7 Questions on Numeracy316 Advanced Discussion

Ted Mitchell

Innumeracy

• The following 7 questions are drawn from standardized math exams at the grade 10 level

• The questions are designed to explore the student’s ability to identify and manipulate percentages as ratios rather than whole numbers.

• People who are innumerate at the grade 10 level make errors associated with whole number dominance

Question #1

• The ratio of the profit returned on your investment, ROI, has increased from 5% last month to 10% this month. The amount of dollar profit you earned from last month to this month has _________________A) decreased B) remained the same C) increased D) not enough information to know the answer

• This is a classic question that forces us to recognize the assumptions that we all tend to make

Question #1• The ratio of the profit returned on your investment, ROI, has increased

from 5% last month to 10% this month. The amount of dollar profit you earned from last month to this month has _________________A) decreased B) remained the same C) increased D) not enough information to know the answer

• Answer: • The correct answer is D) not enough information• You need to know the amount of the investment each period to

answer the question• The popular answer is that profit increased because it common to

assume the amount of investment is constant across time.

To understand a percentage

• You need a picture of a simple two-factor model.

• Simple mechanical models in which the two things that determine profit from an investment are

• 1) the amount of the investment and • 2) the rate of return on the investment.

To understand any percentage

• $ Profit = (conversion rate at which the dollars of investment are being converted into dollars of profit) x ($ Investment)

• $ Profit = ($ Profit / $ Investment) x ($ Investment)• $ Profit = Return on Investment x ($ Investment)• If you only know one component of a two factor

model, then you can not establish the complete model.

Two Factor Models

• Output of the Machine =• Input of the Machine x Conversion efficiency

• Conversion efficiency = Output/Input

There are thousands of two factor models that we use every day

• We are far too quick to assume that one of the terms in the model is constant

• Miles per gallon: Output is distance and we tend to assume that the size of the fuel tank on the car is constant.

• A car that get better miles per gallon goes furtherand is a better more efficient car

• Miles Covered = Miles per gallon x number of gallons• Miles = (M/G) x Gallons• Miles = mpg x Gallons

Miles per Hour

• We are far too quick to assume that one of the terms in the model is constant

• Miles per hour: Output is Distance and we tend to assume that all trips are the same length of time

• A runner that has a higher speed is the better runner and will go further

• Distance = miles per hour x number of hours• Distance = mph x number of hours

Intrest Rate at The Bank

• We are far too quick to assume that one of the terms in the model is constant

• Bank interest payment: Output is interest payment and we tend to assume the size of the investment remains constant.

• I will make more at the end of the month because the bank increased my interest rate

• Payment = (payment ÷ amount invested) x (amount invested)

• Payment = % Interest rate x amount invested

In the New Shoes Game ROS shows up

• You have made a higher Return on Sales, ROS, this period compared to last period. Have you made more profit this period?

• Profit = (Profit / Sales Revenue) x Sales Revenue• Z = (Z/R) x R• Z = ROS x R• You can not answer the question without

assuming or knowing what the sales revenue, R.

In the New Shoes Game ROME shows up

• You have a higher Return on Marketing Expenses, ROME, this period compared to last period. Have you made more profit this period?

• Profit = (Profit / Marketing Expenses) x marketing expenses• Z = (Z/E) x E• Z = ROS x E• You can not answer the question without assuming

or knowing what the marketing expenses are.

In the New Shoes Game Markup %

• You have a higher % markup, Mp, this period compared to last period. Are you making more profit per sale?

• Unit Profit = (Unit Profit / Selling price) x Selling Price • Unit profit = markup on price x Selling Price• Unit Profit = ((P-V))/P) x P• Unit Profit = Mp x P• You can not answer the question without

assuming or knowing what the selling price is.

What we learn from Question #1

• 1) There are at least two factors that determine the size of an output

• a) amount of the input• b) degree of efficiency to which input is being

converted into output• 2) Do not assume you know the size of

the input • 3) Do not assume that denominator of the

efficiency ratio remains constant

Question # 2• 2) You are in charge of the firm's promotion campaign. You

have maintained a ratio of 3 to 2 in print advertising to radio advertising. What is the percentage of print advertising?A) 3/2 or 150% B) 2/3 or 66.67% C) 3/5 or 60% D) 1/3 or 33.33% E) not enough information to calculate a solution

• Answer: Advertising’s percentage of total promotion is 60 %

• The total amount of promotion is 5 units with a ratio of 3 to 2• The percentage of advertising to the total is 3/5 = 60%

Ratios and Proportions

• Are not as popular in the business lexicon as they once were

• They show up once in a while when allocation budgets are based on the power function describing the phenomena

• Q = Price-1.5 x Adv 0.6 x Sales force 0.3

• The promotion budget is allocated in the proportion of 0.6 to 0.3 (i.e. 2 to 1)

What we learn from Question #2

• That there are business problems in which • 1) we have to convert ratios and proportions

to percentages• 2) we have to convert percentages into ratio

or proportions

Question #3• You work in a call center selling products over the telephone.

When you make 20 sales a week, then you can take the rest of the time off from work. Last week to reach the quota of 20 sales you closed sales at a rate 40%. This week you closed sales at the rate of 50%. What is your average rate of closing sales for the two weeks?A) 30% B) 44.44% C) 45% D) 46.67% E) not enough information to calculate a solution

• Answer: • The close rate is 44.44%• The most popular answer is (40% + 50%)/2 = 45%

Answer to Question #3• You work in a call center selling products over the telephone.

When you make 20 sales a week, then you can take the rest of the time off from work. Last week to reach the quota of 20 sales you closed sales at a rate 40%. This week you closed sales at the rate of 50%. What is your average rate of closing sales for the two weeks?

• Answer: Closing Rate is 44.44%• You should NOT treat percentages as whole numbers• Brute force Solution: The average rate for two weeks is the total

number sales closed divided by the total number of calls made• Total calls made last week = 20/.4 = 50• Total calls made this week = 20/.5 = 40• Total of 90 calls and 40 closes = close rate of 40/90 = 44.44%

General Solution to Question #3• You work in a call center selling products over the telephone. When you

make 20 sales a week, then you can take the rest of the time off from work. Last week to reach the quota of 20 sales you closed sales at a rate 40%. This week you closed sales at the rate of 50%. What is your average rate of closing sales for the two weeks?

• Answer: Closing Rate is 44.44%• What is the General solution to this type of problem?

Last week This week 2 week total

Total calls 20/0.4 = 50 calls 20/.5= 40 calls 90 calls

Closing Rate 40% 50% 40/90 = 44.44%

Sales Completed 20 20 40 completed

General Solutions to Questions like #3

Last week This week 2 week total

Input Made 50 calls Made 40 calls Total of 90 calls

Efficiency = Output/input

40% closing rate 50% closing rate 40/90 = 44.44% closing rate

Output 20 completed calls 20 completed calls 40 completed calls

• Remember you have • 1) Input (Do Not Assume it Remains Constant)• 2) percentage of conversion efficiency ( Do Not assume

the denominator remains constant)• 3) Output• If you have any 2 of the 3 you can calculate the third

What is learned from Question #3

• 1) rates and percentages are NOT whole numbers (50% and 40% are rates they are not whole numbers like 50 tons and 40 tons)

• 2) Do Not take the simple average of rates or percentages to be the average rate (average 50 tons and 40 tons I 90/2 = 45 tons but never assume the average 50% and 40% is 45%.

Question #4

• You are a store manager and you have mailed your customers a coupon that gives your customers a 25% discount off the price. You have also put a coupon in the local newspaper that gives customers an additional 20%. What is the total discount a customer will get when they present both coupons?A) a total discount of 45% B) a total discount of 40% C) a total discount of 30%D) not enough information to calculate an answer

• Answer: • The total discount is 40% • The most popular answer is 45%

Brute Force Approach to Question #4

• You are a store manager and you have mailed your customers a coupon that gives your customers a 25% discount off the price. You have also put a coupon in the local newspaper that gives customers an additional 20%. What is the total discount a customer will get when they present both coupons?Answer: The total discount is 40%

• assume any price (e.g. $100 initial price)• Discount 1, D1 = $100 x 0.25 = $25 • Discount 2, D2 = ($100-$25) x 0.20 = $75 x 0.20 = $15• (D1+ D2)/100 = ($25+ $15)/$100 = 40/100 = 40%

General Solution to Question #4

• You are a store manager and you have mailed your customers a coupon that gives your customers a 25% discount off the price. You have also put a coupon in the local newspaper that gives customers an additional 20%. What is the total discount a customer will get when they present both coupons?Answer: C1 + C2 + (C1 x C2)– Where C1 = Rate of First Discount or negative change– Where C2 = Rate of the Second Discount

• Calculation is (-0.25 + (-0.20)) + (-0.25 x -0.20)• - 0.45 + 0.5 = -0.40 rate of change or a 40% discount

Definitions in the General Formula for Calculating Total Coupon Discounts

• Po = the original priceP1 = the price after the first change in Po

• P2 = the price after the second change • ∆Po = P1– Po = size of the first change from Po• %∆Po = (P1-Po)/Po = size of first coupon• ∆P1= P2– P1= size of the 2nd change from P1• %∆P1 = (P2-P1)/P1 = size of the 2nd coupon• %Total Change = (P2-P1)/Po = total %∆

Derivation of the Total Discount Formula = C1 + C2 + (C1 x C2)

• P1 = Po +%∆Po(Po)• P1 = Po (1+%∆Po) E1• P2 = P1 + %∆P1(P1)• P2 = P1(1 + %∆P1) E2• Substitute E1 into E2• P2 = Po (1+%∆Po) (1+%∆P1)• P2/Po = (1+%∆Po) (1+%∆P1)• P2/Po = 1 + %∆Po + %∆P1 + (%∆Po x %∆P1)• (P2/Po) – 1 = %∆Po + %∆P1 + (%∆Po x %∆P1)• (P2/Po) – (Po/Po) = %∆Po + %∆P1 + (%∆Po x %∆P1)• (P2 – Po)/Po = %∆Po + %∆P1 + (%∆Po x %∆P1)• Total % change = %∆Po + %∆P1 + (%∆Po x %∆P1)

There are two related questions

• 1) What is the total discount that the store is giving with a 20% coupon and a 25% coupon?

• Answer 40% off• 2) Hey! Marketing Manager what is the

average size of the coupons we are offering• Answer; 20% + 25% = 45/2 = 22.5%• We use percentages as labels and numbers

in special cases

What is learned from questions like #4

• Do not add percentage changes or discounts together to find the total discount or change

• Calculate total change rates or discount by applying the changes sequentially

• If you do add them up, then subtract the joint effect from the total

• In special cases the sizes of the coupon is treated as a number or as a signal to customers

• The General Solution is quicker and easier to use

Question #5

• You are a store owner and last month you got $60,000 of the total retail sales for a 30% market share. This month you got $96,000 of the total sales for a 40% market share. What is your average market share over the two months?A) less than 35% B) 35% C) more than 35% D) not enough information to calculate an answer

• Answer: • more than 35%• The most popular answer is 35%

Answering Question #5

• You are a store owner and last month you got $60,000 of the total retail sales for a 30% market share. This month you got $96,000 of the total sales for a 40% market share. What is your average market share over the two months?

Last Month This Month Total

Total Market

Share of Market 30% 40% (40 + 30)/2 = 35

Sales Revenue $60,000 $96,000

You can NOT treat percentages and ratios like

whole numbers

Answering Question #5

• You are a store owner and last month you got $60,000 of the total retail sales for a 30% market share. This month you got $96,000 of the total sales for a 40% market share. What is your average market share over the two months?

Last Month This Month Total

Total Market

Share of Market 30% 40% Average = Sales Revenue / Total Market Revenue

Sales Revenue $60,000 $96,000

You need the firms sales divided by the industry sales

Answering Question #5• You are a store owner and last month you got $60,000 of the

total retail sales for a 30% market share. This month you got $96,000 of the total sales for a 40% market share. What is your average market share over the two months?

• We remember that • Sales Revenue = Market Share x Total Market Revenue• Market Share = (Sales Revenue) / (Total Market Revenue)• Total Market Revenue = (Sales Revenue) / (Market Share)

Last Month This Month Total

Total Market 60,000/.3 = 200,000 96,000/.4=240,000

Share of Market 30% 40%

Sales Revenue $60,000 $96,000

Answering Question #5

• You are a store owner and last month you got $60,000 of the total retail sales for a 30% market share. This month you got $96,000 of the total sales for a 40% market share. What is your average market share over the two months?

Last Month This Month Total

Total Market $200,000 $240,000 $440,000

Share of Market 30% 40%

Sales Revenue $60,000 $96,000 $156,000

Monthly Market Revenue = (Sales Revenue) / (Market Share)

Answering Question #5

• You are a store owner and last month you got $60,000 of the total retail sales for a 30% market share. This month you got $96,000 of the total sales for a 40% market share. What is your average market share over the two months?

Last Month This Month Total

Total Market $200,000 $240,000 $440,000

Share of Market 30% 40%

Sales Revenue $60,000 $96,000 $156,000

Average Monthly Size of the Market = $440,000/2 = $220,000

Average Monthly Revenue = $156,000/2 = $78,000

Answering Question #5

• You are a store owner and last month you got $60,000 of the total retail sales for a 30% market share. This month you got $96,000 of the total sales for a 40% market share. What is your average market share over the two months?

Last Month This Month Total

Total Market $200,000 $240,000 $440,000

Share of Market 30% 40% 156/440 = 35.45%

Sales Revenue $60,000 $96,000 $156,000

Average monthly market share = $78,000/$220,000 = 35.45%

What is learned from Question #5

• 1) rates and percentages are NOT whole numbers (40% and 30% are rates they are not whole numbers like 40 pints and 30 pints)

• 2) Do Not take the simple average of rates or percentages to be the average rate (average 40 pints and 30 pints = 70/2 = 35 pints but never assume the average 40% and 30% is 35%.

• 3) The average output / the average input is the average conversion percentage or efficiency rate

Question #6

• You are a store manager and you reduced the original selling price on an item by 20% last week as part of a special promotion. You wish to return the price to the original selling price. What percentage increase will you need to return the price to its original amount?A) a 15% increase B) a 20% increase C) a 25% increase D) not enough information to calculate an answer

• Answer: • A 25% increase is necessary• The most popular answer is 20%

20% Another Classic Case of Whole Number Dominance

• You are a store manager and you reduced the original selling price on an item by 20% last week as part of a special promotion. You wish to return the price to the original selling price. What percentage increase will you need to return the price to its original amount?A) a 15% increase B) a 20% increase C) a 25% increase D) not enough information to calculate an answer

• Answer: A 25% increase is necessaryNot 20%

20% Another Classic Case of Whole Number Dominance

• You are a store manager and you reduced the original selling price on an item by 20% last week as part of a special promotion. You wish to return the price to the original selling price. What percentage increase will you need to return the price to its original amount?

• You can not assume a % is a whole number• You don’t subtract 20% from the price and then add back 20%• You subtract (20% x price) from the price leaving you with a

new price = 80% of original price• Solve by brute force pick any convenient price• Assume the Price was $100,you drop it by 20% to $80• You must add back $20 to return the original price but the

rate of change $20/$80 is 25% increase

A Formal Solution• You are a store manager and you reduced the original selling

price on an item by 20% last week as part of a special promotion. You wish to return the price to the original selling price. What percentage increase will you need to return the price to its original amount?

• You can not assume a percentage change or the change ratio, %∆, is a whole number

• The Formal Solution is

• %∆Nn = -%∆No/(1+%∆No)

• %∆Nn = -(-20%)/(100% +(-20%) = 0.2/(1-0.2)

• %∆Nn = 0.2/0.8 = 0.25 or 25%

Formal Definitions• You are a store manager and you reduced the original selling price, Po

on an item by %∆Po last week as part of a special promotion. You wish to return the current price, Pc, back to the original selling price, Po. What percentage increase on the current price will you need to return the current price to its original amount?

• You can not assume a percentage change or the change ratio, %∆, is a whole number

• The Definition of %∆ is∆ is the symbol for a change or a difference between two numbers

• ∆No = (Nn – No) = the change or the difference that a new number, Nn, is from the original number, No

• %∆No = the ratio of the change that a new number, Nn, is from an original number, No, based on the original number, No

• %∆No = (Nn – No)/No

Derivation of the General Formula%∆Pn = -%∆Po/100% + %∆Po)

• Po = original price• Pn = the new on-sale price• ∆Po = Pn – Po = the dollar markdown off the

original price• %∆Po = (Pn-Po)/Po = the percentage markdown• ∆Pn = Po – Pn = the dollar markup from the on-sale

price to return to the original price• %Pn = (Po-Pn)/Pn = the percentage markup on the

on-sale price

Derivation of the General Formula%∆Pn = -%∆Po/100% + %∆Po)

• Use %∆Po to mean the ratio of (Pn-Po)/Po not the percentage• Pn = Po + %∆Po(Po) = Po(1+%∆Po) E1• To change Pn back to Po• Po = Pn +%∆Pn(Pn) = Pn(1+%∆Pn) E2• Subsitute E2 into E1• Pn = Pn(1+%∆Pn) (1+%∆Po)• Pn/Pn = (1+%∆Pn) (1+%∆Po)• 1 = 1 + %∆Pn + %∆Po + (%∆Pn x %∆Po)• 0 = %∆Pn + (%∆Pn x %∆Po) + %∆Po • 0 = %∆Pn (1 + %∆Po) + %∆Po• %∆Pn (1 + %∆Po) = - %∆Po• %∆Pn = - %∆Po / (1 + %∆Po)

What is learned from question #5

• 1) Do Not Treat Ratios and Percentages as whole numbers!• You do NOT lose 20% the way you lose 20 pounds• And then you do NOT regain 20% in the way you regain 20

pounds • 2) The Denominator of a change is not constant, it depends

on the current input• 3) People, who do NOT have whole number dominance

issues, can solve business problems by brute force calculations

• 4) General Solutions are easier

Whole Number Dominance

• We wrongly treat ratios and percentages as whole numbers We try to add them and subtract them as whole numbers

• We wrongly assume that the denominator in the ratio or percentage must remain constant

• We wrongly assume that ratios and percentages are stand alone measures of size and performance

Try to Think in Terms of Simple Mechanical Models of Business

Inputs converted to Outputs

• Try to Avoid Thinking only in terms of Whole Numbers

• Relative Numbers and Rates are very important to Managers

• Sometimes rates and ratios are hard to spot because they are talked about as whole numbers

Example of a Hard Ratio to Spot

• A hard Ratio to Spot in the Biz-Café Game is the price of a cup of coffee

• 1) Marketers treat selling price like a number we send as a signal to customers

• 2) Accountants think of price as the average revenue per cup!

• Revenue = Revenue per cup x number of cups sold• R = R/Q x Q• Revenue per cup is a ratio = Revenue/(number of cups)• Accountant’s “Price” per cup is a ratio!

Average Price per Cup

• Large cup of coffee sells for a price of $2.00• Small cup of coffee sells for a price of $1.00• What is the average selling price of coffee?• $2 + $1 = $3/2 = $1.50 per cup

Question #7

• Your average price per cup was $2 last month and is $3 per cup this month. Did you make more total sales revenue this month?

• Yes• No• Not enough information• Does average price mean average revenue

per cup or the average selling price?

If it means average selling price asked from the customer

Quantity Sold

Selling price per Cup$2 $3

Higher price implies a smaller quantity sold

If it means average selling price asked from the customer

Revenue

Selling price per Cup$2 $3

Higher price does not implies anything

about the amount of revenue

If it means price per cup means average revenue per cup

• The revenue, R, in a two factor model is determined by • 1) Input: the number of cups sold, Q, and • 2) the efficiency, P, at which the sales process converts

quantity into revenue, P = R/Q where P is called the Price per cup but is the average revenue generated per cup

• Output: Revenue = (Revenue/Quantity sold) x Quantity• R = (R/Q) x Q • R = P x Q• To calculate the amount of revenue you need

both the price (ratio of R/Q) and the quantity sold, Q.

In a perfect world.

• In a perfect world, the price tag on the product is equal to the average revenue per product sold.

• In a perfect world, price “the signal” is the same as price “the average revenue per unit”

• Any Questions