Difference between revisions of "Proportion"

 
(Inverse 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.
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==Direct Proportion==
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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:
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:<math>y=kx</math>
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where <math>k</math> is some [[real number]].
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The graph of a direct proportion is always [[line]]ar.
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Often, this will be written as <math>y \propto x</math>.
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==Inverse Proportion==
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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:
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:<math>xy=k</math>
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==Exponential Proportion==
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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 = k^x\,</math> or
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:<math>y = \log_k (x).\,</math>
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for some real number <math>k</math>, where <math>k</math> is not zero or one.
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==Problems==
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===Introductory===
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*Suppose <math>\frac{1}{20}</math> is either <math>x</math> or <math>y</math> in the following system:
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<cmath>\begin{cases}
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xy=\frac{1}{k}\\
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x=ky
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\end{cases} </cmath>
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Find the possible values of <math>k</math>. ([[proportion/Introductory|Source]])
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===Intermediate===
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*<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)
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===Olympiad===
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==See Also==
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*[[Ratio]]
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*[[Fraction]]
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[[Category:Algebra]]
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[[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.

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:

$xy=k$

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

  • Suppose $\frac{1}{20}$ is either $x$ or $y$ in the following system:

\[\begin{cases} xy=\frac{1}{k}\\ x=ky \end{cases}\] Find the possible values of $k$. (Source)

Intermediate

  • $x$ is directly proportional to the sum of the squares of $y$ and $z$ and inversely proportional to $y$ and the square of $z$. If $x = 8$ when $y = \frac{1}{2}$ and $z = \frac{\sqrt {3}}{2}$, find $y$ when $x = 1$ and $z = 6$, what is $y$? (Source) (Thanks to Bicameral of the AoPS forum for this one)

Olympiad

See Also