REVIEW!!! REMEMBER FROM TOPIC 4….
Fraction or Percent:
• Fractions or percents are used when comparing part to total of the same type
of variable. (example: percent of adults with AIDS/HIV) Percents can also
be used to show the relative change. Percent change is calculated by dividing
the absolute change by the original amount.
[Reminder: Percent change (new value –old value)/old value]
Rate:
• Rates are used compare different types of variables (example: tickets per
person, miles per hour, or crimes per 1000 people)
Ratio:
• Ratios are used to compare the same type of variable from two sources. For
example: California’s population is 33,872,000 and Oregon’s population is
3,421,000. Clearly CA’s population is larger but how many times larger?
33,872,000/3,421,000 = 9.90 Calculating the ratio of the populations tells us
that CA’s population is almost 10 times as large as OR’s population.
• The type of data you have will determine what type of relative quantity
is appropriate.
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REVIEW!!! REMEMBER FROM TOPIC 4….
Absolute and Relative Change
 We use absolute change to describe the actual increase or decrease
from a reference (or old/earlier) value to a new (or later) value:
 Absolute Change = new value – reference value
 We use relative change to compare the absolute change to the
reference value:
 Relative Change =
=
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REVIEW!!! REMEMBER FROM TOPIC 4….
• For communication purposes, we convert relative change, which
is a fraction (converted to a decimal number) to a percentage
(percentage change). The following are three ways to convert a
fraction (decimal number) to a percentage:
•
•
•
Move the decimal place to the right two places
Multiply by 100%
Use the button
in Excel
• For this course, we will generally show percentages formatted to
two (2) decimal places. (Right click on the cell, format cell)
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Percentage of…
 Understanding “Percentage of” in 3 ways:
I. The Formula:
part
%
whole
(where the % is written as a decimal)
II. Visually: Whole means the entire pie. Part means one of the shaded regions or
pieces of the pie.
III. There are three ways to think about this relationship:
part
%
whole
part
 whole
%
% * whole  part
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IV. Deriving the formulas: Can you figure out (algebraically) why all three of
these are just different versions of the same relationship?
a.
Starting with the formula:
Derive:
b.
Starting with the formula:
Derive:
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V. Solving Problems
 There are two approaches to solving the following problems. The first
approach is to identify the two given numbers. Then decide which version of
the part/whole relationship will help you answer the question.
 If you are given part and whole, then use the first version.
 If you are given part and % then use the second version.
 And, finally, if you are given whole and % then use the third version.
 The second approach is to remember the first formula, fill in the information
you are given and then solve for the missing variable.
 For all problems, remember to use the decimal version of the %.
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 2 is what percentage of 10?
 20% of what number is 2?
 What is 20% of 10?
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a) 17 is 32% of what number?
e) 35 is 9% of what number?
b) 67.2 is what percentage of 150? f) 10,003 is what percentage of
1,762,325?
c) What is 233% of 71?
d) What is .7% of 50?
g) one million three hundred
thousand is what percentage of
one billion?
h) one thousand is what
percentage of two thousand
three hundred and six?
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VII. Applications:
 In Chicago in the year 2000, there were approximately 1.053
million African Americans, 907 thousand whites (nonHispanic), and 754 thousand Hispanics, and 181 thousand
others (other races or two or more races). What percentage
of Chicagoans in 2000 were of Hispanic origin?
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
 DePaul’s undergraduate student body is approximately
21,000 students. 54% of the student body is female.
Approximate how many females attend DePaul?
Prepared by Ozlem Elgun
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 In 1993, 248.7 million people in the United States were born
in the United States, and the rest, 19.8 million were foreign
born. What percentage of the total population of the US was
foreign born?
Prepared by Ozlem Elgun
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 The sales tax is 8.75% in most counties of Illinois. If you
purchase a new car for $15,000, what is the sales tax you will
pay?
Prepared by Ozlem Elgun
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 You are in another state (not Illinois). You are buying a
computer at Best Buy. The price before taxes is $949. When
the cashier wrings up your purchase you owe $1005.94. What
is the sales tax in this state? (You might be in Connecticut or
Pennsylvania)
Prepared by Ozlem Elgun
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 At one point, the Tribune article refers to a subtotal of
murders “with only 10% of the year yet to go.” 10% of the year
is how many months?
Prepared by Ozlem Elgun
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Successive Percents
 The process:
 Goal: Our goal is to calculate the overall percentage change between the
Final Value (in this example, Final Price with Coupon) and the Beginning
Value (in this example, Retail Price) when you are given two intermediate
percentage decreases, increases or a mixture. In this example, we are
given two intermediate decreases.
 Example: Jeans are on sale for 40% off the retail price. The retail price is
$40.00. If you have a coupon, you can receive an additional 20% off the
sale price. What is the overall percentage savings?
 II. Visually:
45.00
40.00
$40.00 x .40 = $16.00
35.00
$40.00 x.52 = $20.80
30.00
25.00
$24.00 x.20 = $4.80
20.00
$40.00 - $16.00 = $24.00
15.00
$24.00 - $4.80 = $19.20
10.00
$40.00 - $20.80 = $19.20
5.00
0.00
Retail Price
Sale Price
Price
Final Price with
Coupon
Overall Percentage
Change
Amount Saved
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Example: Jeans are on sale for 40% off the retail price. The retail price is
$40.00. If you have a coupon, you can receive an additional 20% off the sale
price. What is the overall percentage savings?
 III. Mathematically:
Determine the sale price:
Determine the final price with coupon:
Determine the overall percentage change:
 which is an overall savings of 52%.
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Example: Jeans are on sale for 40% off the retail price. The retail price is
$40.00. If you have a coupon, you can receive an additional 20% off the sale
price. What is the overall percentage savings?
IV. The Formula: (1 ± P1) ∙(1 ± P2) – 1 = % (where the % is
written as a decimal)
P1 = First percentage increase/decrease
P2 = Second percentage increase/decrease
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V.
Deriving
the
Successive
Percent
Formula:
Goal: Our goal is to calculate the overall percentage change between the Final Value (F) and the Beginning Value (B) when
you are given the intermediate percentage decreases.
Deriving the Formula: The overall percentage change doesn’t depend on the Beginning Value (B). Can we show this by
determining a process (formula) that includes just the two percents?
Variables:
B = Beginning Value
I = Intermediate Value
F = Final Value
P1 = First percent decrease
P2 = Second percent decrease
Determining the process:
With Variables
With Numbers (this is considered the “long way”)
First Equation
B - B∙P1 = I
(40 – 40 ∙ 0.40) = 24
Second Equation
I - I∙P2 = F
(24 – 24 ∙ 0.20) = 19.20
Final Equation
(F – B) / B
(19.20 – 40) / 40 = -0.52 or 52% savings
Rewrite the first equation:
B - B∙P1 = I as:
B∙(1 – P1) = I
Rewrite the second equation as:
I - I∙P2 = F as:
I∙(1 - P2) = F
Using the final equation, the goal is to get the entire equation in terms of B, P1 and P2.
Substitute F with
I∙(1 - P2) to get:
Substitute I with
B∙(1 – P1) to get:
The B’s cancel out to arrive at:
(1 – P1) ∙(1 - P2) – 1
For percents that increase, substitute “+” for “-“. The final formula that works for all successive percent problems is:
Overall Percentage Change (Successive Percent) = (1 ± P1) ∙(1 ± P2) – 1
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VI. Solving Problems:
1. Situation to discuss in class: 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 of 60%.” Evaluate the
promise.
2. Solve: Spot prices for crude oil are rather
volatile. From 1998 to 1999, spot prices for crude oil
decreased by 28%. From 1999 to 2000, they increased by
106%. What was the percentage change over the two
year period from 1998 to 2000?
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Also see:
 New ways of thinking about Percentage Change.doc
 Size Comparisons Using Percentages.doc
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