Greenwood press

84
PROPORTIONS
It is difficult to track animals due to
their mobility and ability to hide well. A
capture–recapture method can be used
to tag and predict their population
without having to count them all.
A scaled model of a tyrannosaurus can
be used to predict its actual mass with
information about its actual length.

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In baseball, an earned run average (ERA) is a statistic that describes how
many runs a pitcher would be expected to give up in a nine-inning game. How-
ever, a pitcher hardly lasts that long during a single game. On a good day, a major
league pitcher will play for about six to nine innings. However, if the pitcher is
giving up a lot of runs he will play considerably less, perhaps from one to five
innings. Regardless of the length of the pitching performance, an ERA is used to
compare the different pitchers. Typically, better pitchers have lower ERAs. For
example, a pitcher that gives up two runs in seven-and-a-third innings has an
ERA of 2.45, because the ERA is determined by the proportion 
number or runs allowed
number of innings pitched
=
number of runs in an entire game (or ERA)
number of innings in a nonextended game
.
Substituting the numbers, the proportion is 
2
7
1
3
=
ERA
9
. A bad pitcher may give
up five runs after two-and-two-thirds innings and have an ERA of 16.88. Usually,
the ERA is a statistic that represents a player’s performance over an entire sea-
son, and is updated after each pitching performance.
In 1619, Johannes Kepler determined a proportion relating the mean dis-
tances,
d, that planets were from the sun and the period of their revolution, p.
This proportion, 
d
3
1
d
3
2
=
p
2
1
p
2
2
, was determined through data collection, and can be 
proven using Newton’s theory of gravitation. At the time, this information was
helpful to astronomers to predict the approximate distance that another planet is
from the sun. For example, Mars is observed in the night sky to have a period of
687 days revolving around the sun. There are 365 days in an earth year, so the
ratio of periods of these orbits is 1.882. Kepler’s proportional formula can be re-
written as 


d
1
d
2

3
=


p
1
p
2

2
because exponents distribute in any expression in-
volving a quotient of two numbers. When the Mars-to-earth ratio is substituted
into the equation, the ratio of the distances will be approximately 1.524, as a 
result of the solution to the equation 


d
1
d
2

3
= (1.882)
2
. This means that Mars is 
about 50 percent further from the sun than the earth, perhaps one reason in
understanding why most of Mars regularly maintains temperatures below 0°
Fahrenheit. (See Inverse Square Function for another explanation.) This infor-
mation is also helpful for astronomers to predict when a spacecraft can be
launched from earth so that its trajectory would come in close contact with a
planet. For example, the Voyager ships launched in 1977 had trajectories that
placed them near Jupiter, Saturn, Uranus, and Neptune in order to take photos
that could be sent back to earth for further study. 
The strength of an animal or object is proportional to its surface–area-to-
weight ratio. Small insects can carry objects much greater than their mass, while
humans can only carry small percentages of their mass. If an ant were to grow in
size, it could not maintain its surface area-to-weight ratio. Suppose a giant ant
were twenty times longer than a tiny ant. Since area is related to the square of
length in an object, the ant’s surface area would increase by a factor of 20
2
, or
400. The volume of the ant, which is proportional to its mass, would increase by
a factor of 20
3
, or 8,000, because volume is related to the cube of an object’s