UNIT 3A. Uses and Abuses of Percentages

Transcription

1 UNIT 3A Uses and Abuses of Percentages

2 PERCENTAGES Per Cent.Per 100.divided by 100 Uses symbol % 25% is read 25 per cent and means 25/100 = 0.25 P% = P/100 Examples 40% = 40/100 = % = 100/100= 1 230% = 230/100 = % is just another way of writing 1

3 WORKING WITH PERCENTAGES Convert percentage to common fraction Convert percentage to decimal 75% = 75/100 =3/4 75% = 0.75 *Move decimal two places to left because dividing by 100 (per 100)

4 WORKING WITH PERCENTAGES Convert decimal to percentage Convert common fraction to percentage 0.36 = 36% *Multiply by 100 and use % sign 2/5 = 0.4 = 40% * Divide numerator by denominator, then multiply by 100 and use % sign

5 THREE WAYS OF USING PERCENTAGES 1. As Fractions 2. To Describe change 3. For Comparisons

6 1. Percentages as Fractions Example 1 Express the first number as a percentage of the second. 220 men at a conference with 430 people %

7 2. TO DESCRIBE CHANGE How a quantity changes with time Absolute Change Describes the actual increase or decrease from a reference value to a new value Absolute change = new value reference value Relative Change A fraction that describes the size of the absolute change in comparison to the reference value: Relative change = absolute change = new value reference value reference value reference value Can be converted from a fraction to a percentage by multiplyng by 100%

8 Example 2 The population of the United States grew from 76 million in 1900 to 300 million in Find a) Absolute Change b) Relative Change (as a percent) A. Absolute change = = 224 million B. Relative Change = 224 = 295% 76

9 TIME OUT TO THINK In your own words, explain why doubling in value means a relative increase of 100% (as opposed to say 200%). Extend this thinking to explain why tripling means a 200% increase, quadrupling means a 300% increase, and so on. (Write your response in your notes)

10 3. FOR COMPARISONS Absolute Difference The actual difference between the compared value and the reference value (follows word than) Absolute difference = compared value reference value Relative Difference (can be written as a percent) Describes the size of the absolute difference as a fraction of the reference value Absolute difference = compared value reference value reference value reference value

11 Example 3 The daily circulation of USA Today is about 2.20 million. The daily circulation of the New Your Times is about 1.12 million Assume the first quantity is the compared value and the second quantity is the reference value. Find the: A. absolute difference B. Relative difference A. Absolute Difference = = 1.08 million newspapers B. Relative Difference = 1.08 / 1.12 = 96.4%

12 OF VERSUS MORE THAN 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

13 Example 4 Will is 22% taller than Wanda, so Will s height is % of Wanda s height. Will and Wanda Taller than implies more than (100 + P)% ( )% 122%

14 TIME OUT TO THINK Use the relative change formula to confirm that a population that grows from 200 to 600 increases in size by 200% (Write your response in your notes)

15 HOMEWORK P : even even Read p

16 BELL WORK P. 150: 17 23

17 PERCENTAGE POINTS VERSUS % When you see a change or difference expressed In Percentage Points Assume it is an absolute change or difference With a % sign or the word percent Assume it is a relative change or difference

18 BETWEEN 1960 AND 2010, THE PERCENTAGE OF THE U.S. POPULATION COMPOSED OF PEOPLE OVER 65 INCREASED FROM 9.2% TO 13.2%. Express each change as 1) An absolute change in terms of percentage points 2) A relative change in terms of a percentage % - 9.2% = % - 9.2% = = 43.5% 9.2% The population increased by 43.5% in relative terms or by 4 percentage points in absolute terms.

19 ASSUME THAT 30% OF CITY EMPLOYEES IN CARSON CITY RIDE THE BUS TO WORK. CONSIDER THE FOLLOWING TWO STATEMENTS: Case 1: The percentage of city employees who ride the bus to work is 10% higher in Freetown than in Carson City. Case 2: The percentage of city employees who ride the bus to work is 10 percentage points higher in Freetown than in Carson City.

20 FOR EACH CASE, STATE THE PERCENTAGE OF CITY EMPLOYEES IN FREETOWN WHO RIDE THE BUS TO WORK. Case 1 Case 2 The 10% is interpreted as a relative difference. 10% of 30% 10% x 30% is 3% 0.10 x 0.30 = 0.03 = 3% The percentage points is interpreted as the absolute difference. 30% + 10% = 40% The percentage of city employees in Freetown who ride the bus to work is 33% The percentage of city employees in Freetown who ride the bus to work is 40%

21 SOLVING PERCENTAGE PROBLEMS If the compared value is P% more than the reference value, then Compared value = (100 + P)% x reference value and Reference value = compared value (100 + P)%

22 SOLVING PERCENTAGE PROBLEMS If the compared value is P% less than the reference value, then Compared value = (100 - P)% x reference value and Reference value = compared value (100 - P)%

23 YOU PURCHASE A BICYCLE WITH A LABELED (PRE-TAX) PRICE OF $760. THE LOCAL SALES TAX RATE IS 7.6%. WHAT IS THE FINAL COST? 760 x ( )% The final cost of the bike is $ x

24 YOUR TOTAL COST FOR DINNER IN A RESTAURANT AFTER THE TIP WAS $ ASSUMING YOU LEFT A 20% TIP, WHAT WAS THE BILL FOR FOOD AND TAXES. Reference Value = ( )% The bill for food and tax before the tip was $63.64 = = $63.64

25 ABUSES OF PERCENTAGES Shifting reference Values Due to the economy you are given a 10% pay cut. Your employer promises to give you a 10% pay raise the following year. Will the pay raise restore your original salary? Less than Nothing or More than Everything Can an athlete give 110%? Can a glass of juice have 110% of the minimum daily requirement for viatmin C? Don t Average Percentages Midterm 10 questions (7/10 = 70%) Final 100 questions (90/100 = 90%) Average 80% (actual score (97/110 = 88%

26 HOMEWORK P : even Read p