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answer these questions in the Investigation 3A and use the other file for understanding
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MTH 116 Foundations of Quantitative Literacy
Section 3A Investigation Exercise: Uses and Abuses of Percentages
Please work through the following items. For questions referencing items online, please do a general
search using your laptop, phone, or tablet. For items found online, please list the publication
information, or the site’s URL. Be able to answer and discuss each of the items presented.
Terminology:
Please define the terminology using your own words.
Percent:
Absolute Change:
Relative Change:
Reference Value:
Introduction:
News reports often include quantitative statements that include or reference percents and percentages.
Consider the following statements:
•
•
•
“For newspapers, typically 15% or less of total revenues come from online operations.”
“Operating income for Coca-Cola increased 2.6% from 2009 to 2010.”
“The rate of e-cigarette usage among middle school students was up 36 percent, to 5.3
percent.”
Each of these statements quotes percentages in context to a larger story. What does each value cited
mean within the statement given? How are there usages different? Likewise, can you explain what the
third statement is indicating, given that it cites two different percentages?
In what different ways can percentages be used? Examine the question below to see if you can answer
this question, and gain some insight to the uses as indicated in the three statements above.
Question: Find three recent news reports that quote percentages. In each case, describe the use of the
percentage and explain its context. If any of the news reports are unclear, rewrite the report clarifying
the use of the percentage. If the article identifies a reference value, cite it. Remember to cite the
source of the article.
News Source 1:
Use of Percent:
News Source 2:
Use of Percent:
News Source 3:
Use of Percent:
Follow-up
Answer the following questions:
1. In what different ways can percentages be used? Provide examples you might encounter in
your daily activities.
2. How can a percentage be the same as using a fraction to describe an amount? Construct a
statement involving an item being on sale and equate the percent off to fraction of original
price.
3. How can a percentage describe change? Find the number of wins for a professional sports
team for two consecutive years and discuss the percent change in wins from one year to the
next.
4. How can percentages be used to compare numbers? Compare the US population to the world
population using figures you find online. Be certain to list the sources of your information.
Chapter 3
Numbers in
the Real
World
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Unit 3A
Uses and Abuses
of Percentages
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Three Ways of Using Percentages
◼
As fractions:
A total of 13,000 newspaper employees, 2.6% of
the newspaper work force, lost their jobs.
◼
To describe change:
Citigroup stock fell 15% last week, to $44.25.
◼
For comparisons:
The advanced battery lasts 125% longer than the
standard one, but costs 200% more.
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Example: Presidential Survey
An opinion poll finds that 35% of 1069 people
surveyed said that the President is doing a good
job. How many said the President is doing a good
job?
Solution
Because of indicates multiplication, 35% of the 1069
respondents is
35% × 1069 = 0.35 × 1069 = 374.15 ≈ 374
About 374 people said the President is doing a good
job. We rounded the answer to 374 to obtain a
whole number of people.
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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 =
new value − reference value
100%
reference value
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Example: Stock Price Rise
During a 6-month period, Lunar Industry’s stock
doubled in price from $7 to $14. What
were the absolute and relative changes in the
stock price?
absolute change = new value – reference value
= $14 – $7 = $7
new value − reference value
100%
relative change =
reference value
$14 − $7
=
100% = 100%
$7
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Example: Depreciating a Computer (1 of 2)
You bought a new laptop computer three years ago
for $1000. Today, it is worth only $300. Describe the
absolute and relative change in the laptop’s value.
Solution The reference value is the original price of
$1000, and the new value is its current worth of
$300. The absolute change in the computer’s value
is
absolute change = new value − reference value
= $300 − $1000 = −$700
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Example: Depreciating a Computer (2 of 2)
The negative sign tells us that the computer’s
current worth is $700 less than the price you paid
three years ago. The relative change is
new value − reference value
rel chg =
100%
reference value
$300 − $1000
=
100% = −70%
$1000
Again, the negative sign tells us that the laptop is
now worth 70% less than it was three years ago.
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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
=
compared value − reference value
100%
reference value
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Example: Income Comparison (1 of 4)
Recent data showed that California ranked first
among the 50 states in average income, at about
$68,900 per person, and West Virginia ranked last
at $46,600 per person.
a. How much lower is average income in West
Virginia than in California?
b. How much higher is average income in California
than in West Virginia?
Answer both questions in both absolute and relative
terms.
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Example: Income Comparison (2 of 4)
Solution
a. absolute difference = compared value – reference value
= $46,600 − $68,900
= −$22,300
compare value − reference value
rel diff =
100%
reference value
$46,600 − $68,900
=
100%
$68,900
= −32.4%
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Example: Income Comparison (3 of 4)
The negative signs tell us that average income in
West Virginia is less than that in California by
$22,300, or by about 32.4%.
b. This time West Virginia follows than, so we use
the West Virginia income as the reference value and
the California income as the compared value:
absolute difference = compared value – reference value
= $68,900 − $46,600
= $22,300
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Example: Income Comparison (4 of 4)
Relative difference:
compare value − reference value
=
100%
reference value
$68,900 − $46,600
=
100%
$46,600
−47.9%
Average income in California is more than that in
West Virginia by $22,300, or by about 47.9%.
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Of versus More Than (or Less Than)
◼
If the new or compared value is P% more than the
reference value, it is (100 + P)% of the reference
value.
◼
If the new or compared value is P% less than the
reference value, it is (100 − P)% of the reference
value.
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Example: Income Difference
Carol earns 200% more than William. What is
Carol’s income as a percentage of William’s? How
many times as large as William’s income is Carol’s?
Solution
We use the rule that P% more than means
(100 + P)% of. Because Carol’s income is 200%
more than William’s, we set P = 200. Therefore,
Carol’s income is (100 + 200)% = 300% of William’s
income. Because 300% = 3, Carol earns 3 times as
much as William.
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Example: Sale!
A store is having a “25% off” sale. How does an
item’s sale price compare to its original price?
Solution
The “25% off” means that an item’s sale price is
25% less than its original price. The sale price is
(100 – 25)% = 75% of the original price. For
example, if the original price is $100, the sale price
is $75.
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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
3% to 4%, the interest rate increased by
1 percentage point.
4% − 3%
relative change =
100% = 33%
3%
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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.
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Example: Tax Calculations
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
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Abuses of Percentages
◼
Beware of Shifting Reference Values
A 10% pay cut is followed by a 10% pay raise.
◼
Less than Nothing
Decrease caloric intake by 150% to lose weight.
◼
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.
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Example: Shifting Investment Value (1
of 2)
A stockbroker offers the following defense to angry
investors: “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!” Evaluate the stockbroker’s
defense.
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Example: Shifting Investment Value (2
of 2)
Solution Imagine that you began with an investment
of $1000. During the first year, your investment lost
60% of its value, or $600, leaving you with $400.
During the second year, your investment gained 75%
of $400, or 0.75($400) = $300. Therefore, at the end
of the second year, your investment was worth
$400 + $300 = $700, which is still less than your
original investment of $1000 and certainly not a 15%
gain overall. We can trace the problem with the
stockbroker’s defense to a shifting reference value: It
was $1000 for the first calculation and $400 for the
second.
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Example: Impossible Sale
A store advertises that it will take “150% off” the
price of all merchandise. What should happen when
you go to the checkout to buy a $500 item?
Solution
If the price were 100% off, the item would be free.
So if the price is 150% off, the store should pay you
half the item’s cost, or $250. More likely, the store
manager did not understand percentages.
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Example: Batting Average (1 of 3)
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
during the first half of the season and .400 during
the second half of the season. 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.
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Example: Batting Average (2 of 3)
Solution
No. For example, suppose the player had 300 atbats during the first half of the season and 200 atbats during the second half, for a total of 500 atbats. His first-half batting average of .200 means he
got hits in 20% of his 300 at-bats, or 0.2 ×300 = 60
hits. His second-half batting average of .400 means
he got hits in 40% of his 200 at-bats, or
0.4 × 200 = 80 hits.
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Example: Batting Average (3 of 3)
For the season, he got a total of 60 + 80 = 140 hits
in his 500 at-bats, so his season batting average
was 140/500 = 28%, or .280—not the .300 found by
averaging his first-half and second-half batting
percentages. (In fact, the only case in which his
season average would be .300 is if he had precisely
the same number of at-bats in each half of the
season.)
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