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.
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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 '''x''' and '''y''' can be expressed as:
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:<math>y=kx</math>
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where '''k''' 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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where '''k''' is some real number that does not equal zero.
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The graph of an inverse proportion is always a [[hyperbola]], with [[asymptote]]s at the x and y axes.
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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 '''k''', where k 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 '''x''' or '''y''' in the following system:<br />
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:<math>\begin{cases}
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xy=\frac{1}{k}\\
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x=ky
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\end{cases}</math> <br />
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Find the possible values of '''k'''. ([[Proportion/Introductory|Source]])
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===Intermediate===
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===Pre-Olympiad===
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===Olympiad===

Revision as of 18:22, 24 September 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

  • 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

Pre-Olympiad

Olympiad

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