Difference between revisions of "Proportion"
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==Problems== | ==Problems== | ||
===Introductory=== | ===Introductory=== | ||
− | *{{ | + | * |
+ | </noinclude> | ||
+ | Suppose <math>\frac{1}{20}</math> is either '''x''' or '''y''' in the following system: | ||
+ | <cmath>\begin{cases} | ||
+ | xy=\frac{1}{k}\\ | ||
+ | x=ky | ||
+ | \end{cases} </cmath> | ||
+ | Find the possible values of '''k'''. ([[proportion/Introductory|Source]]) | ||
===Intermediate=== | ===Intermediate=== | ||
+ | *<math>x</math> is directly proportional to the sum of the squares of <math>y</math> and <math>z</math> and inversely proportional to <math>y</math> and the square of <math>z</math>. If <math>x = 8</math> when <math>y = 1/2</math> and <math>z = \sqrt {3}/2</math>, find <math>y</math> when <math>x = 1</math> and <math>z = 6</math>, what is <math>y</math>? | ||
===Pre-Olympiad=== | ===Pre-Olympiad=== | ||
===Olympiad=== | ===Olympiad=== |
Revision as of 17:53, 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.
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)
Intermediate
is directly proportional to the sum of the squares of
and
and inversely proportional to
and the square of
. If
when
and
, find
when
and
, what is
?