Proportion
Two numbers are said to be in proportion to each other if some numeric relationship exists between them. There are several types of proportions, each defined by a separate class of function.
Contents
[hide]Direct Proportion
Direct proportion is a proportion in which one number is a multiple of the other. Direct proportion between two numbers and
can be expressed as:
where is some real number.
The graph of a direct proportion is always linear.
Often, this will be written as .
Inverse Proportion
Inverse proportion is a proportion in which as one number's absolute value increases, the other's decreases in a directly proportional amount. It can be expressed as:
Exponential Proportion
A proportion in which one number is equal to a constant raised to the power of the other, or the logarithm of the other, is called an exponential proportion. It can be expressed as either:
or
for some real number , where
is not zero or one.
Problems
Introductory
- Suppose
is either
or
in the following system:
Find the possible values of
. (Source)
Intermediate
is directly proportional to the sum of the squares of
and
and inversely proportional to
and the square of
. If
when
and
, find
when
and
, what is
? (Source) (Thanks to Bicameral of the AoPS forum for this one)