Review and Examples of Percentages and Percentage

Changes

Ted Mitchell

In-Class problem

• You have one coupon that provides a 20% discount

• You have a second coupon that provides a 25% discount

• You can use both coupons at the same time on a single purchase.

• What is the total percentage discount?Your answer is __________?

Do NOT Treat Rates as Whole NumbersDiscount Percentages are Rates

• You have one coupon that provides a 20% discount

• You have a second coupon that provides a 25% discount

• You can NOT add the two numbers together!• The answer is NOT 45%

Brute Force Solution

• You have one coupon that provides a 20% discount• You have a second coupon that provides a 25%

discount• Assume a price tag of $100• First discount is 20% of $100 = $20

price after first discount = $100-$20 = $80• Second discount is 25% of $80 = $20• Total dollar discount is $20 +$20 = $40• Total Percentage discount = $40/$100 = 40%

The vast majority of Consumers

• Are sufficiently innumerate to think that you add 20% + 25% to get a total discount of 45%

• You are to be marketing managers You are NOT consumer behaviorists

• You have to choose the coupon rates!

The Need to Choose Rates

• Last week you sent out a coupon worth 20% to your customers

• This week your competitor just sent the customers in your market a coupon worth 40% off

• You want to sent out a second coupon that will match your competitor’s discount and give your customers a total discount of 40% using two coupons

• What should be the size of the discount on the second coupon?

Solution Requires You

• To understand percentages and percentage changes• The general solution for the total discount using two

discounts (D1 and D2) is

• Total Discount = D1 +D2 –(D1 x D2)• The total discount has to be 40% and D1 = 20%• 40% = 20% +D2 –(20%xD2) (get rid of the %)

• 0.4 =0.2 +D2 –.2D2

• 0.4-0.2 = (1-.2)D2

• 0.2=0.8D2

• D2 = 0.2/0.8 = .25 or make it a percent, 25% Discount

Review Percentage Rates and Percentage Rates of Change

Ted Mitchell

Examples of Three Types of Percentage Calculations

• that are possible given the basic Two-Factor Model of an Initial Price, Input = current price, P, and a Final Price Point, F

• The General Two-Factor Model is• Output, O = O/I x Input, I

The percentage conversion rate, r = O/I = %iOutput, O = %i x Input I

Consider Percentage Conversion rate

• Size of Output, O = O/I x Size of Input, IThe conversion rate, r =O/I = %i

• O = %i x I• Final Price, F = F/P x Current price, P

The conversion rate, r = F/P = %p• F = %p x P

1) Calculate Output or Final State, F

• Biz-Café announces a decrease in the current price of its coffee. The current price is P=$4 a cup. The new price will be %p = 90% of the current price. What is the new price, F?

• Output, F = (Conversion rate, %p) x Input, P• New Price, F = %p x Current Price, P• New Price, F = 90% x $4• New Price, F = 0.90 x $4 = $3.60• The New Price, F, is $3.60

2) 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 F = $4 a cup. The current selling price is P = $5 a cup. What percentage is the new price of the current price?

• Output, O = (conversion rate, %i) x Input, IConversion rate from I, %i= O/I

• New Price, F = %p x Old Price, P• %p = New Price, F / Old Price, P• %p= $4 / $5 = 0.80 = 80%• The new price is 80% of the old price• New price, F = 80% x Old Price, P

3) Calculate the Price or Initial state, I

• Biz-Café reviews last week’s decision to reduce the selling price of its coffee. The newly determined selling price is F=$3.50 a cup. The new price is 70% of the old price. What was the old price, P?

• Output, F = (conversion rate, %i) x Input, I• New Price, F = %p x Old Price, P• Old Price, P = (New Price, F) / %p• Old Price, P = $3.50/0.70 = $5.00• The Old Price, P, was $5.00 a cup

Percentage Changes in a Price

Ted Mitchell

Examples of Percentage Change Calculations

• that are possible given the basic Two-Factor Model where theSize of Change from the Input is now the Output,

• Output, Size of Change from the Input, (O–I) = ∆i• Percentage Change in size is the Conversion Rate

conversion rate = (O-I)/I = %∆iconversion rate = ∆i/I = %∆i

• The Two-Factor Model is written asOutput, ∆i = %∆i x Input I

General Model of Percentage Change Calculations

• that are possible given the basic Two-Factor Model of an Initial Price, Input = current price, P, and the amount of the change between the Current Price, P, and the Final Price Point, F

• Size of Change from the Current Price is now the Output, • Output is the Size of Change from the Input, P

Output = (Final price) – (current Price) Output = (F–P) = ∆p

• The conversion rate is Output/Input(F-P)/P = ∆p/P ∆p/P = %∆p

• As a Two-Factor Model Output, change from current price=(%∆p) x (Input, Current Price, P)Change from current price, ∆p = (∆p/P) x (Current Price, P)

• ∆p = %∆p x P

Example #1 Calculating Percentage Changes in Prices

• Biz-café gives customers a coupon that reduces the current selling price of P = $4.00 a cup to a new lower price of F =$3.20. What is the percentage size of the discount?

• Remember a coupon is a price discount or a reduction in price, ∆p a percentage discount is a negative percentage change in the price, %∆p

• %∆p = ∆p/P• %∆p = (F-P)/P• %∆p = ($3.20-$4.00)/$4.00 = $0.80/$4.00 = 20%

Example #2 Calculating Percentage Changes in Prices

• Biz-café is providing a coupon that gives customers a 20% discount on the current selling price of P = $4.00 a cup. How much money will the customer save, ∆p? By how much will the current price be reduced, ∆p?How much money will be subtracted from the current price, –∆p?

• Remember a coupon is a price discount or a reduction in pricea percentage discount is a negative percentage change in the price, %∆p = –20%

• ∆p = %∆p x P• ∆p= –0.20 x $4 = –$0.80• 80 cents off the current price

Example #3 Calculating Percentage Changes in Prices

• Biz-café is providing a coupon that gives customers a %∆p = 20% discount on the current selling price of P = $4.00 a cup. What is the amount of the final price, F, that customers actually pay?

• ∆p = %∆p x P• Remember The output is the difference from the current

price!• (F-P) = %∆p x P• (F-$4) = –0.20 x $4• (F-$4) = –$0.80• F = $4–$0.80 = $3.20

The first place we used %∆P

First place we %∆P was

• Elasticity, e • by Popular definition

Percentage change in quantity sold divided by the percentage change in the price

• e = %∆Q/%∆P• e = (∆Q/Q) / (∆P/P)• e = (Q2-Q1)/P1)) / ((P2-P1)/P1)