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Section 3A Uses and Abuses of Percentages Reprise. Pages 133-147. 3-A. 3 Ways of Using Percentages. As fractions – “Percent of” To describe change over time For comparison. 3-A. 2. Percents are often used to describe how a quantity changes over time. Given: original value and new value.

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3 ways of using percentages

3-A

3 Ways of Using Percentages
  • As fractions – “Percent of”
  • To describe change over time
  • For comparison
2 percents are often used to describe how a quantity changes over time

3-A

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

Given: original value and new value

slide4

3-A

3. Percents are often used to compare two values.

Given:comparedvalue andreferencevalue:

slide5

3-A

The daily circulation of the Wall Street Journal is ≈2.7 million. The daily circulation of the New York Times is ≈ 1.14 million

[Find the absolute and relative difference. Assume that the first quantity is the compared value and the second is the reference value.]

slide6

3-A

The daily circulation of the Wall Street Journal is ≈2.7 million. The daily circulation of the New York Times is ≈ 1.14 million

Absolute difference = 2,700,000-1,140,000 = 1,560,000

The WSJ has 1,560,000 more readers than the NYT.

Relative difference = 1,560,000/1,140,000 = 1.37 = 137%

The WSJ has 137% more readers than the NYT.

slide7

3-A

The daily circulation of the Wall Street Journal is ≈2.7 million. The daily circulation of the New York Times is ≈ 1.14 million

Absolute difference = 1,140,000-2,700,000 = -1,560,000

The NYW has 1,560,000 fewer readers than the WSJ.

Relative difference = -1,560,000/2,700,000 = -.577 = -57.8%

The NYT has 57.8% fewer readers than the WSJ.

solving percentage problems

3-A

Solving Percentage Problems
  • You purchase a bicycle with a labeled (pre-tax) price of $699. The local sales tax rate is 7.6%. What is your final cost?

final cost

= 100% of labeled price + 7.6% of labeled price

= (100 + 7.6)%  labeled price

= 107.6%  $699 = 1.076×$699

=$752.12

solving percentage problems1

3-A

Solving Percentage Problems
  • The final cost of your new shoes is $107.69. The local sales tax rate is 6.2%. What was the labeled (pre-tax) price.

final cost

= 100% labeled price + 6.2% of labeled price

= (100 + 6.2)%  labeled price

$107.69 = 106.2%  labeled price

$107.69 / 1.062 = labeled price

= $101.40

solving percentage problems2

3-A

Solving Percentage Problems
  • Your dinner bill is $18.75. You leave $22. What percent tip did you leave?

Total bill

$22 = dinner bill + tip

tip = $22 - $18.75 = $3.25

$3.25 is what percent of 18.75?

$3.25/18.75 = .1733

= 17.33%

percentages of percentages

3-A

Percentages of Percentages
  • Interest rate increases from 3% to 4%
  • Please DON’T say “my interest rate increased by 1%”
  • Do you mean absolute interest rate? Or relative interest rate?
slide12

3-A

  • Interest rate increased from 3% to 4%
  • Absolute change= 1percentage point
  • Relative change
slide13
Example:

“The percentage of all bachelor’s degrees awarded to women rose from 44% in 1972 to 58% in 2000.”

The percentage of degrees awarded to women rose by14 percentage points.

The percentage of degrees awarded to women rose by31.8%.

abuses of percentages

3-A

Abuses of Percentages
  • Beware ofShifting Reference Values
  • Less than Nothing
  • Don’t Average Percentages
slide15
1. Shifting Reference Values:

Example:

If you accept a 10% pay cut now

And get a 10% pay raise in 6 months . . .

In six months – will you be back to your originalsalary?

slide16
Starting salary=$40,000/year

Ifyou take a 10% pay cut – your salary willbecome(100-10)% $40,000/year

=90% $40,000/year

=.9 $40,000/year

=$36,000/year

slide17
Six months later, salary = $36,000/year

You get a 10% pay raise – your salary will become(100+10)% $36,000/year

=110% $36,000/year

=1.10 $36,000/year

=$39,600/year

Which is not as much ($40,000/year) as you started with!

slide18
absolute change is -$400.
  • relative change is - 400/40000 =

-.01 = -1%.

  • Your new salary is 1% less than original.
slide19
“I admit that the value of your investments fell 60% during my first year on the job. This year, however, their value has increased by 75%, so you are now 15% ahead!”

Is the stock broker correct?

slide20
Starting investment = $10,000

First year– lost 60% (retained 40%)

40% $10,000

=.4 $10,000 =$4,000

Second year – gained 75%

(of $4,000)

175% $4,000 = 1.75  $4,000

= $7,000

slide21
absolute change is -$300.
  • relative change is -300/1000 = -.3 = -30%
  • The new value is 30% less than original.
slide22
A pair of boots was originally marked 20% off. Then they were marked down an additional 30%. The sales clerk tells you this means the boots are now 50% off the original price.

Is she correct?

suppose the boots initially cost 100
Suppose the boots initially cost $100

To take 20% off means the boots now cost (100-20)% = 80% of their original price

So, they cost 80%  $100 = .8  $100 = $80

Now take another 30% off.

So the boots will cost (100-30)% = 70% of the $80 sale price.

That is, 70%  $80

= .7  $80 = $56

slide24
Original Price = $100

Final sale price = $56

  • absolute change is -$44.
  • relative change is -44/100 = -.44 = -44%.
  • The final price is 44% less than original.

Saleslady said the boots would be 50% off (i.e. $50).

She was wrong!

Percentages don’t add!

slide25
2. Less than Nothing:

Example:

A store advertises that it will take “120% off” all red-tagged items.

You take a red-tag blouse marked $15.97 to the counter. How much should it cost you?

slide26
Less than Nothing:

120% of 15.97

= 1.2 × $15. 97

= $19.16

You should get $19.16 OFF the $15.97 price.

The store should pay you $3.19!

slide27
Less than Nothing:

Can an athlete give a 110% effort?

Can a glass of juice have 125% of the minimum daily requirement of vitamin C?

Can Mary be 100% shorter than her older sister Vivian?

Can Vivian be 110% taller than her younger sister Mary?

slide28
3. Don’t Average Percentages:

Example:

You answered 80% of the midterm questions correctly.

You answered 90% of the final exam questions correctly.

Conclusion: You answered (80%+90%)/2

= 85% of the test questions correctly.

Right?

slide29
Not so fast:

10 questions on the midterm

80% correct … 8 correct questions

30 questions on the final

90% correct … 27 correct questions

(8+27) / (10+30) = 35/40= 87.5%

30 questions on the midterm

80% correct … 24 correct questions

10 questions on the final

90% correct … 9 correct questions

(24+9) / (10+30) = 33/40= 82.5%

slide31

3-A

Homework

Pages 147-151

# 10, 11, 58, 73, 79, 82, 87, 89, 92, 94, 101, 106, 108