LSP 120: Quantitative Reasoning and Technological Literacy

Report
LSP 120: Quantitative Reasoning and
Technological Literacy
Topic 6: Percentages
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REVIEW!!! REMEMBER FROM TOPIC 4….
Different types of Relative Quantities
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
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 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.
by Ozlem
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• For all problems, remember toPrepared
use the
decimal
version of the %.
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VI. Number Drills
• 2 is what percentage of 10?
2/10= 1/5= 0.2 = 20%
• 20% of what number is 2?
20%= 0.2
2/0.2= 10
• What is 20% of 10?
20% = 0.2
0.2*10= 2
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• 17 is 32% of what number?
17/0.32 = 53.125
• 67.2 is what percentage of 150?
67.2/150= 0.448= 44.8%
• What is 233% of 71?
233%=2.33
2.33*71= 165.43
• What is .7% of 50?
0.7% = 0.007
0.007*50= 0.35
• 10,003 is what percentage of
1,762,325?
10003/1762325= 0.005676 =
0.5676%
• one million three hundred
thousand is what percentage of
one billion?
one million three hundred
thousand =1.3 million
one billion= 1000 million
1.3/1000= 0.0013 = 0.13%
• one thousand is what percentage
of two thousand three hundred
and six?
• 35 is 9% of what number?
0.433651= 43.37%
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35/0.09= 388.8889
VII. Applications:
• In Chicago in the year 2000, there were
approximately 1.053 million African Americans, 907
thousand whites (non-Hispanic), 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?
Total: 1.053 + 0.907 +0.754 + 0.181= 2.895 million
0.754/2.895 = 0.260449 = 26%
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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?
54% = 0.54
0.54* 21,000= 11,340
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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?
Total population = 248.7 + 19.8 = 268.5
19.8/268.5 = 0.073743 = 7.37%
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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?
8.75%= 0.0875
0.0875*15000= $1312.5
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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)
1005.94-949=56.94
56.94/949=0.06= 6%
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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?
10% = 0.1
0.1*12= 1.2 months
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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:
40 - 40∙(0.40) = 24
Determine the final price with coupon:
24 - 24∙(0.20) = 19.20
Determine the overall percentage change: (19.20-40)/40 = -0.52
– 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
First Equation
Second Equation
Final Equation
Rewrite the first equation:
With Numbers (this is considered the “long way”)
B - B∙P1 = I
(40 – 40 ∙ 0.40) = 24
I - I∙P2 = F
(24 – 24 ∙ 0.20) = 19.20
(F – B) / B
(19.20 – 40) / 40 = -0.52 or 52% savings
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, P 1 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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