Ratio and Proportion Basics

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Ratio and Proportion Basics
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Additive versus Multiplicative Comparisons
Cora makes 12 per hour and Laura makes 8 per
hour.
One could say that Cora makes 4 per hour more
than Laura, or that Laura makes 4 less than
Cora. These are additive comparisons.
One could also say that Cora makes one and one
half times what Laura makes. Or one could say
that Cora makes 50 more than Laura. These are
multiplicative comparisons.
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In upper elementary school, students transition
from making simple additive comparisons to
working with multiplicative ones. Ratios,
proportions and percents are based on
multiplicative comparisons.
four blocks higher
three times as many blocks
Multiplicative Comparison
Additive Comparison
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Ratio

• A ratio is a comparison between two quantities.

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Ratios and Fractions
Consider the ratios involved in the following
situation
The first string of the varsity water polo team
is made up of two seniors and five juniors.
This information produces the following ratios
25 52 27 57
Answer the following questions about each
ratio a. What does each number represent? b.
How would you express the ratio in words?
Include what the numbers represent. c. Does the
ratio represent a part/whole or a part/part
relationship?
For example 25 a) 2 is the number of seniors
and 5 is the number of juniors b) The ratio of
seniors to juniors is two to five. c)
part/part relationship
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The first string of the varsity water polo team
is made up of two seniors and five juniors.
Here are the rest
52 a) 5 is the number of juniors and 2 is the
number of seniors b) The ratio of juniors to
seniors is five to two. c) part/part
relationship
27 a) 2 is the number of seniors and 7 is the
total number of players b) The ratio of seniors
to players is two to seven. c) part/whole
relationship
57 a) 5 is the number of juniors and 7 is the
total number of players b) The ratio of juniors
to players is five to seven. c) part/whole
relationship
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Here is a diagram of the first string.
Two fractions can be made from this
two-sevenths of the players are seniors.
five-sevenths of the players are juniors.
Note that both fractions represent a part/whole
relationship. 2/5 would make no sense in the
context as a fraction.
Fractions always represent a part/whole
relationship ratios may or may not represent a
part/whole relationship.
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Rate
A rate is simply a ratio which involve two units.
Examples 5 for two pounds 120 miles
per 4 gallons 17.1 births
per 1,000 population
These can also be written in fraction form
Note that the unit in the numerator is different
from that in the denominator.
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Unit Rate
A unit rate is rate in which the denominator (or
the value of the second unit) is one.
Examples 2.50 per pound (that is,
2.50 per one pound) 30 miles per gallon
(It is understood that we mean one gallon.)
0.0171 birth per person (This is a bit hard to
comprehend which is why birth rates are
normally done per 1,000 people.)
These can also be written in fraction form
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Proportion
A proportion is a statement that two ratios are
equal.
5 by 7 units
7.5 by 10.5 units
10 by 14 units
The ratios of width to length are equal. The
sizes of the pictures are different, but they are
proportionate.
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How can you tell if two ratios are proportionate?
It is apparent from ones knowledge of equivalent
fractions that
Determining whether two ratios are proportionate
is like determining whether two fractions are
equivalent. A common method of doing this is to
apply what is sometimes referred to as the seeing
if the product of the means is equivalent to the
product of the extremes. In the case above, we
see that 5 x 14 7 x 10 70. This demonstrates
that the fractions are equivalent and the related
ratios are proportionate. (Note that you get a
similar result if you multiply both sides of the
equation by the common denominator, 14.)
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Using Proportions to Solve Problems
In algebra, you used proportions to solve
problems. In older arithmetic books this process
was called the Rule of Three. Here is an example
to help you remember.
The incarceration rate in Texas in 2001 was 711
per 100,000 residents. Transform this
statement to the following form One in ______
Texans were incarcerated In 2001.
Set up a proportion
Solve
711x (100,000)(1) x 140.6
One in 141 Texans were incarcerated in 2001.
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Another problem
Four-sevenths of the students in Mr. Cantus
class are boys. If there are five more boys than
girls in the class, how many boys are in Mr.
Cantus class?
One could use a rectangular model and solve the
problem with fractions as we did in MATH 1350.
However, we will use proportions. Still, there
are a number of ratios here and quite a few ways
to solve the problem with proportions.
The ratio given is four boys per seven students.
From this we can derive other ratios including,
there are three girls for every four boys. Try
to solve the problem by setting up a proportion
using this ratio.
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Four-sevenths of the students in Mr. Cantus
class are boys. If there are five more boys than
girls in the class, how many boys are in Mr.
Cantus class?
Let x be the number of girls.
Set up a proportion
Solve
3(x 5) 4x 3x 15 4x 15 x
There are fifteen girls in the class. There are
five more boys than girls, so there are twenty
boys in the class. The ratio 15 girls to 20 boys
is proportionate to the ratio 3 girls to 4 boys.
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• Proportion problems can take many forms and
  get rather complicated. Make sure that you study
  the example problems in the text. Attempt to do
  all the exercises on your own. If you are able,
  after a few attempts, to solve a problem
  yourself, it is more likely that you will have
  success solving problems on the test. If you
  remember little of this from your prior math
  classes, you may want to consult a basic math or
  introductory algebra book for more examples.

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