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
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:
is either x or y in the following system:
Find the possible values of k. (Source)
