Examples of R ewritting Traditional Math Questions as Two-Factor Marketing Modesl

1. Examples of Rewritting Traditional Math Questionsas Two-Factor Marketing Modesl Ted Mitchell

2. The Three Types of Problems • Are often presented as “Math problems” • 1) Find the amount that is some percentage of some numberWhat is 10% of 35? • 2) Find a percentage rate or ratioWhat percentage is 20 of 50? • 3) Find the amount for which a number represents some percentage of. • What is the number for which $20 is 60% of?

3. 1) Find the amount that is some percentage of some number For example: What is 12% of 300?

4. Rewriting Traditional Problems • As Two-Factor Models • A Traditional Question is • What is 12% of 300? • How to write it as a Two-factor Model?

5. What is 12% of 300? • The question implies that some Two-Factor Model has an input of I =300 and conversion rate of %I =12% and is asking for the amount of the output, O. • Output = %Ix Input • Output = 12% x 300 units • Output = 0.12 x 300 units • Output = 36 units • Answer: 12% of 300 is 36

6. Rewriting a Traditional Problem • As a Two-Factor Model • The Traditional Question is • What is 12% of 300? • Better written as a Two-Factor Model • There is an input of 300 and a conversion process of 12%. What is the output?

7. Using the Two-Factor Model Looks like Over-kill • Why is it better to frame the question as a two-factor model? • 1) The Two-Factor Model provides a context. The traditional question “What is 12% of 300?” has no context. It is an abstract formulation for practicing mathematical manipulation. • 2) The Two-Factor Model makes complex problems are easier to conceptualize.

8. Finding the number that is a percentage of another number • Given a percentage and the other number • What is the 90% of 12? • Nobody does numbers for the sake doing numbers! • All business problems have the mathematics in a context! • The normal selling price of the shirt is $12. You have sent your customers a coupon that will allow them to buy the shirt for a special sale price that is 90% of its normal selling price. What is the special sale price? • Output = %I x Input • Special Price = Coupon Conversion x Normal Price • Special Price = 90% x $12 = $10.80

9. What is 118% of 60? • Rewrite the question in the context of a Two-Factor model. • The output of the process will be the result of 118% conversion of the 60 units used in the input. What is the output? • Output = %I x Input • Output = 118% x 60 units • Output = 1.18 x 60 = 70.8 units

10. Rewrite of • What is 12% of 20? • Add the concept of an output • What is the output of taking12% of 20? • Provide a context • What is the dollar value of the savings generated by a 12% discount on a $20 price?

11. Any discussion • on putting traditional Type 1 percentage math questions into the context of a two-factor model? • What is 10% of 1,600? • What is the output of a process that generates 10% of a 1,600 unit input?

12. Type 2 Traditional Math Questions Finding a percentage rate or ratio What percentage is 20 of 50? What percent of 30 is 6? 15 is what percent of 60?

13. The basic percent, rate or ratio • That converts an input, I, into an output, O, is written as %Iand is simply read as the percentage change from the input, I. • %Iimplies that the output, O, is a percentage of the input, I • Output = %I x Input • Output = (Output/Input) x Input • O = (O/I) x I • The rate (O/I) is written as %I and indicated the the input is the denominator of the percentage

14. The basic Two-Factor Model • that defines rates, ratios and percentages • is an identity • Output = Factor 2 x Factor 1 • Output = (Output/Input) x Input • Output = %I x Input • O = %I x I • If you know two of the three components then you can calculate the third.

15. Basic Percentage Calculations • Usually the traditional question leaves the Two-Factor model and the percent or rate implicit • Question: 74 is what percent of 200? • Answer: 74/200 = 0.37 or 37% • However, the question implies a Two-Factor Model of • Output = %I x Input • If the output is 74 and the input is 200 what is the conversion ratio or percentage rate? • 74 = %I x 200 • %i = 74/200 = 0.37 or 37%

16. Rewriting the basic percentage question as a Two-Factor Model • What percent of 18 is 6? • In context the question is implying that the output of some conversion process is 6 units and the input is 18 units. What is the rate at which the input is being converted to output? • Output = %I x Input • %I = Output/Input • %i = 6 units/18 units = 0.3333 = 33.33% • The percentage rate of 6 to 18 is 33.33%

17. A Traditional Question Rewritten • A total of 50 students enter the examination room and 38 students left with a passing grade. What percentage passed the test? • The traditional solution is to to determine what percentage of 50 is 38. • The question should be written as a two factor with the input being 50 students wrote the test. The examination process generated an output of 38 successful students. What percentage of the 50 students who started the process successfully completed the process? • Output = %I x Input • 38 students = %I x 50 students • %i = 38 students/50 students = 38/50 = 0.76 = 76%

18. What percentage of 18 is 6? • The question implies that some Two-Factor Model has an input of 18 and an output of 6 • Output = %I x Input • 6 = %I x 18 • %I = O/I • %I = 6/18 = 0.3333 = 33.33%

19. Rewrite of • What percentage of 50 is 15? • Add the word rate to the description • What is the percentage rate of 15 to 50? • Add the context • The conversion process has transformed 50 units of input into 15 units of output. What is the rate of conversion?

20. Any questions • about rewriting traditional percentage rate question in which the problem is to find the relative percentage size of one thing relative to another? • By making the Two-Factor model explicit? • Output = %I x Input • %I= Output/Input

21. Type 3) Find the amount for which a number represents some percentage ofit What is the number for which $20 is 60% of? 30 is 10% of what number? 8% of what number is 60?

22. The traditional question leaves • Two-Factor that defines the input implicit. • 35 is 10% of what amount? • The implied Two-Factor model is that an output of 35 units has been produced from an input through a 10% conversion process. What is the amount of the input? • Output = %I x Input • Input = Output / %I • Input = 35 units / 10% = 35 units/0.10 = 350 units

23. Rewrite the Traditional Question • 8% of what number is 60? • Add the word Input • 8% of what input amount is 60? • Add the context • A conversion process produces 60 units of output using a rate of 8% of the input. How much is the input? • Output = %I x Input • Input = Output/%I = 60 units/8% • Input = 60 units/.08 = 750 units

24. A classic to rewrite • A salesman is making a commission of 24% and received his commission check of $480. How much sales revenue did the sales man have to generate to earn the $480 in commission? • Output is $480 in commissions and the commission process converts 24% of the revenue input into commissions. What is the revenue input? • Output = %I x Input • Input = Output/%I = $480/24% • Input = $480/0.24 = $2,000 in sales revenues

25. Any discussion • on the rewriting of traditional question in which the whole amount of the input is found when given the output as a partial amount of the input and the percent the partial amount of the whole that the output represents? • Output = %I x Input • Input = Output / %I

26. In summary the Three Types of Basic Percent Problems • made explicit with Two-Factor Models are • Output = Factor 2 x Factor 1 • Factor 1 = the Input • Factor 2 =the conversion rate %I • Type 1) Output = %I x Input • Type 2) %i = Output /Input • Type 3) Input = Output / %I

27. Any questions • on the use of a Two-Factor Model to define a percent as number that represents a ratio. • Decimals that represent a percent, a rate or a ratio should never be treated as whole numbers • Never use a percent, rate or ratio as a standalone measure of performance because it assumes that the base or input of the percent or rate remains constant