Percentage Comparisons and a short review of percentage of problems

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Percentage Comparisons(and a short review of
percentage of problems)
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I. Percentage Of Problems A / B
Or, PART / WHOLE
Example There are 2 females in a class of 6
students. What percent of the class is female?
A/B 2/6 1/3 .3333 Or 33.33, which could
also be understood as 33 females out of 100
students
2 females
6 total students
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This problem could be posed three different ways
Note All three contain the key phrase percent
of and, in particular, the key word of.

• There are 2 females in a class of 6 students.
  What percent of the class is female?
• There are 6 students in a class. 33.33 of them
  are female. How many women are in the class?
• There are 2 women in a class. This represents
  33.33 of the class. How many students altogether
  are in the class?

Given A and B find the .
Given B and , find the part, A.
Given A and , find the whole, B.
For more explanation see Worksheet 2.
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All of these problems involved a part, A, and a
larger whole, B.
What if we asked the question one class has 10
students and a second class has 75 students. How
many times as large as the first class is the
second?
The key words how many times indicate that we
should divide 75/10 7.5 or the second class
is 7.5 times as large as the first.
Note This is the same calculation (A/B) that we
did in the part over whole problems, even though
this does not involve a part and a whole.
So, this problem suggests a different category of
percent problems
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Percentage Comparisons

• Often instead of looking a the relationship
  between a part and a whole (or a subgroup and a
  total)
• we want to compare two similar things.
• For example
• Two states (or 50 states)
• Two countries (or all the world nations)
• Two classes (or all the classes at DePaul)
• The price of two cars (or maybe the prices of all
  the top-ranked small SUVs)
• The price of gas at the leading gas station
  chains
• The population of the US in two different years
• The murder rate in 1970 versus 2005.

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Categorize the following scenarios as percentage
of (or part/whole) relationships or percentage
comparisons

• The number of cases of tuberculosis in the US and
  the U.S. population
• The number of English majors and the number of
  students at DePaul
• The number of students in the class of 1985 at
  DePaul and the number of students in the class of
  2004 at DePaul.
• The number of hybrid cars produced in 2004 and
  all the cars produced in 2004.
• The number of cars made in the U.S. in 2004 and
  the number of cars made in Germany in 2004.
• The average price of gas in Chicago and the
  average price of gas in Houston.
• The number of fatal accidents involving alcohol
  in 2004 and the number of fatal accidents in 2004.

Part/whole
Part/whole
Percentage comparison
Part/Whole
Percentage comparison
Percentage comparison
Part/whole
The Good News?
We calculate percentage comparisons almost the
same way we calculate part/whole (or percentage
of) problems
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Lets say we want to compare the number of houses
in two different small developments. Suppose
Development One only has two houses and
Development Two has 10.
We could ask Development Two has ____ times as
many houses as Development One?Answer A/B
10/2 5 times as many
The first questions just asks how many times
the first development will fit inside the second
(see animation above).
We can think about this another way
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Since this is 100 of Development One
we can think of Development Two as follows
So, Development Two is 500 of Development One,
or Development Two is 400 larger than
Development One.
This is simply A/B 1 or 10/2 -1 5 1 4 or
400.
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II. Percentage Comparison Problems

• The second type of percentage problem you will
  most often encounter is a percentage comparison
  problem. It will be phrased something like one of
  the following statements
• By what percent is Development Two larger than
  Development One?
• By what percent is the population of the U.S.
  smaller than the population of India?
• By how many percent did the price of gasoline
  rise or fall over the last two months?
• If the life expectancy of Canada is 79.1 years
  and the life expectancy of the U.S. is 77 years,
  by how many percent is Canadas life expectancy
  higher than that of the U.S.?

What do these have in common? In other words, can
you identify any key words?
Key words percentage a growth word
(increase/decrease, up/down, higher/lower,
greater/smaller,
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The formula for percentage comparison problems
is A/B 1
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Lets review the house example which
demonstrates two ways to make comparisons one
more time
We can think of
as being A/B 10/2 5 times as large as
Or
as being 10/2 1 5 1 4 or 400 larger than
We can think of
For more explanation see Worksheet 3a. We will
now do Worksheet 3b in class.