Difference between arithmetic and algebraic fraction

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.

In 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.

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