into dollars of sales revenue
Sales Revenue, $R = (% conversion from advertising) x $ advertising, $A
Revenue, R = %A x Advertising, A
Vaguely called ‘Return on Advertising’
2) with a conversion rate of the advertising into dollars of advertising profit Advertising Profit = (Sales Revenue-Advertising) = R-A
Advertising Profit = (% of $ change from Advertising) x Advertising, A
Advertising Profit, (R–A) = ((R-A)/A) x Advertising, A
Advertising Profit, (R–A) = (%∆A) x Advertising, A
Also Vaguely called ‘Return on Advertising’

14. Confusion Yes! 1) %A is called “Return on Marketing” Should be called
Sales Revenue being Returned on Advertising
2) %∆A is called the “Return on Marketing” Should be called
Marketing Profit being Returned on Marketing

15. To Reduce Confusion
1) Always report and record value-free rates as percents
2) be as specific as you can about the context of the conversion process
Focus on the output! Ensure you have stated the output as either
The size of the output as a proportion of the input
The size of the output as the difference between the output and the input

16. Examples of Outputs that are relatively ‘concrete’ amounts of input
Output, O = (Output, O)/(Input, I) x Input, I
Output, O = %i x Input, I
Output, 3 = a percent of input x Input, 5 Output, 3 = 60% x Input, 5
Output, $3 Cost = 60% x Input, $5 Revenue
Cost is 60% of Revenue
Output, 3 returning customers = 60% x Input, 5 total customers
60% Retention rate
Output, 3 sales = 60% x Input, 5 total industry sales
60% Market Share
Output, 3 satisfied customers = 60% x Input, 5 total customers
60% satisfaction rate
Output, 3 aware customers = 60% x Input, 5 total customers
60% awareness level

17. Two Ways to Measure Output
1) The amount of output measured in concrete terms, O = dollars, customer, units sold and described as a proportion of the Input, Output, O described as %I
2) The measure the output as the size of the difference between output and input, ∆I = (O-I),
the difference in dollars, customers, units sold and described as a percent difference from the Input Output, ∆I described %∆I

18. Examples of abstract outputs that are the size of the difference from input
Output, (O-I) = ((O–I)/ I) x Input, I
Output, ∆I = (∆I/I) x Input, I
Output, ∆I = (%∆i) x Input, I
Output ∆I, 3–5 = (percent ∆ from input) x Input, 5 Output ∆I, –2 = (3-5)/5 x Input, 5
Output ∆I, –2 = -40% x Input, 5
-2 total customers = -40% x 5 total customers
Customer Loss Rate, (%∆i) = 40%
-$2 from price= -40% x $5 original price
Coupon Discount Rate, (%∆i) = 40%
-$2 from book value = -40% x $5 original book value
Depreciation Rate, (%∆i) = 40%

19. Examples of abstract outputs that are the size of the difference from input
Output, (O-I) = (O–I) / (Input, I) x Input, I
Output, ∆I = ∆I / I x Input, I
Output, ∆I = %∆i x Input, I
Output, 23–20 = (percent ∆ from input) x Input, 20 Output ∆I, 3 = (23-20)/20 x Input, 20
Output ∆I, 3= 15% x Input, 20
3 customer gain = 15% x 20 total customers
Customer gain rate, (%∆i) = 15%
$3 interest= 15% x $20 principal invested
Interest Rate, (%∆i) = 15%
$3 profit= 15% x $20 sales revenue
Return on Sales (Profit Margin), (%∆i) = 15%

20. The Two Types of Outputs and Rates
Are confusing because people in the profession know the context of their conversations
“That is a great return.”
Without context you don’t know if ‘great return’ means
the size of dollar gain, ∆I = (F-I) or
the percentage rate of the dollar gain, %∆I = (F-I)/I

21. Students like Concrete rates, customers per hour
There are no percents, %I, and no percent differences, %∆I
In marketing there are a large number of performance measure that use concrete rates
However, the strategic performance measures are invariably, value-free or ‘context -free’ rates using percents to imply a rate, %I, or a rate of difference, %∆I is being used