Uses and Abuses of Percentages Chapter 2, Unit A Uses and Abuses of Percentages

Essential Question Where do we see percentages in the real world? What must we look out for when using percentages?

Decimal, Fraction, Percent (REVIEW) (17-32) Percent to Decimal: drop the % symbol and divide by 100 Decimal to Fraction: The place value of last digit is the denominator (ex: .02 means the denominator is 100 because 2 is in the “hundredths” place value) The digits from the decimal are the numerator (ex: .02 means that the numerator is 2 because you don’t write “02”) Fraction to Decimal: Divide the numerator by the denominator Decimal to Percent: Multiply by 100 and insert the % symbol

Using Percentage to Describe Change ABSOLUTE change = new value – reference value RELATIVE change = (new value –reference value)/(reference value) If a quantity: Doubles in value, its relative change is 100% Triples in value, its relative change is 200% Quadruples in value, its relative change is 300% Halved in value, its relative change is -50%

Example #50 (49) Between the 2010 US Census and 2011, the official population of Denver increased from 600,158 to 619, 968. During the same period the population of San Antonio increased from 1,327,407 to 1, 359,758. Which city had the greater absolute change in population? Which city had the greater relative change in population?

EX (from #50) Absolute Change San Antonio = 1,359,758-1,327,407 = 32,351 people Denver = 619-968 – 600,158 = 19,810 people Relative Change San Antonio = (1,359,758 – 1,327,407)/(1,327,407)=0.024 = 2.4% Denver = (619,968 – 600,158)/(600,158) = 0.033 = 3.3%

“OF” versus “MORE THAN” (or “LESS THAN”) MORE THAN: ________ is _____% more than the original population. Relative change = (new value – ref. value)/(ref. value) Ex: A population triples in size from 200 to 600 Relative change = (600-200)/(200)=2.0=200% The new population is 200% more than the original population

“OF” versus “MORE THAN” (or “LESS THAN”) OF: The new population is ____% of the original population Ratio of new pop. to original pop. = (new pop.)/(original pop.) Ex: A population triples in size from 200 to 600 Ratio of new pop. to original pop. = (600)/(200) = 3 The new population is three times the size of the original population

OF versus MORE THAN (or LESS THAN) (61-68) If the new value is P% MORE THAN the reference value, it is (100 +P)% OF the reference value Ex (#62): The area of Norway is 24% more than the area of Colorado, so Norway’s area is _____% of Colorado’s area (100 + 24)% = 124% If the new value is P% LESS THAN the reference value, it is (100-P)% OF the reference value Ex (#64): The net worth of Zelda is 6.4% less than the net worth of Alicia, so Zelda’s net worth is ___% of Alicia’s net worth (100 – 6.4)% = 93.6%

PERCENTAGE POINTS (69-72) Percentage points = absolute change or difference % or “Percent” = relative change or difference Ex (#70): The percentage of Republicans in the House of Representatives increased from 40.9% in 2010 to 55.6% in 2012 Absolute change = 55.6% - 40.9% = 14.7 percentage points Relative change = (55.6 – 40.9)/(40.9)= 0.359 = 35.9%

Care in wording (#74) (73) The average annual precipitation on Mt. Washington, New Hampshire, is 90 in. During one particularly wet year, different news reports carried the following statements. The precipitation this year is 200% of normal OF: % = (new)/(original) = 2.00 = (new/90) = 180 in. The precipitation this year is 200% above normal MORE THAN: % = (new – ref.)/(ref.) = 2.00 = (new- 90)/(90) = 270 in.

Tax Calculations (#76) (75) The final cost of your new shoes is $107.69. The local sales tax rate is 6.2%. What was the retail (pre-tax) price? Final cost = (100 + tax)% x retail price 107.69 = (100 +6.2)% x retail price 107.69 = 1.062 x retail price Retail price = $101.40

Solving Percent Problems (#78) (77) Between 2000 and 2010, the percentage of fatal automobile accidents due to speeding decreased by 34% to 16%. What percentage of fatal automobile accidents were due to speeding in 2000? “New rate is 34 LESS THAN the previous rate” = (100 – P)% of the previous rate = (100 – 34)% = 66% = .66 OF: % = new rate/old rate 0.66 = 16/old rate Old rate = 16/0.66 = 24.2%

Shifting Reference value (#80) (79-82) True or False: You receive a pay raise of 5%, then receive a pay cut of 5%. After the two changes in pay, your salary is unchanged. Pay raise = (100 + P)% = (100 + 5)%= 105% Pay cut = (100 – P)% = (100 – 5)% = 95% 95% of 105%: % = new/old 0.95= new/105% New = 0.95 x 105% = 99.75% False. Your salary has changed, it is .25% less than two years ago

Shifting Reference Value (#82) (79-82) True or False: A high school reports that its students’ SAT scores were down by 20% for one year. The next year, however SAT scores rose by 30%. The high school principal announces, “Overall, test scores have improved by 10% over the past two years.” First year: (100-P)% = (100-20)% = 80% Second year: (100+P)% = (100+30)% = 130% 130% of 80%: % = new/old 1.30 = new/80 New = 1.30 x 80 = 104 False. The scores did not improve by 10%, but by 4%