10th Class Math Chapter 5 Notes PDF 2026 – Algebraic Fractions

Factorize both numerator and denominator fully, then cancel every common factor.

Example 1 — reducing a fraction

Reduce $\frac{4 x ^{2} - 9 y ^{2}}{4 x ^{2} + 6 x y}$ to lowest terms.

Numerator: $(2 x - 3 y) (2 x + 3 y)$ .
Denominator: $2 x (2 x + 3 y)$ .
Cancel $(2 x + 3 y)$ .

Answer: $\frac{2 x - 3 y}{2 x}$

Factorize every denominator, find their LCM as the common denominator, then combine numerators.

Example 2 — subtracting with factored denominators

Simplify $\frac{3}{x - y} + \frac{1}{y - x}$ .

$y - x = - (x - y)$ , so $\frac{1}{y - x} = \frac{- 1}{x - y}$ .
$\frac{3}{x - y} - \frac{1}{x - y} = \frac{2}{x - y}$ .

Answer: $\frac{2}{x - y}$

Factorize everything first, cancel common factors across the whole product, and for division multiply by the reciprocal of the second fraction.

Example 3 — multiplying algebraic fractions

Simplify $\frac{x ^{2} - 3 x + 2}{x ^{2} + 3 x - 4} \times \frac{2 x ^{2} + 8 x}{3 x + 6}$ .

Factorize: $\frac{( x - 1 ) ( x - 2 )}{( x + 4 ) ( x - 1 )} \times \frac{2 x ( x + 4 )}{3 ( x + 2 )}$ .
Cancel $(x - 1)$ and $(x + 4)$ : $(x - 2) \times \frac{2 x}{3 ( x + 2 )}$ .

Answer: $\frac{2 x ( x - 2 )}{3 ( x + 2 )}$

Some higher-degree or exponential equations become quadratic after a substitution.

Example 4 — a biquadratic equation

Solve $5 x^{4} - 19 x^{2} + 12 = 0$ .

Let $y = x^{2}$ : $5 y^{2} - 19 y + 12 = 0$ .
Factorize: $(5 y - 4) (y - 3) = 0 \Rightarrow y = \frac{4}{5}$ or $y = 3$ .
$x^{2} = \frac{4}{5} \Rightarrow x = \pm \frac{2}{5}$ ; $x^{2} = 3 \Rightarrow x = \pm 3$ .

Answer: $x = \pm \frac{2}{5}, \pm 3$

Example 5 — an exponential equation via substitution

Solve $4 \cdot 2^{2 x + 1} - 9 \cdot 2^{x} + 1 = 0$ .

$2^{2 x + 1} = 2 \cdot (2^{x})^{2}$ , so the equation is $8 (2^{x})^{2} - 9 (2^{x}) + 1 = 0$ .
Let $y = 2^{x}$ : $8 y^{2} - 9 y + 1 = 0 \Rightarrow (8 y - 1) (y - 1) = 0$ .
$y = \frac{1}{8} \Rightarrow 2^{x} = 2^{- 3} \Rightarrow x = - 3$ ; $y = 1 \Rightarrow 2^{x} = 1 \Rightarrow x = 0$ .

Answer: $x = - 3$ or $x = 0$

Example 6 — a symmetric equation

Solve $(x^{2} + \frac{1}{x ^{2}}) + 2 (x + \frac{1}{x}) = 11$ (outline).

Let $y = x + \frac{1}{x}$ , so $x^{2} + \frac{1}{x ^{2}} = y^{2} - 2$ .
Substitute: $(y^{2} - 2) + 2 y = 11 \Rightarrow y^{2} + 2 y - 13 = 0$ , now an ordinary quadratic in $y$ .
Solve for $y$ , then solve $x + \frac{1}{x} = y$ (itself quadratic in $x$ ) for each value.

Answer: Reduces to two quadratics in sequence: one for $y$ , one for $x$

Remember

A job done in $h$ hours has a rate of $\frac{1}{h}$ of the job per hour. Rates of workers or pipes acting together add: $\frac{1}{h _{1}} + \frac{1}{h _{2}} = \frac{1}{h _{together}}$ .

Example 7 — two workers together

Arshia and Ibraheem finish a job together in $4$ hours; Arshia alone takes $6$ hours. How long would Ibraheem alone take?

$\frac{1}{6} + \frac{1}{x} = \frac{1}{4}$ .
$\frac{1}{x} = \frac{1}{4} - \frac{1}{6} = \frac{3 - 2}{12} = \frac{1}{12}$ .

Answer: $12$ hours

Quick Recap

●Reduce/multiply/divide: factorize everything first, then cancel
●Add/subtract: factorize denominators, use their LCM as the common denominator
●Biquadratic $a x^{4} + b x^{2} + c = 0$ : substitute $y = x^{2}$
●Exponential equations: substitute $y = a^{x}$ to get a quadratic in $y$
●Symmetric $x + \frac{1}{x}$ equations: substitute $y = x + \frac{1}{x}$ , using $x^{2} + \frac{1}{x ^{2}} = y^{2} - 2$
●Work problems: rate $= \frac{1}{time}$ ; combined rates add

10th Class Math Chapter 5 Notes

Reducing to Lowest Terms

Adding and Subtracting Algebraic Fractions

Multiplying and Dividing Algebraic Fractions

Equations Reducible to Quadratic Form

Applications: Work and Rate Problems

More 10th Class Math notes

Complex Numbers — Solved Exercises

Quadratic Equations and Inequalities

Matrices and Determinants

Algebraic Fractions

Vectors in Plane

Vectors in Plane