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. | 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 | + | ==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: | + | 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: |
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| − | ==Inverse | + | :<math>y=kx</math> |
| − | 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]]. |
| − | where k is some real number that does not equal zero. | + | |
| − | The graph of an inverse proportion is always a hyperbola, with | + | The graph of a direct proportion is always [[line]]ar. |
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| + | Often, this will be written as <math>\displaystyle y \propto x \displaystyle</math>. | ||
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| + | ==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: | ||
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| + | :<math>y = k^x\,</math> or | ||
| + | :<math>y = \log_k (x).\,</math> | ||
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for some real number '''k''', where k is not zero or one. | for some real number '''k''', where k is not zero or one. | ||
Revision as of 17:48, 14 September 2007
| This is an AoPSWiki Word of the Week for Sep 13-19 |
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:
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
or
for some real number k, where k is not zero or one.
