Section 3A Uses and Abuses of Percentages Pages

Where do we use percents?

Percents - A brief review

per cent = “per 100” = “divided by 100” 47% = 47/100 =.47 47% = 47/100 =.47 2% = 2/100 =.02 2% = 2/100 = % = 813/100 = % = 813/100 = 8.13 To divide by 100, move the decimal 2 places to the left.

To convert a percentage to a fraction: Divide by 100: Divide by 100: 45% = 45/100 = 9/20 2% = 2/100 = 1/50 2% = 2/100 = 1/50 525% = 525/100 = 21/4 525% = 525/100 = 21/4

To convert a percentage to a decimal: Divide by 100 and write as a decimal (so move the decimal 2 places to the left): Divide by 100 and write as a decimal (so move the decimal 2 places to the left): 45% = 45/100 =.45 2% = 2/100 =.02 2% = 2/100 = % = 525/100 = % = 525/100 = 5.25

To convert a decimal to a percentage: Multiply by 100 and add the % symbol (so move the decimal 2 places to the right): Multiply by 100 and add the % symbol (so move the decimal 2 places to the right):.38 = 100% .38 = 38%.075 = 100% .075= 7.5% 1.98 = 100%  1.98 = 198% 1.98 = 100%  1.98 = 198%

3 Ways of Using Percentages 1. As fractions – “Percent of” 2. To describe change over time 3. For comparison

1. Percent “of “ -> multiply In 2000, 14.4% of the citizens in Alabama lived in poverty. The population of Alabama was 642,000. How many people were living in poverty? In 2000, 14.4% of the citizens in Alabama lived in poverty. The population of Alabama was 642,000. How many people were living in poverty? “(14.4% )  (642,000)” = (.144)  (642,000) = 92,448 people

2. Percents are often used to describe how a quantity changes over time 3-A absolute change Example: A diversified portfolio grows from $1,500 to $2,250. = new value – original value = $2,250 – $1,500 = $750 Absolute Change vs. Relative Change

Absolute Change vs. Relative Change 3-A relative change Example: A diversified portfolio grows from $1,500 to $2,250. = $750 / $1,500 =.50 = 50%

During the last year, my stock doubled in price from $3500 to $7000.  Absolute change in price = $7000-$3500 = $3500 = $3500  Relative change in price =

Suppose my stock tripled in price from $3500 to $10,500.  Absolute change in price = $10,500-$3500 = $7000 = $7000  Relative change in price =

3. Percentages Used for Comparisons  Compare the value/quantity of two different items  As with change – we can compare Absolutely or  Relatively (as a percent) Pick one value to “compare to” – this is your “reference value”

Absolute and Relative Difference The absolute difference 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: 3-A

Daily circulation of the Wall Street Journal ≈ 1.77 million Daily circulation of the New York Times is ≈ 1.07 million Compare the circulation of the WSJ to the NYT. Reference value = 1.07 million (NYT) Compared value = 1.77 million (WSJ)

Reference value = 1.07 million (NYT) Absolute difference = compared-reference = 1.77 million – 1.07 million =.7 million =.7 million = 700,000 = 700,000 WSJ has about 700,000 more readers than the NYT.

Compared value = 1.77 million (WSJ) Reference value = 1.07 million (NYT) Relative difference = (absolute difference) reference value reference value = (.7 million) / 1.07 million =.654 = 65.4% The Wall Street Journal’s circulation is about 65.4% more than the New York Times’.

Now, Compare the circulation of the NYT to the WSJ. Reference value = 1.77 million (WSJ) Compared value = 1.07 million (NYT)

Reference value = 1.77 million (WSJ) Absolute difference = compared-reference = 1.07 million – 1.77 million = -.7 million = -.7 million = -700,000 = -700,000 NYT has about 700,000 fewer readers than the WSJ.

Compared value = 1.07 million (NYT) Reference value = 1.77 million (WSJ) Relative difference = (absolute difference) reference value reference value = (-.7 million) / 1.77 million = = -39.6% The New York Times’ circulation is about 39.6% less than the Wall Street Journal’s.

Solving Percentage Problems Example: You purchase a shirt with a labeled (pre-tax) price of $21. The local sales tax rate is 6%. What is your final cost? final cost final cost = 100% of labeled price + 6% of labeled price = 100% of labeled price + 6% of labeled price = ( )%  labeled price = ( )%  labeled price = 106%  $21 = 1.06  $21 = 106%  $21 = 1.06  $21 = $22.26 = $ A

Example: Your receipt shows that you paid $19.35 for a DVD, tax included. The local sales tax rate is 7.5%. What was the labeled (pre-tax) price of the DVD? final cost final cost = 100% labeled price + 7.5% of labeled price = 100% labeled price + 7.5% of labeled price = ( )%  labeled price = ( )%  labeled price $19.35 = 107.5%  labeled price $19.35 / = labeled price $19.35 / = labeled price = $18.00 = $ A

Homework for Friday: Pages # 11, 28, 30, 46, 52, 56, 62, 78, 82