Geoscience Reference
In-Depth Information
In the above, notice that the terms in the denominator were inverted before the fractions were mul-
tiplied. This is a standard rule that must be followed when dividing fractions.
Another example is
2
2
mm
mm
m
m
mm
2
becomesmm
×
2
2
2
2.9.3 b
asiC
o
peration
: C
anCel
or
d
ivide
n
umerators
and
d
enominators
We must know how to cancel or divide terms in the numerator and denominator of a fraction. After
fractions have been rewritten in the vertical form and division by the fraction has been re-expressed
as multiplication, as shown above, then the terms can be canceled (or divided) out.
Key Point:
For every term that is canceled in the numerator of a fraction, a similar term must be
canceled in the denominator and
vice versa
, as shown below:
kg
d
d
kg
×=
min
min
2
m
mm
2
2
mm
×
=
m
2
3
3
gal
min
ft
gal
ft
×=
min
Question:
How do we calculate units that include exponents?
Answer:
When written with exponents, such as ft
3
, a unit can be left as is or put in expanded form,
(ft)(ft)(ft), depending on other units in the calculation. The point is that it is important to ensure that
square and cubic terms are expressed uniformly (e.g., sq ft, ft
2
cu ft, ft
3
). For dimensional analysis,
the latter system is preferred.
For example, to convert a volume of 1400 ft
3
to gallons, we will use 7.48 gal/ft
3
in the conver-
sions. The question becomes do we multiply or divide by 7.48? In this instance, it is possible to use
dimensional analysis to answer this question of whether we multiply or divide by 7.48.
To determine if the math setup is correct, only the dimensions are used. First, try dividing the
dimensions:
3
3
ft
gal/ft
ft
gal
ft
=
3
3
Multiply the numerator and denominator to get
6
ft
gal
So, by dimensional analysis, we have determined that if we divide the two dimensions (ft
3
and
gal/ft
3
) then the units of the answer are ft
6
/gal, not gal. It is clear that division is not the right
approach to making this conversion.
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