Proportion

Revision as of 18:22, 24 September 2007 by Temperal (talk | contribs) (Introductory)

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

Pre-Olympiad

Olympiad

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