Difference between revisions of "Proportion"

(Introductory)
(template)
Line 31: Line 31:
 
==Problems==
 
==Problems==
 
===Introductory===
 
===Introductory===
*Suppose <math>\frac{1}{20}</math> is either '''x''' or '''y''' in the following system:<br />
+
*{{proportion/Introductory}} ([[proportion/Introductory|Source]])
:<math>\begin{cases}
 
xy=\frac{1}{k}\\
 
x=ky
 
\end{cases}</math> <br />
 
Find the possible values of '''k'''. ([[Proportion/Introductory|Source]])
 
  
 
===Intermediate===
 
===Intermediate===
 
===Pre-Olympiad===
 
===Pre-Olympiad===
 
===Olympiad===
 
===Olympiad===

Revision as of 17:51, 9 October 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.

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

Intermediate

Pre-Olympiad

Olympiad

Invalid username
Login to AoPS