Difference between revisions of "Proportion"
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==Problems== | ==Problems== | ||
===Introductory=== | ===Introductory=== | ||
| − | < | + | Suppose <math>\displaystyle \frac{1}{20}</math> is an answer to the system of equations:<br /> |
| + | :<math>\begin{cases} | ||
| + | xy=\frac{1}{4}\\ | ||
| + | x=ky | ||
| + | \end{cases}</math> <br /> | ||
| + | Find '''k'''. | ||
===Intermediate=== | ===Intermediate=== | ||
===Pre-Olympiad=== | ===Pre-Olympiad=== | ||
===Olympiad=== | ===Olympiad=== | ||
Revision as of 17:37, 15 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.
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 an answer to the system of equations:
Find k.
