In this chapter we will learn the difference between ” simple fraction ” and “arithmetic fraction” with solved examples.

## Arithmetic and Algebraic fraction

Arithmetic Fraction

In arithmetic fraction, **both numerator and denominator contain constant values** that can be integer, whole number or decimal.

Some examples of arithmetic fractions are;

\mathtt{( i) \ \ \frac{2}{3}}\\\ \\ \mathtt{( ii) \ \frac{6}{21}}\\\ \\ \mathtt{( iii) \ \frac{-11}{100}}\\\ \\ \mathtt{( iv) \ \frac{1}{31}}\\\ \\ \mathtt{( v) \ \frac{5^{2}}{3^{4}}}

### Algebraic Fraction

In algebraic fraction **both numerator and denominator contains both constants and variables** in the form of polynomials.

The algebraic expression can take form of monomial, binomial, trinomial and so on.

The **exact value of algebraic fraction is not known since it contain variable values**.

Given below are some **examples of algebraic expressions**;

(a) **Both numerator and denominator are monomials**

\mathtt{( i) \ \ \frac{xy}{7}}\\\ \\ \mathtt{( ii) \ \frac{2}{x^{2} y}}\\\ \\ \mathtt{( iii) \ \frac{6xyz}{5y^{2} z}}\\\ \\ \mathtt{( iv) \ \frac{13a^{2} b}{c^{2}}}

(b) **Numerator is binomial and denominator is monomial**

\mathtt{( i) \ \ \frac{x^{2} +y^{2}}{5}}\\\ \\ \mathtt{( ii) \ \frac{xy+x}{5x}}\\\ \\ \mathtt{( iii) \ \frac{x^{3} y+z^{3} y}{xyz}}\\\ \\ \mathtt{( iv) \ \frac{a+2b}{6a^{2}}}

**(c) Both numerator and denominator is binomial**

\mathtt{( i) \ \ \frac{5x^{2} +2x}{6x+7}}\\\ \\ \mathtt{( ii) \ \frac{9xyz-8}{2+7z}}\\\ \\ \mathtt{( iii) \ \frac{x^{3} +y^{3}}{6x+5y}}

**(d) Both numerator and denominator are polynomial of different size**

\mathtt{( i) \ \ \frac{3x^{3} +2x^{2} +7}{8x+3}}\\\ \\ \mathtt{( ii) \ \frac{10x-10y}{x^{5} +9x^{2} +3x^{3} +4}}\\\ \\ \mathtt{( iii) \ \frac{16x^{2} yx}{8x+9y+10z}}\\\ \\ \mathtt{( iv) \ \frac{5}{x^{7} +10x^{4} +9x^{2}}}

## Proper & Improper fractions

**In arithmetic fraction**, given below are definition of proper and improper fractions.**Proper Fractions**

The arithmetic fraction in which numerator is smaller than denominator is called proper fraction.

**Examples are;**

\mathtt{\frac{2}{5} ,\frac{1}{3} \ and\ \frac{11}{15}}

**Improper fractions**

Arithmetic fraction in which numerator is greater than denominator are called improper fractions.

\mathtt{\frac{11}{2} ,\frac{5}{3} \ and\ \frac{16}{10}} are some of the examples.

I**n algebraic fraction**, the proper and improper fractions have different definitions.

**Proper Fraction**

In algebraic fraction, when degree of numerator is less than denominator then the fraction is called proper algebraic fraction.

**Examples of proper algebraic fractions are;**

\mathtt{( i) \ \ \frac{x+3}{x^{2} +6x}}

Numerator = polynomial with degree 1

Denominator = polynomial with degree 2

Since **denominator degree > numerator**, the above fraction is proper algebraic fraction.

\mathtt{( ii) \ \ \frac{x^{2} +9x+8}{x^{4} +1}}

Numerator = polynomial with degree 2

Denominator = polynomial with degree 4

Since, denominator degree > numerator degree, it is a proper algebraic fraction.

**Improper Fraction**

In algebraic fraction, when degree of numerator is greater than degree of denominator then it is called improper algebraic fraction.

**Examples of Improper algebraic fraction**

\mathtt{( i) \ \ \frac{x^{2} -12x-13}{x-1}}

Numerator degree = 2

Denominator degree = 1

Since numerator degree > denominator, it is improper algebraic fraction.

\mathtt{( ii) \ \ \frac{x^{5} -1}{x^{3} +1}}

Numerator degree = 5

Denominator degree = 3

Since, numerator degree > denominator, it is improper algebraic fraction.