# 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.

## Proportions

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 .${\ displaystyle {\ frac {a} {a + b}} = {\ frac {c} {c + d}}}$ ${\ displaystyle {\ frac {ab} {b}} = {\ frac {cd} {d}}}$ ${\ displaystyle {\ frac {a} {ab}} = {\ frac {c} {cd}}}$ ${\ displaystyle {\ frac {a + b} {ab}} = {\ frac {c + d} {cd}}}$ ### 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
• ${\ displaystyle b: v = c: w}$ .