Arithmetic Fraction and Algebraic Fraction

What is Arithmetic Fractions?

All arithmetic fractions are expressed in the form of p/q (Where q ≠ 0), p is known as ‘numerator’ and q is known as ‘denominator’. That means p/q = numerator/denominator; it can be expressed as p ÷ q also.

For example: 2/3, 5/7, 8/17 etc.

Note:

(i) If the ‘numerator’ and ‘denominator’ of the fractions are multiplied by the same quantity, then the value of the fraction remains unchanged.

(ii) If the ‘numerator’ and ‘denominator’ of the fractions are divided by the same quantity, then the value of the fraction remains unchanged.

Arithmetic quantities are mostly monomial quantities or they can be reduced to monomials.

For example: 4/8 = ½

27/81 = 1/3

12/16 = ¾, etc.

What is Algebraic Fractions?

Algebraic quantities may be monomials, binomials, polynomials. So, algebraic fractions expressed in the form of p/q can be of different types.

Some examples if algebraic fraction:

(i) When both the ‘denominator’ and ‘numerator’ are monomials,

For example: $\frac{p}{q}, \frac{m}{n}, \frac{xy}{z}, \frac{- ax^{2}}{uv}, \frac{2m^{2}}{n}$ , etc.

(ii) When ‘denominator’ is monomial and ‘numerator’ is binomial/polynomial,

For example: $\frac{a + b}{c}, \frac{x^{2} + xy + y^{2}}{xy}, \frac{2m^{2} + n}{m}, \frac{ab + bc + ca}{d}$ , etc.

(iii) When ‘denominator’ is binomial/polynomial and ‘numerator’ is monomial,

For example: $\frac{x}{y - z}, \frac{a}{b + c}, \frac{m}{2m^{2} + 5}, \frac{d}{ab + bc + ca}$ , etc.

(iv) When ‘denominator’ and ‘numerator’ both are binomial/polynomial,

For example: $\frac{m + n}{m - n}, \frac{x + y + z}{x + z}, \frac{m^{2} + 4mn + 4n^{2}}{m + n}$ , etc.

Note: When the denominator is equal to 0 an algebraic fraction is said to be undefined.

For example: The algebraic fraction $\frac{5}{x - 2}$ is undefined when x = 2 since, $\frac{5}{2 - 2}$ = $\frac{5}{0}$ which have no meaning. Thus, when the denominator is 0 then the algebraic fraction is said to be undefined.