quotient

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

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

Ratios of similar sizes are often given in percent , whereby the value of the ratio does not change, e.g. B. . To get the percentage, multiply the ratio fraction by one, where . In the example: . ${\displaystyle {\tfrac {1}{5}}=20\,\%}$ ${\displaystyle 1=100\,\%}$ ${\displaystyle {\tfrac {1}{5}}={\tfrac {100}{5}}\,\%=20\,\%}$

Special quotients in this sense are e.g. B .:

• 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 the ratio of two lengths .

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

• Expansion coefficient , efficiency ,
• Specific quantities such as density .

Proportions

→ Main article : rule of three

Ratio equations or proportions are equations that equate two ratios:

${\displaystyle a:b=c:d}$

${\displaystyle a}$and are also called forelimbs , and hind limbs of proportion. In addition, and are called outer links and and inner links . The proportion can be transformed into an equation of the form by cross multiplication . By swapping the inner links or the outer links of a proportion, new proportions are created: and . In addition, the laws of corresponding addition and subtraction apply : ${\displaystyle c}$ ${\displaystyle b}$ ${\displaystyle d}$ ${\displaystyle a}$ ${\displaystyle d}$ ${\displaystyle b}$ ${\displaystyle c}$ ${\displaystyle a\cdot d=c\cdot b}$ ${\displaystyle a:c=b:d}$ ${\displaystyle d:b=c:a}$

Laws of corresponding addition and subtraction

Let the proportion be given. Then the proportions also apply ${\displaystyle a:b=c:d}$

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

Continuous proportions

Occasionally there is also the spelling

${\displaystyle a:b:c=u:v:w}$,

as " , , how behave to be pronounced." These continuous proportions , also called chain proportions or ratio chains, are not to be understood as a single equation, but rather are a short form for the two equations ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle c}$ ${\displaystyle u}$ ${\displaystyle v}$ ${\displaystyle w}$

• ${\displaystyle a:b=u:v}$ and
• ${\displaystyle b:c=v:w}$

or equivalent

• ${\displaystyle a:u=b:v}$ and
• ${\displaystyle b:v=c:w}$.

Examples

• The definition of the golden ratio
• The law of sines
• The ray sentences
• The law of refraction in optics
• The octave of music

Web links

Wiktionary: quotient  - explanations of meanings, word origins, synonyms, translations

• The mathematical term "ratio"

Individual evidence

1. ^ Walter Gellert, Herbert Kästner , Siegfried Neuber (eds.): Lexikon der Mathematik , VEB Bibliographisches Institut Leipzig, 1979. S 447, Proportion .