Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "Proportion"

(→‎Introductory)
(→‎Inverse Proportion)
 
(25 intermediate revisions by 5 users not shown)
Line 1: Line 1:
−
{{WotWAnnounce|week=Sep 13-19}}
 
 
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 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>
 
:<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.
 
The graph of a direct proportion is always [[line]]ar.
  
−
Often, this will be written as <math>\displaystyle y \propto x \displaystyle</math>.
+
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:
 
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>
  
−
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:
 
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 28: Line 23:
 
:<math>y = \log_k (x).\,</math>
 
:<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==
 
==Problems==
 
===Introductory===
 
===Introductory===
−
<!-- edit: problem did not really involve proportions -->
+
*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===
 
===Intermediate===
−
===Pre-Olympiad===
+
*<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====
+
 
 +
===Olympiad===
 +
 
 +
==See Also==
 +
*[[Ratio]]
 +
*[[Fraction]]
 +
 
 +
[[Category:Algebra]]
 +
[[Category:Definition]]

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)

Olympiad

See Also

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=Proportion&oldid=174224"