ICSE Class 8 Maths Algebra Chapter 07 Algebraic Fractions

Algebraic Fractions

Fractions in which the numerator or denominator or both are algebraic expressions are called algebraic fractions.

Example \(\frac{x}{2}\), \(\frac{2}{x^2+3}\) and \(\frac{a-6}{a^2+3a+2}\) are algebraic fractions.

Algebraic fractions can be reduced, added, subtracted, multiplied and divided according to the same rules that govern arithmetic fractions.

Reduction To Lowest Terms

When the numerator and denominator of an algebraic fractions do not have any common factor except 1, the algebraic fraction is said to be expressed in the lowest terms.

To reduce an algebraic fraction to the lowest terms, take the following steps.

Steps 1. Factorize the numerator and the denominator.

2. Cancel the factors that are common to the numerator and the denominator.

Examples (i) \(\frac{15x^2}{25x^3} = \frac{3 \times 5 \times x \times x}{5 \times 5 \times x \times x \times x} = \frac{3}{5 \times x} = \frac{3}{5x}\)

Alternatively, \(\frac{15x^2}{25x^3} = \frac{15}{25} \times \frac{x^2}{x^3} = \frac{3}{5} \times \frac{1}{x^{3-2}}\) \(\left[\because \frac{x^m}{x^n} = \frac{1}{x^{n-m}}, n > m\right]\)

\(= \frac{3}{5x}\)

(ii) \(\frac{2a^2b}{6ab^4} = \frac{2 \times a \times a \times b}{2 \times 3 \times a \times b \times b \times b} = \frac{a}{3 \times b \times b \times b} = \frac{a}{3b^3}\)

Alternatively, \(\frac{2a^2b}{6ab^4} = \frac{2}{6} \times \frac{a^2}{a} \times \frac{b}{b^4} = \frac{1}{3} \times \frac{a^{2-1}}{b^{4-1}} = \frac{a}{3b^3}\)

(iii) \(\frac{2a^2-6a}{9b-3ab} = \frac{2a(a-3)}{3b(3-a)} = \frac{2a(a-3)}{-3b(a-3)} = \frac{2a}{3b}\)

(iv) \(\frac{x^2-5x+6}{x^2-6x+9} = \frac{x^2-3x-2x+6}{x^2-2 \times x \times 3 + 3^2} = \frac{x(x-3)-2(x-3)}{(x-3)^2} = \frac{(x-3)(x-2)}{(x-3)^2} = \frac{x-2}{x-3}\)

Teacher's Note

When simplifying fractions in recipes or adjusting medication dosages, we use the same cancellation principle to reduce quantities proportionally.

Addition And Subtraction

If algebraic fractions have the same denominator then add or subtract the fractions as follows.

\(\frac{a}{b} + \frac{c}{b} = \frac{a+c}{b}\) and \(\frac{a}{b} - \frac{c}{b} = \frac{a-c}{b}\)

To add or subtract algebraic fractions that do not have the same denominator, first find the LCM of their denominators and then proceed as in arithmetic.

\(\frac{a}{b} + \frac{c}{d} = \frac{a(bd+b)+c \times (bd+d)}{bd} = \frac{(a \times d)+(c \times b)}{bd}\)

Thus, \(\frac{a}{b} + \frac{c}{d} = \frac{ad+bc}{bd}\) and \(\frac{a}{b} - \frac{c}{d} = \frac{ad-bc}{bd}\)

Examples (i) \(\frac{4}{5a} + \frac{6}{5a} = \frac{4+6}{5a} = \frac{10}{5a} = \frac{2}{a}\)

(ii) \(\frac{5}{x-1} + \frac{x-6}{x-1} = \frac{5+x-6}{x-1} = \frac{x-1}{x-1} = 1\)

(iii) \(\frac{7}{3m^2} - \frac{1}{3m^2} = \frac{7-1}{3m^2} = \frac{6}{3m^2} = \frac{2}{m^2}\)

(iv) \(\frac{2x}{x^2-4} - \frac{4}{x^2-4} = \frac{2x-4}{x^2-4} = \frac{2(x-2)}{(x+2)(x-2)} = \frac{2}{x+2}\)

(v) \(\frac{13}{x} + \frac{24}{y} = \frac{13 \times y + 24 \times x}{xy} = \frac{13y+24x}{xy}\)

(vi) \(\frac{3a}{a+5} - \frac{8a}{a-5} = \frac{3a(a-5)-8a(a+5)}{(a+5)(a-5)} = \frac{3a^2-15a-8a^2-40a}{(a+5)(a-5)} = \frac{-5a^2-55a}{(a+5)(a-5)} = \frac{-5a(a+11)}{a^2-5^2} = \frac{-5a(a+11)}{a^2-25}\)

Teacher's Note

Finding common denominators in fractions is similar to synchronizing schedules - we need a common timeframe to combine different activities.

Multiplication

To find the product of two algebraic fractions, multiply the numerator of one by the numerator of the other and the denominator of one by the denominator of the other. Then reduce the fraction to the lowest terms.

\(\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d} = \frac{ac}{bd}\), where \(b \neq 0, d \neq 0\)

Examples (i) \(2a^2 \times \frac{3b^2}{5a^3} \times \frac{5}{4ab} = \frac{2a^2 \times 3b^2 \times 5}{5a^3 \times 4ab} = \frac{2a^2 \times 3b^2 \times 5}{1 \times 5a^3 \times 4ab} = \frac{2 \times a \times a \times 3 \times b \times b \times 5}{5 \times a \times a \times a \times 2 \times 2 \times a \times b} = \frac{3 \times b}{2 \times a \times a} = \frac{3b}{2a^2}\)

(ii) \(\frac{3a+12}{a^4} \times \frac{a^2}{a^2-16} = \frac{3(a+4) \times a \times a}{a \times a \times a \times a \times (a+4) \times (a-4)} = \frac{3}{a \times a \times (a-4)} = \frac{3}{a^2(a-4)}\)

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