# Difference between revisions of "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: | ||

+ | |||

+ | :<math>y=kx</math> | ||

+ | |||

+ | where '''k''' is some [[real number]]. | ||

+ | |||

+ | The graph of a direct proportion is always [[line]]ar. | ||

+ | |||

+ | Often, this will be written as <math>y \propto x</math>. | ||

+ | |||

+ | ==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: | ||

+ | |||

+ | :<math>xy=k</math> | ||

+ | |||

+ | where '''k''' is some real number that does not equal zero. | ||

+ | |||

+ | The graph of an inverse proportion is always a [[hyperbola]], with [[asymptote]]s 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: | ||

+ | |||

+ | :<math>y = k^x\,</math> or | ||

+ | :<math>y = \log_k (x).\,</math> | ||

+ | |||

+ | for some real number '''k''', where k is not zero or one. | ||

+ | |||

+ | ==Problems== | ||

+ | ===Introductory=== | ||

+ | *Suppose <math>\frac{1}{20}</math> is either '''x''' or '''y''' in the following system:<br /> | ||

+ | :<math>\begin{cases} | ||

+ | xy=\frac{1}{k}\\ | ||

+ | x=ky | ||

+ | \end{cases}</math> <br /> | ||

+ | Find the possible values of '''k'''. ([[Proportion/Introductory|Source]]) | ||

+ | |||

+ | ===Intermediate=== | ||

+ | ===Pre-Olympiad=== | ||

+ | ===Olympiad=== |

## Revision as of 18:22, 24 September 2007

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

## 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:

where **k** 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:

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:

- or

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

## Problems

### Introductory

- Suppose is either
**x**or**y**in the following system:

Find the possible values of **k**. (Source)