# Fundamental Mathematics/Arithmetic/Division

## Operation

${\displaystyle {\frac {A}{B}}=C}$

Where

${\displaystyle A}$ is the dividend.
${\displaystyle B}$ is the divisor.
${\displaystyle C}$ is the quotient.

## Divisibility

Division without remainder

${\displaystyle {\frac {A}{B}}=C}$ without remainder then ${\displaystyle A=BC}$

Division with remainder

${\displaystyle {\frac {A}{B}}=C}$ with remainder R then ${\displaystyle A=BC+R}$

## Division Table

 / 1 2 3 4 5 6 7 8 9 10 11 12 1 1 0.5 1/3 0.25 0.2 1/6 1/7 0.125 1/9 0.1 1/11 1/12 2 2 1 2/3 0.5 0.4 1/3 2/7 0.25 2/9 0.2 2/11 1/6 3 3 1.5 1 0.75 0.6 0.5 3/7 0.375 1/3 0.3 3/11 0.25 4 4 2 1 1/3 1 0.8 2/3 4/7 0.5 4/9 0.4 4/11 1/3 5 5 2.5 1 2/3 1.25 1 5/6 5/7 0.625 5/9 0.5 5/11 5/12 6 6 3 2 1.5 1.2 1 6/7 0.75 2/3 0.6 6/11 0.5 7 7 3.5 2 1/3 1.75 1.4 1 1/6 1 0.875 7/9 0.7 7/11 7/12 8 8 4 2 2/3 2 1.6 1 1/3 1 1/7 1 8/9 0.8 8/11 2/3 9 9 4.5 3 2.25 1.8 1.5 1 2/7 1.125 1 0.9 9/11 0.75 10 10 5 3 1/3 2.5 2 1 2/3 1 3/7 1.25 1 1/9 1 10/11 5/6 11 11 5.5 3 2/3 2.75 2.2 1 5/6 1 4/7 1.375 1 2/9 1.1 1 11/12 12 12 6 4 3 2.4 2 1 5/7 1.5 1 1/3 1.2 1 1/11 1

## Example

• Definition: ${\displaystyle {\frac {a}{b}}=a\times {\frac {1}{b}}}$ , whenever ${\displaystyle b\neq 0}$ .

Let's look at an example to see how these rules are used in practice. Of course, the above is much longer than simply cancelling ${\displaystyle x+3}$ out in both the numerator and denominator. But, when you are cancelling, you are really just doing the above steps, so it is important to know what the rules are so as to know when you are allowed to cancel. Occasionally people do the following, for instance, which is incorrect:

${\displaystyle {\frac {2\times (x+2)}{2}}={\frac {2}{2}}\times {\frac {x+2}{2}}=1\times {\frac {x+2}{2}}={\frac {x+2}{2}}}$ .

The correct simplification is

${\displaystyle {\frac {2\times (x+2)}{2}}=\left(2\times {\frac {1}{2}}\right)\times (x+2)=1\times (x+2)=x+2}$ ,

where the number ${\displaystyle 2}$ cancels out in both the numerator and the denominator.