## Fractions

As a fraction is called the quotient of two numbers this is the division problem m divided by n, commonly written with a fraction bar. The value above the fraction bar is called numerator and the value below is the denominator.

General fraction with the numbers m and n
$$m:n=\frac{m}{n}$$
here is m the numerator and n the denominator of the fraction.

#### Examples for fractions

Fraction with the numerator 7 and the denominator 8

General example with a sum in the numerator of the fraction

### Reducing

Reducing of a fraction means the dividing of the numerator and the denominator by the same factor unequal zero. The value of the fraction is unchanged by this reducing. Reducing with the greatest common divisor of numerator and denominator results in a irreducible fraction

General reducing of a fraction:
$$\frac{a\cdot c}{b\cdot c}=\frac{a}{b}\cdot \frac{c}{c}=\frac{a}{b}\cdot 1=\frac{a}{b}$$

Are in the numerator and / or denominator sums then the common factor must be present in all summands and must divided in each summand respectively:
$$\frac{a\cdot c+b\cdot c}{x\cdot c+y\cdot c}=\frac{c\left(a+b\right)}{c\left(x+y\right)}=\frac{c}{c}\cdot \frac{a+b}{x+y}=\frac{a+b}{x+y}$$

#### Examples for the reducing of fractions

$\frac{12}{16}=\frac{3\cdot 4}{4\cdot 4}=\frac{3}{4}\cdot \frac{4}{4}=\frac{3}{4}$

In 12 and 16 is the common factor 4 and with this factor the fraction can be reduced. 4 is the greatest common divisor of 12 and 16. Separation of the fraction in a product gives the faktor 4/4 and this is 1.

$\frac{a{x}^{2}+axy}{ax}=\frac{ax\left(x+y\right)}{ax}$
$=\frac{ax}{ax}\cdot \frac{x+y}{1}=x+y$

In the sum is the common factor a⋅x and can be reduced.

The expanding of a fraction is the opposite of reducing. The multiplying of the numerator and denominator by the same factor does not change the value of the fraction it is the same as the multiplication of the total fraction with one.

### Addition

The addition of fractions is done by expanding the fraction so that they get the same denominator.

Addition of fractions in general:
$$\frac{a}{b}+\frac{c}{d}=\frac{a\cdot d}{b\cdot d}+\frac{c\cdot b}{d\cdot b}=\frac{a\cdot d+c\cdot b}{b\cdot d}$$

#### Example for the addition of fractions

$\frac{1}{2}+\frac{2}{3}$
$=\frac{1\cdot 3}{2\cdot 3}+\frac{2\cdot 2}{3\cdot 2}$
$=\frac{1\cdot 3+2\cdot 2}{2\cdot 3}=\frac{7}{6}$

Expanding the fraction to the main denominator (LCM) 6 and summarized. Each denominator must be multiplied by a factor to the main denominator. In order not to change the value of the fraction, the factor is multiplied also in the numerator. The multiplication of a factor in the numerator and denominator of the fraction is referred to expanding the fraction.

### Subtraction

The subtraction of fractions is analogous to the addition.

General subtraction of two fractions:
$$\frac{a}{b}-\frac{c}{d}=\frac{a\cdot d}{b\cdot d}-\frac{c\cdot b}{d\cdot b}=\frac{a\cdot d-c\cdot b}{b\cdot d}$$

#### Example of the subtraction of fractions

$\frac{1}{2}-\frac{2}{3}$
$=\frac{1\cdot 3}{2\cdot 3}-\frac{2\cdot 2}{3\cdot 2}$
$=\frac{1\cdot 3-2\cdot 2}{2\cdot 3}=-\frac{1}{6}$

Expanding the fractions to the main denominator 6 analogous to addition and summarized under consideration of the sign

#### Example of the addition / subtraction of fractions in steps

$\frac{4}{9}+\frac{2}{-15}$

In this example all steps are explained step by step.

$=\frac{4}{9}-\frac{2}{15}$

Step 1: Negative signs in the numerator or denominator can be moved in front of the fraction. It is - time - gives + and - times + gives -. The minus in the denominator of the second fraction is multiplied plus in front of the fraction and + times - results in -.

$=\frac{4\cdot 5}{9\cdot 5}-\frac{2\cdot 3}{15\cdot 3}=\frac{4\cdot 5}{45}-\frac{2\cdot 3}{45}$

Step 2: Determine the least common multiple of the denominators.

Multiple of 9 are 9; 18; 27; 36; 45

Multiple of 15 are 15; 30; 45

The LCM is 45. That is so both fractions must be adapted so that the denominator is 45. For this purpose, the first break with 45/9 = 5 is extended and the second break with 45/15 = 3 extended.

$=\frac{4\cdot 5-2\cdot 3}{45}$

Step 3: The fractions are now brought to the common denominator and can be written to a common fraction bar.

Step 4: Evaluate the numerators returns the result. It remains to be examined whether the fracture has a common divisor and can be shortened.

Divisors of 14 are 1; 2; 7; 14

Divisors of 45 are 1; 3; 5; 9; 15; 45

The greatest common divisor is 1. That is so the fraction can not be further reduced. Otherwise you would numerator and denominator divide by the gcd.

### Multiplication

The multiplication of fractions carried by the numerator and denominator are multiplied respectively.

General multiplying two fractions:
$$\frac{a}{b}\cdot \frac{c}{d}=\frac{a\cdot c}{b\cdot d}$$

#### Example for the multiplication of fractures

$\frac{1}{2}\cdot \frac{2}{3}$
$=\frac{1\cdot 2}{2\cdot 3}$
$=\frac{2}{6}=\frac{1}{3}$

Multiply the numerator and denominator and then reducing the fraction

### Division

The division of fractions is carried out by the first fraction is multiplied by the reciprocal value of the second.

General division of two fractions:
$$\frac{a}{b}:\frac{c}{d}=\frac{a}{b}\cdot \frac{d}{c}=\frac{\frac{a}{b}}{\frac{c}{d}}=\frac{a\cdot d}{b\cdot c}$$

#### Example of the Division of fractions with main fraction bar

$$\frac{\frac{a+b}{x}+\frac{1}{{x}^{2}}}{1+\frac{1}{x}}$$

In this example all steps are explained step by step.

$$=\frac{\frac{x\left(a+b\right)}{{x}^{2}}+\frac{1}{{x}^{2}}}{1+\frac{1}{x}}$$

Step 1: The fractions in the numerator are brought to the common denominator. That means the first fraction is expanded with x.

$$=\frac{\frac{x\left(a+b\right)}{{x}^{2}}+\frac{1}{{x}^{2}}}{\frac{\mathrm{x}}{x}+\frac{1}{x}}$$

Step 2: The fractions in the denominator are brought to the common denominator.

$$=\frac{\frac{x\left(a+b\right)+1}{{x}^{2}}}{\frac{\mathrm{x}+1}{x}}$$

Step 3: Now the fractions in the numerator and the denominator in the fraction can be written on their common denominator.

$$=\frac{\left(x\left(a+b\right)+1\right)x}{{x}^{2}\left(\mathrm{x}+1\right)}$$

Step 4: Execution of the division.

$$=\frac{x\left(a+b\right)+1}{x\left(\mathrm{x}+1\right)}$$

### Power

A fraction is raised to the power in the way that numerator and denominator potentiated seperatly.

$${\left(\frac{a}{b}\right)}^{p}=\frac{{a}^{p}}{{b}^{p}}$$

### Roots

The root of a fraction is obtained by dividing the roots of the numerator and denominator of the fraction.

$$\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$$

#### Example of a fraction with roots

$\sqrt{\frac{a{x}^{2}}{\left(a-b\right)}}=\frac{\sqrt{a{x}^{2}}}{\sqrt{\left(a-b\right)}}=\frac{\sqrt{a}x}{\sqrt{\left(a-b\right)}}$

The root is applied to the numerator and denominator.