Numbers in the Real World
This section will include practice problems.
Please include them in your notes.
Copyright © 2011 Pearson Education, Inc.
Unit 3A
Uses and Abuses
of Percentages
Copyright © 2011 Pearson Education, Inc.
Slide 3-3
3-A
Brief Review of Percentages p.129

per cent means per 100

Convert a percentage to a common fraction- replace the % with
division by 100

Convert a percentage to a decimal- drop the % a move decimal 2
places to left

Convert decimal to percentage- move the decimal 2 places to the right
and add %

Convert a common fraction to a percentage- convert fraction to
decimal, then decimal to percentage
Copyright © 2011 Pearson Education, Inc.
P% = P/100
Slide 3-4
3-A
Practice

Express the following numbers in 3 forms:
reduced fraction, decimal, and percentage

1. 2/5
2. 30%
3. .20
4. .85
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5. 150%
6. 2/3
7. 4/9
8. 1.25

9. 5/8
10. 44%
11. 69%
12. 4.25
Copyright © 2011 Pearson Education, Inc.
Slide 3-5
3-A
Three Ways of Using Percentages

As fractions:
15% of the of the 850 students in a school were
absent.

To describe change:
The price of a stock increased 75% from $50 per
share.

For comparisons:
A Mercedes costs 25% more than a Lexus.
Copyright © 2011 Pearson Education, Inc.
Slide 3-6
Class Notes (1) Example 1
Newspaper Survey
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3-A
1. A newspaper reports that 64% of 1069 people
surveyed said that the president is doing a good
job.
13. How many said the president is doing a good
job?
(p.129)
Copyright © 2011 Pearson Education, Inc.
Slide 3-7
3-A
Practice
Percentages as Fractions

28 pounds of recyclable trash in a barrel of 52
pounds.

12.0 million metric tons of beef produced annually
in the US out of 65.1 million metric tons of beef
produced annually worldwide.

The median salary for US men in 2007 was
$33,196 and the median salary for US women in
2007 was $20,922.
Copyright © 2011 Pearson Education, Inc.
Slide 3-8
3-A
Absolute and Relative Change

The absolute change describes the actual
increase or decrease from a reference value
(starting number) to a new value:
absolute change = new value – reference value

The relative change is 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
Copyright © 2011 Pearson Education, Inc.
Slide 3-9
3-A
Absolute vs. Relative Change
Example: A diversified portfolio grows from
$1,500 to $2,250.
absolute change = new value – reference value
= $2,250 – $1,500 = $750
absolute change
relative change =
reference value
= $750 / $1,500 = 0.5 = 50%
Copyright © 2011 Pearson Education, Inc.
Slide 3-10
3-A
Class Notes (2) Example 2
Stock Price Rise

During a 6 month period, Nokia’s stock doubled in
price from $10 to $20.



2. What were the absolute and relative changes
in the stock price?
(p.130)
Copyright © 2011 Pearson Education, Inc.
Slide 3-11
3-A
Practice
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Salary Comparisons
Clint’s salary increased from $20,000 to 28,000
over a three-year period. Helen’s salary
increased from $25,000 to $35,000 over the same
period.
Who’s salary increased more in absolute terms?
In relative terms? Explain.
Copyright © 2011 Pearson Education, Inc.
Slide 3-12
3-A
Class Notes (3) Example 3
World Population Growth

World population was 2.6 billion in 1950 and 6.0
billion in 2000.

3. Describe the absolute and relative change in
world population from 1950 – 2000.
(p.131)

Copyright © 2011 Pearson Education, Inc.
Slide 3-13
3-A
Practice
Percentage Change

The population of the US increased from 249
million in 1990 to 308 million in 2010.

The congressional delegation of California
increased from 30 in 1950 to 53 in 2010.

The number of daily newspapers in the US was
2226 in 1990 and 1420 in 2010.
Copyright © 2011 Pearson Education, Inc.
Slide 3-14
3-A
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:
absolute difference
relative difference 
reference value

Copyright © 2011 Pearson Education, Inc.
compared value  reference value
reference value
Slide 3-15
3-A
Class Notes (4) Example 4
Depreciating a Computer

You bought a computer three years ago for
$1000. Today it is worth only $300.

4. Describe the absolute and relative change in
the computer’s value.

(p.131)
Copyright © 2011 Pearson Education, Inc.
Slide 3-16
3-A
Practice
Percentage Change

The number of daily newspapers in the US was
2226 in 1900 and 1420 in 2010.

The number of music CDs shipped in the US
decreased from 942 million in 2000 to 511 million
in 2008.
Copyright © 2011 Pearson Education, Inc.
Slide 3-17
3-A
Class Notes (5)Example 5
Pay Comparison
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Average pay for full time wage earners varies
from state to state. Recent data (2006 data,
compiled in 2009) showed that New York ranked
first in average pay at $55,479 per person. S.
Dakota had the lowest average pay, at $30,291
per person.
5. Compare average pay in S. Dakota to that in
New York in both absolute and relative terms.
(p.133)
Copyright © 2011 Pearson Education, Inc.
Slide 3-18
3-A
Practice

The gestation period of humans (266 days) is
________% longer than the gestation period of
grizzly bears (220 days).

The 2009 life expectancy in Canada (81.2 years)
is __________% greater than the 2009 life
expectancy in Russian (65.9 years).
Copyright © 2011 Pearson Education, Inc.
Slide 3-19
3-A
Of versus More Than (or Less 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.
Copyright © 2011 Pearson Education, Inc.
Slide 3-20
Class Notes (6) Example 6
Salary Difference
3-A

Carol earns 50% more than William.

6. How many times larger is her income than his?
(p.134)
Copyright © 2011 Pearson Education, Inc.
Slide 3-21
3-A
Practice


Will is 22% taller than Wanda, so Will’s height is
_____% of Wanda’s height.
The area of Norway is 24% more than the area of
Colorado, so Norway’s area is ______% of
Colorado’s area.
Copyright © 2011 Pearson Education, Inc.
Slide 3-22
3-A
Class Notes (7) Example 7
Prices and Sales Sale!
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A store is having a 25% off sale.
32. How does an item’s sale price compare to its
original price?
(p.134)
Copyright © 2011 Pearson Education, Inc.
Slide 3-23
3-A
Practice

The wholesale price of a TV is 40% less than the
retail price. Therefore, the wholesale price is
_______ times the retail price.

A store is having a 50% off sale. Therefore, the
original price of an item is _____ times as much
as the sale price.
Copyright © 2011 Pearson Education, Inc.
Slide 3-24
3-A
Percentages of Percentages
When a change or difference is expressed
 in percentage points, assume it is an absolute
change or difference.
 with the % sign or the word percent, it is a relative
change or difference.
Example: If a bank increases its interest rate from
4% to 5%, the interest rate increased by
1 percentage point.
5%  4%
relative change 
 0.25  25%
4%
Copyright © 2011 Pearson Education, Inc.
Slide 3-25
Class Notes (8) Example 8
Newspaper Readership Declines
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3-A
The percentage of adults who report reading a daily
newspaper fell from about 78% in 1980 to 49% in 2009.


8. Describe this change in newspaper readership.
(p.135)
Copyright © 2011 Pearson Education, Inc.
Slide 3-26
3-A
Practice


The annual interest rate for Jack’s savings
account increased from 2.3% to 2.8%
The percentage of Republicans in the House of
Representatives decreased from 53.3% in 2007 to
46.4% in 2009.
Copyright © 2011 Pearson Education, Inc.
Slide 3-27
Class Notes (9a-b)
Care in Wording
3-A

9. Assume that 40% of the registered voters in Carson
City are Republicans. Read the following questions
carefully, and give the most appropriate answers.

The percentage of voters registered as Republicans is
25% higher in Freetown than in Carson City.
a. What percentage of the registered voters in Freetown
are Republicans?


The percentage of voters registered as Republicans is 25
percentage points higher in Freetown than in Carson City.
b. What percentage of the registered voters in Freetown
are Republicans?
Copyright © 2011 Pearson Education, Inc.
Slide 3-28
3-A
Solving Percentage Problems

If the compared value is P% more than the
reference value, then
compared value  (100  P)%  reference value
and

compared value
reference value 
(100  P)%
If the compared value is less than the reference
value, use (100 – P) instead of (100 + P) in the
above calculations.
Copyright © 2011 Pearson Education, Inc.
Slide 3-29
3-A
Practice
Solving Percentage Problems
You purchase a shirt with a labeled (pre-tax) price of
$21. The local sales tax rate is 6%. What is your
final cost (including tax)?
final cost = labeled price + (6% of labeled price)
= (100 + 6)%  labeled price
= 106%  $21 = 1.06  $21 = $22.26
Copyright © 2011 Pearson Education, Inc.
Slide 3-30
Class Notes (10a-b) Example 10
Tax Calculations
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3-A
You purchase a shirt with a labeled (pre-tax) price
of $17. The local sales tax rate is 5%.
a. What is your final cost (including tax)?
Your receipt shows that you paid $19.26 for a Bluray disc, tax included.
b. The local sales tax rate is 7%. What was the
labeled (pre-tax) price of the disc?
Slide 3-31
Practice
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3-A
You purchase a bicycle with a retail (pre-tax) price of $760.
The local sales tax rate is 7.6%
What is the final cost?
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?
The 2410 women undergraduates at the college comprise
54% of all undergraduates.
How many undergraduates attend the college?
3-A
Class Notes (11) Example 11
Up 44%, to 10.4%
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Consider the following statement from the
introduction to this unit:
The rate of smoking for eighth graders is up 44
percent, to 10.4 percent.
What was the previous smoking rate for eighth
graders.
Copyright © 2011 Pearson Education, Inc.
Slide 3-33
3-A
Abuses of Percentages

Beware of Shifting Reference Values
A 10% pay cut is followed by a 10% pay raise.
wow!@!
Copyright © 2011 Pearson Education, Inc.
Slide 3-34
3-A
Class Notes (12) Example 12
Shifting Investment Value


A stockbroker offers the following defense to
angry investors: “I admit that the value of your
investments fell 60% of its value during my first
year on the job. This year, however, their value
has increased by 75%, so you are now 15%
ahead.
12. Evaluate the stockbroker’s defense.
Copyright © 2011 Pearson Education, Inc.
Slide 3-35
3-A
Class Notes (13) Example 13
Tax Cuts


A politician promises, “if elected, I will cut your
taxes by 20%, for each of the first three years of
my term, for a total cut of 60%.
13. Evaluate the promise
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Slide 3-36
3-A
Another wow!@!

Less than Nothing
Decrease caloric intake by 150% to lose weight.
We often see numbers that represent large “more
than” percentages. However in most cases it is
not possible to have a “less than” percentage that
is greater than 100%.
Copyright © 2011 Pearson Education, Inc.
Slide 3-37
3-A
Class Notes (14) Example 14
Impossible Sale


A store advertises that it will take “150% off” the
price of all merchandise.
14. What should happen when you go to the
counter to buy a $500 item?
Copyright © 2011 Pearson Education, Inc.
Slide 3-38
3-A
Don’t Average Percentages
If 70% of the boys and 60% of the girls in a class
voted to go to a water park, then 65% of the
students in the class voted to go to the water
park.
It is tempting to say yes, but it would be wrong.
Unless both happened to have the same number
of kids.
Copyright © 2011 Pearson Education, Inc.
Slide 3-39
Class Notes (15) Example 15
Batting Average
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3-A
In baseball, a player’s batting average represents the
percentage of at-bats in which he got a hit. For example,
a batting average of .350 means the player got a hit 35%
of the times he batted.
Suppose a player had a batting average of .200 and .400.
Can we conclude that his batting average for the entire
season was .300 (the average of .200 and .400)?
Why or why not?
Give an example that illustrates your reasoning.
Copyright © 2011 Pearson Education, Inc.
Slide 3-40
3A Homework
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3-A
Class Notes
Quick Quiz p. 140:1-10
Exercises p.141 1-16
1 world
 108. Percentages
 109. Percentage Change
 110. Abuse of Percentages
Copyright © 2011 Pearson Education, Inc.
Slide 3-41