What is Proportionality
This pages describes rates and proportionality constants with units.
Rates are in fact proportionality constants. Fractions with units may
appears in calculations below as proportionality constants, and as the
given or to be found values of variables or quantities below.
Different kinds of Proportionality
There might be more
Direct Proportionality (Direct Variation): A number or quantity y
is directly proportional to another quantity x in several circumstances
when and only when the quotient y ÷ x = z/x has a constant value k,. or
equivalently, there is a constant k, a number or quantity, such that y =
k x. That is, in each instance where we find or measure the value of x,
the value of y will be kx.
The proportionality constant k here is called a rate if it is a
fraction with units or if it equals an irrational number. It is called
a ratio if it equals (possible after cancellation of units), a rational
number or fraction.
That is, a single quantity y is directly proportional to a second
quantity x when and only when there is a non-zero constant k such that y
= k x -the proportionality relation.
The direct use of the proportionality relation y = kx is to calculate
the value of y from those of k and x. But tin practice, the a problem
gives the values (x1, y1) first, from which the
value of the proportionality k can be computed via a backward use of
the formula. And after k is known, the formula y = k x can be used
directly or indirectly to compute y or x respectively. In practice, the
problem may ask for the latter value of y and x in some circumstance,
while including values (x1, y1) which can be used
to find k. The typical proportional relation problem involves two
steps. The first is to find the proportionality constant k. The second
is to use proportionality relation forwards or backwards, taking
advantage of the just calculated value of k.
Joint Proportionality (Joint Variation). A number or quantity z is
directly proportional to quantities x and y in several circumstances when
and only when there is a constant k, a number or quantity, such that z =
k xy That is, in each instance where we find or measure the value of x
and y, the value of z will be kxy
That is, a single quantity z is jointly proportional to a second
quantity x and a third quantity y when and only when there is a
non-zero constant k such that z = kxy = the product of k, x and y. The
backward use of the equation z = k x y may give the value of the
proportionality constant k in terms of the quantities x, y and z:
Clearly k = z/(xy). Three given values of x, y and z are thus
sufficient to calculate k.
Inverse Proportionality (Inverse Variation): A number or quantity
z is inversly proportional to another quantity x in several circumstances
when and only when the product z×x has a constant value k,.or
equivalently, there is a constant k, a number or quantity, such that z =
k/ x. That is, in each instance where we find or measure the value of x,
the value of z will be k/x. Here z is directly proportional to the
reciprocal of x = the multiplicative inverse of x.
New Proportional Relations from old: Suppose quantity z is
directly proportional to quantities x and y. Then The backward use of
the equation z = k x y may give the value of the proportionality
constant k in terms of the quantities x, y and z, or it may give the
value of one of the two variables x and y, say y in terms of x, z and
k. Clearly z = kxy implies
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y
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=
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z
kx
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=
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1
k
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×
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z
x
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=
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K
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×
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z
x
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Thus y is jointly proportional to z and inversely proportional to x.
There-in lies another kind joint direct and inverse proportionality. In
the one just obtained, the proportionality constant for the new
relation is K = 1/k.
More generally, when product of one group of quantities may be
proportional to the product of another group of quantities, multiple
proportionality relations are implied, one for each quantity or
variable that may appear in either of the products. We assume in the
first instance that the products have no variables in common.
Algebraic Perspective of Proportionality.
In situations involving multiple proportionalities, amounts are
proportional to each other and to any linear function of the amounts in
questions (in which the coefficients are fixed and positive).
Example: The sides of similar triangles and of similar polygons
are proportional to each other. There are multiple proportions here.
Now the simultaneous equations x =kx, y=ky and z =kz can be used
backwards and forwards. If k is unknown, it can be found given from one
of the equations Then it can be use in the other two.
Assume x, y and z are all nonzero. In 3D projective geometry, the point
(x,y,z) with is equivalent to another point (X,Y,Z) when and only when
x:y:z = X: Y: Z when and only when (i) the three ratios x/X, y/Y and z/Z
have a common value k, when and only when (ii) there is a constant k such
that x =kX, y=kY and z =kZ. See the earlier discussion of multiple
rations.
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