Difference between revisions of "Proportion"

Latest revision as of 15:34, 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)

@@ Line 1: / Line 1: @@
 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==
-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:
+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>
-where '''k''' is some [[real number]].
+where <math>k</math> is some [[real number]].
 The graph of a direct proportion is always [[line]]ar.
@@ Line 12: / Line 12: @@
 Often, this will be written as <math>y \propto x</math>.
-==Inverse proportion==
+==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:
 :<math>xy=k</math>
-where '''k''' is some real number that does not equal zero.
+==Exponential Proportion==
-The graph of an inverse proportion is always a [[hyperbola]], with [[asymptote]]s 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:
@@ Line 27: / Line 23: @@
 :<math>y = \log_k (x).\,</math>
-for some real number '''k''', where k is not zero or one.
+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:
-</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]])
+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]])
+*<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)
-===Pre-Olympiad===
 ===Olympiad===
+==See Also==
+*[[Ratio]]
+*[[Fraction]]
+[[Category:Algebra]]
+[[Category:Definition]]

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