Quotient

In mathematics and the natural sciences, the quotient denotes a ratio of two quantities to each other, i.e. the result of a division. The quotient of two integers (dividend and divisor) is always a rational number and can be written as a fraction (e.g. ${\tfrac {2}{3}}=2/3=2:3$ for two thirds).

A quotient is often used to classify a value in an overall scale, e.g. the intelligence quotient, which relates the number determined for a person in an intelligence test to the "average intelligence" corresponding to their age group. The intelligence quotient 100 represents the average. Other examples are the proportions of national flags or aspect ratios.

Ratios of like quantities are often expressed as a percentage, where the value of the ratio does not change, e.g. ${\tfrac {1}{5}}=20\,\%$ . To get the percentage value, multiply the ratio fraction by one, where $1=100\,\%$ . In the example, ${\tfrac {1}{5}}={\tfrac {100}{5}}\,\%=20\,\%$ .

Special quotients in this sense are, for example:

The slope as the ratio of the increase in value on the vertical coordinate axis to the increase in value on the horizontal axis.
The scale as a ratio of two lengths.

Many physical quantities are also defined as quotients, e.g.

Proportions

→ Main article: Rule of three

Ratio equations or proportions are equations that equate two ratios:

$a:b=c:d$

and are also called front members, and rear members of the proportion. Furthermore, and called outer links, and and inner links. The proportion can be $a\cdot d=c\cdot b$ transformed into an equation of the form by cross-multiplication. Swapping the inner members or the outer members of a proportion creates new proportions: $a:c=b:d$ and $d:b=c:a$ . In addition, the laws of corresponding addition and subtraction apply:

Laws of corresponding addition and subtraction

Let the proportion $a:b=c:d$ given. Then the proportions also apply

$\frac{a+b}{b}=\frac{c+d}{d}$ and $\frac{a}{a+b}=\frac{c}{c+d}$ and $\frac{a-b}{b}=\frac{c-d}{d}$ and $\frac{a}{a-b}=\frac{c}{c-d}$ and $\frac{a+b}{a-b}=\frac{c+d}{c-d}$ .

Continuous proportions

Occasionally you will also find the spelling

$a:b:c=u:v:w$ ,

which are defined as " , , behave as to to ". These continuous proportions, also called chain proportions or ratio chains, are not to be understood as a single equation, but rather are shorthand for the two equations