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Mathematics of Percent, Ratios, and Basic Rates of Change,

Mathematics of Percent, Ratios, and Basic Rates of Change, . Ted Mitchell. Learning Objectives. At the end of this lecture students should be able to 1) Define a traditional Percent in terms of base and output 2) Describe the Three elements of a Two-Factor Marketing Model

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Mathematics of Percent, Ratios, and Basic Rates of Change,

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  1. Mathematics of Percent, Ratios, andBasic Rates of Change, Ted Mitchell

  2. Learning Objectives • At the end of this lecture students should be able to • 1) Define a traditional Percent in terms of base and output • 2) Describe the Three elements of a Two-Factor Marketing Model • 3) Understand why a Rate and a Percent should not be used as a Whole Number • 4) 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 • 5) Identify the three types of problems that report rates as a percent • 6) Transform traditional “Math Questions” about percent into problems in a business context

  3. Two-Factor Models • Are the best context in which to learn that a percent is reported as a rate or ratio, that outputs are considered the final states of a process, and bases are discussed as 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 isOutput = 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 x Input • Conversion Rate is defined as the ratio of Output to Input • Conversion Rate = (Output / Input) • and is written as the rate of the Output per Input

  6. Common Outputs for Two Factor Models of Marketing • 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 • Number of Customers Called Upon = Calls per Day x Number of Days Worked

  7. Common Inputs Two Factor Models of Marketing • 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 • Number of Customer Called Upon = Calls per Day x Number of Days Worked

  8. Common Rates of Conversion • 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

  9. Rates of Conversion • Are Usuallyeasy to identify because the rate is described in terms of • Units of Output per Units of Input, Output/Input • They are often presented as ratios such as • Dollars per Pound, Dollars/Pound • Visitors per Night, Visitors/Night • Sales per Square Foot, Sales/Square Foot • GRPs per $1,000, Gross Rating Points/Budget $

  10. When Rates of Conversion • Have the same unit measures in their Input and their Outputs. then the metrics cancel each other out and the rate is value free • There is room for confusion • Dollars of Profit / Dollars of Investment • $10 Profit = $10/$200 x $200 Investment • $10 profit = 0.05 x $200 Investment • To prevent the decimal from being treated as a Whole Number we convert it to a percent • $10 Profit = 5% x $200 Investment • Some rates of percent are very common and have acquired labels to prevent confusion • A 5% Interest Rate

  11. When Rates of Conversion > 1 • Are value free there is room for confusion! • Dollars of Sales / Dollars of Advertisingis a rate of conversionSales Revenues $ = (Sales $)/(Advertising $ spent) x Advertising $ • $300 Sales = $300/$50 x $50 of Advertising • $300 profit = 6 x $50 of Advertising • To prevent the 6 fold rate from being treated as the whole number 6 we convert it to a percent • $300 Profit = 600% x $50 of Advertising • Some rates of percent are very common and have acquired labels to prevent confusion • A 600% Return on Marketing Investment

  12. Rates as Percent Reduce Confusion • Dollars of Net Profit / Dollars of Revenue • $50 Net Profit= $50/$200 x $200 in Sales • $50 Net Profit= 0.25 x $200 in Sales • To ensure the decimal 0.25 is treated as a rate of conversion we convert it to a percent • $50 Net Profit= 25% x $200 in Sales • Some rates of percent are very common and have acquired labels to prevent confusion • A 25% Return on Sales, ROS

  13. Do NOT treat a Percent, as a • Whole Number • If you gain 20 pounds to your normal weight during the holidays, then you must lose 20 pounds to regain your normal weight. • Pounds are whole numbers • If you gain 25% to your normal weight during the holidays, then you must lose 20% of your new weight to regain your normal weight. • A Percent is a Rate NOT a whole number.

  14. Whole Number Dominance Creates a Big Mistake • Your sales revenue dropped from its normal level by 20% last month. This month you expect your sales to grow enough to return your sales revenue to its normal level. By what percentage must the sales revenue increase to regain the normal sales level? • Whole number dominance leads to the wrong answer of 20%

  15. Brute Force and Intuition • Can lead to the correct answer • Your sales dropped from the normal level of $100 by 20% • Size of Sales Drop = 20% x $100 = $20 • Current sales level $100 - $20 = $80 • Increase of $20 is required to return to the normal level of $100 • Required Rate of Increase = $20/$80 = 0.25 • Required Rate of Increase = 25% of current Sales

  16. There is a General solution! • Your sales revenue dropped from its normal level by 20% last month. This month you expect your sales to grow enough to return your sales revenue to its normal level. By what percentage must the sales revenue increase to regain the normal sales level? • %∆Pn = - %∆Po / (1 + %∆Po) • %∆Pn = -(-0.20) / (1 + (-0.20) • %∆Pn = 0.2/1-0.20) = 0.2/0.8 = 0.25 = 25%

  17. Do Not Treat a Rate, a Ratio or a Percent • As a Whole Number • You have one coupon that provides a cash rebate of $10 and a second coupon that provides a cash rebate of $20. What is the total cash rebate if both coupons are used? • The answer is add $10 to $20 for a total cash rebate of $30. You can add dollars to dollars. • You have one coupon that provides a 10% discount and a second coupon that provides a 20%. What is the total percentage discount if both coupons are used? • The answer is NOT 30%. You can not add discount rates together.

  18. Do Not Treat a Rate, a Ratio or a Percent • As a Whole Number • You have one coupon, D1, that provides a 10% discount and a second coupon, D2, that provides a 20%. What is the total percentage discount if both coupons are used? • Correct Answer is 28%. You take 10% off or $10 off the original price of $100. You can take 20% off the remaining $90 or a discount of $18. The two cash discounts $10 + $18 = $28 provide • A Total discount of $28/$100 = 28%

  19. There is a General Solution! • You have one coupon, D1, that provides a 10% discount and a second coupon, D2, that provides a 20%. • What is the total percentage discount if both coupons are used? • The formal solution is Total Percent Discount = D1 + D2 + (D1 x D2) • Total % change = %∆Po + %∆P1 + (%∆Po x %∆P1) • (P2-Po)/Po = %∆Po + %∆P1 + (%∆Po x %∆P1) • Total Discount = -0.10 – 0.20 + (-0.10 x -0.20) • Total Discount = -0.30 + (0.02) = -0.28 = 28%

  20. General Solutions are Easy • To create • if you understand percents, rates and ratios

  21. Traditional Mathematical Elements of Percent

  22. 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, or the size of the output to the size of the input.

  23. 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

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

  25. 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. • Is a value free rate compared to miles per gallon, mpg, or dollars per pound, return on sales

  26. A Percent is 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 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

  27. Using Percent • Output= (Conversion rate or efficiency)x Input • O = (O/I) x I • O = %I x I • The symbol %I is used to 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

  28. 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

  29. The #1 difficulty • In learning to work with inputs, percents, outputs and final states is the abstract manner in which traditional math questions are asked. • What is 12% of 300? • The question should be written in a context • What is the number that results from taking 12% of 300? • What is the output of taking 12% of 300? • What is the value of the final state of a process that takes 12% of an initial state with a value of 300?

  30. 1) Math Problems are Abstract Questions • 70% of 52 is what number? • Business problems are always put into a Two-Factor context • Output = 70% x an Input of 52 • What is the final weight of the 52 pounds of input when the 52 pounds is reduce by 75%? • The input was 52 pounds and the conversion process reduced the weight by 75%. What was the weight of the output?

  31. 2) Math Problems are Abstract Questions • What percent of 9 is 4? • Business problems are always 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?

  32. 3) Math Problems are Abstract Questions • 60% of what number is 12? • Business problems are always in a context • 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?

  33. There are Three Types of questions • That can be asked about using simple percentages described as the efficiency or the conversion or the rate factor in a two factor model • Output = Conversion Factor x Input Factor • Type 1) Given the Rate of the Conversion Factor, %I, and the size of the Input Factor, I, determine the size of the Output, O • Type 2) Given the Rate of Conversion Factor, %I, and the Output, O, determine the size of the Input, I • Type 3) Given the size of the Output, O, and the size of the Input Factor, I, determine the Rate of Conversionas a percent %I

  34. The Three Types of question • In equation form they are • Type 1) find the output, O = %I x I • Type 2) find the input, I = O / %I • Type 3) find the rate, percent or ratio of conversion, %I= O / I

  35. Examples of Three Types of Calculations that are possible given the basic Two-Factor Model

  36. 1) Calculate Output or Final State, O • Biz-Café announces a decrease in the current price of its coffee. The current price is $4 a cup. The new price will be 90% of the current price. What is the new price? • Output, O = (Conversion rate, %I) x Input, I • New Price, O = %I x Current Price, I • New Price, O= 90% x $4 • New Price, O= 0.90 x $4 = $3.60 • The New Price, O, is $3.60

  37. 2) Calculate the Input or Initial state, I • Biz-Café reviews its last decision to reduce the selling price of its coffee. The newly determined selling price is $3.50 a cup. The new price is 70% of the old price. What was the old price, I? • Output = (conversion rate, %I) x Input • New Price = %I x Old Price, I • Old Price, I = New Price / %I • Old Price, I = $3.50/0.70 = $5.00 • The Old Price, I, was $5.00 a cup

  38. 3) Calculate the Percentage Rate or Ratio of Conversion, %I • Biz-Café is preparing to announce its new price as a percentage of its old price for a cup of coffee. The new selling price will be $4 a cup. The current selling price is $5 a cup. What percentage is the new price of the current price. • Output = (conversion rate, %I) x Input • Conversion rate from I, %I= O/I • New Price = %Ix Old Price • %I = New Price / Old Price • %I= $4 / $5 = 0.80 = 80% • The new price is 80% of the old price

  39. Can You • 1) Define a traditional Percent in terms of base and output • 2) Describe the Three Elements of a Two-Factor Marketing Model • 3) Understand why a Rate and a Percent should not be used as a Whole Number • 4) Transform traditional “Math Questions” involving percent into problems in a business context

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