Proportion

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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.

Direct Proportion

Direct proportion is a proportion in which one number is a multiple of the other. Direct proportion between two numbers $x$ and $y$ can be expressed as:

$y=kx$

where $k$ is some real number.

The graph of a direct proportion is always linear.

Often, this will be written as $y \propto x$ .

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:

$xy=k$

where $k$ is some real number that does not equal zero.

The graph of an inverse proportion is always a hyperbola, with asymptotes at the x and y axes.

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:

$y = k^x\,$ or

$y = \log_k (x).\,$

for some real number $k$ , where $k$ is not zero or one.

Problems

Introductory

Suppose $\frac{1}{20}$ is either $x$ or $y$ in the following system:

$\begin{cases}xy=\frac{1}{k}\\x=ky\end{cases}$ Find the possible values of $k$ . (Source)

Intermediate

$x$ is directly proportional to the sum of the squares of $y$ and $z$ and inversely proportional to $y$ and the square of $z$ . If $x = 8$ when $y = \frac{1}{2}$ and $z = \frac{\sqrt {3}}{2}$ , find $y$ when $x = 1$ and $z = 6$ , what is $y$ ? (Source) (Thanks to Bicameral of the AoPS forum for this one)

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