section 3a uses and abuses of percentages

Section 3ASection 3AUses and Abuses of Uses and Abuses of

PercentagesPercentages

Pages 133-147Pages 133-147

Where do we use percents?Where do we use percents?

Percents - Percents - A brief reviewA brief review

per centper cent = “per 100” = “per 100” = “divided by 100” = “divided by 100”

47% = 47/100 = .4747% = 47/100 = .47

2% = 2/100 = .022% = 2/100 = .02

813% = 813/100 = 8.13813% = 813/100 = 8.13

To To dividedivide by 100, move the decimal 2 places by 100, move the decimal 2 places to the to the leftleft..

To convert a percentage to a To convert a percentage to a fraction:fraction:

Divide by 100:Divide by 100:

45% = 45% = 45/10045/100 = 9/20 = 9/20

2% = 2% = 2/1002/100 = 1/50 = 1/50

525% = 525% = 525/100525/100 = 21/4 = 21/4

To convert a percentage to a To convert a percentage to a decimal:decimal:

Divide by 100 and Divide by 100 and write as a decimalwrite as a decimal (so (so move the decimal 2 places to the left):move the decimal 2 places to the left):

45% = 45% = 45/100 = 45/100 = .45.45

2% = 2/1002% = 2/100 = = .02.02

525% = 525% = 525/100525/100 = = 5.255.25

To convert a decimal to a To convert a decimal to a percentage:percentage:

MultiplyMultiply by 100 and add the % symbol (so by 100 and add the % symbol (so move the decimal 2 places to the move the decimal 2 places to the rightright):):

.38 = 100% .38 = 100% .38 = .38 = 38%38%

.075 = 100% .075 = 100% .075 .075= = 7.5%7.5%

1.98 = 100% 1.98 = 100% 1.98 = 1.98 = 198%198%

3 Ways of Using Percentages3 Ways of Using Percentages

1.1. As fractions – “Percent of”As fractions – “Percent of”

2.2. To describe change over timeTo describe change over time

3.3. For comparisonFor comparison

1. Percent “of “ -> multiply1. Percent “of “ -> multiply

In 2000,In 2000, 14.4% of the citizens in Alabama 14.4% of the citizens in Alabama lived in poverty. The population of lived in poverty. The population of Alabama was 642,000. How many people Alabama was 642,000. How many people were living in poverty?were living in poverty?

““(14.4% ) (14.4% ) (642,000)” (642,000)”= = (.144) (.144) (642,000) (642,000) = 92,448 people= 92,448 people

2. Percents are often used to describe 2. Percents are often used to describe how a quantity how a quantity changes over timechanges over time

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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 ChangeAbsolute Change vs. Relative Change

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relative change

Example: A diversified portfolio grows from $1,500 to $2,250.

absolute change= original value

new value original value= original value

= $750 / $1,500 = .50 = 50%

During the last year, my stock During the last year, my stock doubleddoubled in price from $3500 to $7000. in price from $3500 to $7000.

Absolute changeAbsolute change in price = in price = $7000-$3500 $7000-$3500

= $3500= $3500

Relative changeRelative change in price = in price =

$3500 1 100%$3500

Suppose my stock Suppose my stock tripledtripled in price in price from $3500 to $10,500. from $3500 to $10,500.

Absolute changeAbsolute change in price = in price = $10,500-$3500 $10,500-$3500

= $7000= $7000

Relative changeRelative change in price = in price =

$7000 2 200%$3500

3. Percentages Used for 3. Percentages Used for ComparisonsComparisons

Compare the value/quantity of two Compare the value/quantity of two different itemsdifferent items

As with change – we can compare As with change – we can compare AbsolutelyAbsolutely or or

RelativelyRelatively (as a percent) (as a percent)

Pick one value to “compare to” – this is your Pick one value to “compare to” – this is your ““reference value”reference value”

Absolute and Relative DifferenceAbsolute and Relative Difference

The The absolute differenceabsolute difference is the actual difference between is the actual difference between the compared value and the reference value:the compared value and the reference value:absolute difference = compared value absolute difference = compared value ‘reference’ value ‘reference’ value

The The relative differencerelative difference describes the size of the absolute describes the size of the absolute difference as a difference as a fractionfraction of the reference value: of the reference value:

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absolute differencerelative difference =

reference value

compared value reference value

reference value

Daily circulation of the Daily circulation of the Wall Street Journal Wall Street Journal ≈≈ 1.77 million1.77 million

Daily circulation of the Daily circulation of the New York TimesNew York Times is ≈ is ≈ 1.07 million1.07 million

Compare the circulation of the WSJ to the Compare the circulation of the WSJ to the NYT.NYT.

Reference value = 1.07 millionReference value = 1.07 million (NYT) (NYT)

Compared valueCompared value = = 1.77 million1.77 million (WSJ) (WSJ)

Compared valueCompared value = = 1.77 million1.77 million (WSJ) (WSJ)

Reference value = 1.07 millionReference value = 1.07 million (NYT) (NYT)

Absolute differenceAbsolute difference = compared- = compared-referencereference

= 1.77 million – 1.07 million= 1.77 million – 1.07 million

= .7 million= .7 million

= = 700,000700,000

WSJWSJ has about 700,000 has about 700,000 moremore readers than readers than the the NYT.NYT.

Compared valueCompared value = = 1.77 million1.77 million (WSJ) (WSJ)Reference value = 1.07 millionReference value = 1.07 million (NYT) (NYT)

Relative differenceRelative difference = = ((absolute differenceabsolute difference)) reference valuereference value= = (.7 million) / 1.07 million(.7 million) / 1.07 million= .654= .654= = 65.4%65.4%The The Wall Street Journal’sWall Street Journal’s circulation is about circulation is about

65.4% 65.4% more thanmore than the the New York Times’.New York Times’.

Now,Now,

Compare the circulation of the NYT to the Compare the circulation of the NYT to the WSJ.WSJ.

Reference value = 1.77 millionReference value = 1.77 million (WSJ) (WSJ)

Compared valueCompared value = = 1.07 million1.07 million (NYT) (NYT)

Compared valueCompared value = = 1.07 million1.07 million (NYT) (NYT)

Reference value = 1.77 millionReference value = 1.77 million (WSJ) (WSJ)

Absolute differenceAbsolute difference = compared- = compared-referencereference

= 1.07 million – 1.77 million= 1.07 million – 1.77 million

= -.7 million= -.7 million

= = -700,000-700,000

NYTNYT has about 700,000 has about 700,000 fewerfewer readers than readers than the the WSJWSJ..

Compared valueCompared value = = 1.07 million1.07 million (NYT) (NYT)Reference value = 1.77 millionReference value = 1.77 million (WSJ) (WSJ)

Relative differenceRelative difference = = ((absolute differenceabsolute difference)) reference valuereference value= (-.7 million) / 1.77 million= (-.7 million) / 1.77 million= -.396= -.396= = -39.6%-39.6%The The New York Times’New York Times’ circulation is about circulation is about

39.6% 39.6% less thanless than the the Wall Street Journal’s.Wall Street Journal’s.

Solving Percentage ProblemsSolving Percentage ProblemsExample: You purchase a shirt with a labeled Example: You purchase a shirt with a labeled

(pre-tax) price of $21. The local sales tax rate is (pre-tax) price of $21. The local sales tax rate is 6%. What is your final cost?6%. What is your final cost?

final cost final cost

= 100% of labeled price + 6% = 100% of labeled price + 6% ofof labeled price labeled price

= (100 + 6)% = (100 + 6)% labeled price labeled price = 106% = 106% $21 = 1.06 $21 = 1.06 $21 $21

== $22.26$22.26

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ExampleExample: Your receipt shows that you paid : Your receipt shows that you paid $19.35 for a DVD, tax included. The local $19.35 for a DVD, tax included. The local sales tax rate is 7.5%. What was the labeled sales tax rate is 7.5%. What was the labeled (pre-tax) price of the DVD?(pre-tax) price of the DVD?

final cost final cost = 100% labeled price + 7.5% = 100% labeled price + 7.5% ofof labeled price labeled price

= (100 + 7.5)% = (100 + 7.5)% labeled price labeled price$19.35 = 107.5% $19.35 = 107.5% labeled price labeled price

$19.35 / 1.075$19.35 / 1.075 = labeled price = labeled price = $18.00= $18.00

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Homework for Friday:Homework for Friday:

Pages 147-150Pages 147-150

# 11, 28, 30, 46, 52, 56, 62, 78, 82# 11, 28, 30, 46, 52, 56, 62, 78, 82