Chapter 3 Review Math 1030Section A.1: Three Ways of Using PercentagesUsing percentagesWe can use percentages in three different ways:• To express a fraction of something. For example, ”A total of 10, 000 newspaper employees, 2.6% ofthe newspaper work force, lost their jobs“ uses percentage to express a fraction of total newspaperwork force.• To describe a change in something. For example, ”Cisco stock rose 5.7% last week, to $18“ uses per-centage to describe a change in stock price.• To compare two objects. For example, ”High definition television sets have 125% more resolutionthan conventional TV sets, but cost 400% more“ uses percentage to compare the resolutions and thecosts of televisions.Using Percentages as FractionsEx.1If 10% of eighth-graders smoke and there are 50, 000 eighth-graders, how many eighth-graders smoke?1Chapter 3 Review Math 1030Using Percentages to Describe ChangeAbsolute change and relative changeWe can express the change of something in two ways:• The absolute change describes the actual increase or decrease from a reference value to a new value:absolute change = new value − reference value.• The relative change is a fraction that describes the size of the absolute change in comparison to thereference value:relative change =absolute changereference value=new value − reference valuereference value.The relative change can be converted from a fraction to a percentage by multiplying by 100%. The relativechange formula leads to the following important rules:• When a quantity doubles in value, its relative change is 1 =100100= 100%.• When a quantity triples in value, its relative change is 2 = 200%.• When a quantity quadruples in value, its relative change is 3 = 300%. And so on.Note that the absolute and relative change are positive if the new value is greater than the reference valueand the absolute and relative change are negative if the new value is less than the reference value.Ex.2Suppose the population of a town was 2, 000 in 1980 and 7, 000 in 2000. Find the absolute change and therelative change.2Chapter 3 Review Math 1030Ex.3 Depreciating a Computer.You bought a computer three years ago for $1000. Today, it is worth only $300. Describe the absolute andrelative change in the computer’s value.Using Percentages for ComparisonsAbsolute change and relative differencePercentages are commonly used to compare two numbers. There are two different ways to compare twoobjects:• The absolute change is the actual difference between the compared value and the reference value:absolute difference = compared value − reference value.• The relative difference describes the size of the absolute difference as a fraction of the reference value:relative difference =absolute differencereference value=compared value − reference valuereference value.The relative difference formula gives a fraction. We can convert the answer to a percent difference by multi-plying it by 100%.The absolute and relative difference are positive if the compared value is greater than the reference valueand the absolute and relative change are negative if the compared value is less than the reference value.3Chapter 3 Review Math 1030Ex.4Suppose we want to compare the price of a $70, 000 Ferrari to the price of a $40, 000 Lexus. Describe theabsolute and relative difference.4Chapter 3 Review Math 1030Section A.2: ”Of“ versus ”More Than“”Of“ versus ”More Than“There are two different ways to state a change with percentages: ”of“ and ”more than“.In the case of ”more than“ we state the relative change. In the case we are using ”of“, we consider the ratioof the new value and the old value.• If the compared value is P % more than the reference value, it is (100 + P )% of the reference value.• If the compared value is P % less than the reference value, it is (100 − P )% of the reference value.Ex.5 Sale!A store is having 50% off sale. How does an item’s sale price compare to its original price?5Chapter 3 Review Math 1030Section A.3: Percentages of PercentagesPercentage Points versus %When you see a change or difference expressed in percentage points, you can assume it is an absolute change ordifference. If it is expressed with the % sign or the word percent, it should be a relative change or difference.Ex.6Suppose your bank increases the interest rate on your savings account from 2% to 5%. What is the absolutechange? What is the relative change?6Chapter 3 Review Math 1030Section A.4: Solving Percentage ProblemsSolving Percentage ProblemsIf the compared value is P % more than the reference value, thencompared value = (100 + P )% × reference valueandreference value =compared value(100 + P )%.If the compared value is P % less than the reference value, thencompared value = (100 − P )% × reference valueandreference value =compared value(100 − P )%.Ex.7Retail prices are 25% more than whole sale prices. If the whole sale price is $10, how much is the retailprice?Ex.8(1) You purchase a bicycle with a labeled (pre-tax) price of $760. The local sale tax rate is 7.6%. What isyour final cost (including tax)?(2) Your receipt shows that you paid $107.69 for your new shoes, tax included. The local sales tax rateis 6.2%. What was the labeled (pre-tax) price of the shoes?7Chapter 3 Review Math 1030Ex.9Consider the statement:In the past four decades the percentage of bicycle in Italy decreased from 55 per cent, to 3 per cent.What was the previous percentage of bicycle in Italy?8Chapter 3 Review Math 1030Section A.5: Abuses of PercentagesSolving Percentage ProblemsThere are few common abuses of percentages: shifting reference values, less than nothing, average of per-centages.Beware of Shifting Reference ValuesEx.10Because of losses by your employer, you agree to accept a temporary 10% pay cut. Your employerpromises to give you 10% pay raise after six months. Will the pay raise restore your original salary?9Chapter 3 Review Math 1030Less Than NothingEx.11 Impossible Sale.A store advertises that it will take 150% off the price of all merchandise. What should happen when yougo to the counter to buy a $500 item?Don’t Average PercentagesEx.12Suppose you got 70% of the questions correct on a midterm exam and 90% of the questions correct on thefinal exam. Can you conclude that you answered 80% of all the questions correctly?10Chapter 3 Review Math 1030Section B.1: Writing Large and Small NumbersLarge and small numbersWorking with large and small