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 | + | Direct proportion is a proportion in which one number is a multiple of the other. Direct proportion between two numbers <math>x</math> and <math>y</math> can be expressed as: |
:<math>y=kx</math> | :<math>y=kx</math> | ||
− | where | + | where <math>k</math> is some [[real number]]. |
The graph of a direct proportion is always [[line]]ar. | The graph of a direct proportion is always [[line]]ar. | ||
− | Often, this will be written as <math> | + | Often, this will be written as <math>y \propto x</math>. |
− | ==Inverse | + | ==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: | 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> | :<math>xy=k</math> | ||
− | + | ==Exponential Proportion== | |
− | |||
− | |||
− | |||
− | ==Exponential | ||
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: | 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 = \log_k (x).\,</math> | :<math>y = \log_k (x).\,</math> | ||
− | for some real number | + | for some real number <math>k</math>, where <math>k</math> is not zero or one. |
+ | |||
+ | ==Problems== | ||
+ | ===Introductory=== | ||
+ | *Suppose <math>\frac{1}{20}</math> is either <math>x</math> or <math>y</math> in the following system: | ||
+ | <cmath>\begin{cases} | ||
+ | xy=\frac{1}{k}\\ | ||
+ | x=ky | ||
+ | \end{cases} </cmath> | ||
+ | Find the possible values of <math>k</math>. ([[proportion/Introductory|Source]]) | ||
+ | |||
+ | ===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 = \frac{1}{2}</math> and <math>z = \frac{\sqrt {3}}{2}</math>, find <math>y</math> when <math>x = 1</math> and <math>z = 6</math>, what is <math>y</math>? ([[Proportion/Intermediate|Source]]) (Thanks to Bicameral of the AoPS forum for this one) | ||
+ | |||
+ | ===Olympiad=== | ||
+ | |||
+ | ==See Also== | ||
+ | *[[Ratio]] | ||
+ | *[[Fraction]] | ||
+ | |||
+ | [[Category:Algebra]] | ||
+ | [[Category:Definition]] |
Latest revision as of 16:34, 1 June 2022
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 and
can be expressed as:
where 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:
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 , where
is not zero or one.
Problems
Introductory
- Suppose
is either
or
in the following system:
Find the possible values of
. (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
? (Source) (Thanks to Bicameral of the AoPS forum for this one)