Introduction
Unit 5 of the new PECTAA Class 10 Mathematics book covers algebraic fractions, also called rational expressions. A rational expression is simply a fraction whose numerator and denominator are polynomials.
🎓 Online Classes with Experienced Experts
This unit strengthens the factorization skills learned earlier and applies them to simplifying, combining, and solving equations with algebraic fractions. It also covers special equations that look complicated at first but reduce to quadratic equations through substitution. These notes follow the new book for session 2026-27 and suit students of all Punjab boards.
What You Will Learn
By the end of this unit, students will be able to:
- describe rational expressions;
- factorize and simplify rational expressions;
- add, subtract, multiply, and divide algebraic fractions;
- solve exponential equations, reciprocal equations, and other equations reducible to quadratic form;
- apply rational equations to real-world problems such as work rates and shared tasks.
Key Topics Explained
1. Rational Expressions
A rational expression is the quotient P(x) ÷ Q(x) of two polynomials, where Q(x) ≠ 0. Examples include 2 ÷ x, (x + 1) ÷ (x − 2), and similar fractions with polynomial numerators and denominators.
2. Simplification
To simplify a rational expression, factorize both the numerator and the denominator completely, then cancel any common factors. This gives the fraction in its lowest form.
3. Addition and Subtraction
For like denominators, add or subtract the numerators directly. For unlike denominators, first find the LCM of the denominators, rewrite each fraction with that LCM, and then combine the numerators.
4. Multiplication and Division
To multiply rational expressions, factorize each numerator and denominator, cancel common factors across the fractions, and multiply what remains. To divide by a rational expression, multiply by its reciprocal, exactly as with ordinary fractions.
5. Equations Reducible to Quadratic Form
Some equations are not quadratic in their given form, but a substitution turns them into one. The book covers four types:
- ax²ⁿ + bxⁿ + c = 0 (n = 2): substitute y = xⁿ to get a quadratic in y.
- Exponential equations, where the variable is in the exponent, such as 4ˣ − 3·2ˣ + 128 = 0: substitute y = 2ˣ.
- Reciprocal equations, which contain a term and its reciprocal, such as x + 1/x.
- (x + a)(x + b)(x + c)(x + d) = k, where a + b = c + d: substitute y = (x + a)(x + b) or a similar product.
In each case, after solving for y, substitute back to find x.
6. Real-World Applications
Rational equations model situations such as the combined work rate of two people, the time to fill a tank with two pipes, or shared task problems, where individual rates are added as fractions.
Worked Examples
Example 1 (Simplification): Reduce (18a³x² − 12a²x) ÷ (12a²x² − 8ax) to lowest form.
Factor: 6a²x(3a − 2x) ÷ 4ax(3a − 2x) = 3a ÷ 2.
Example 2 (Addition, unlike denominators): Simplify x/4 + (x + 1)/6.
LCM of 4 and 6 is 12. = (3x + 2(x + 1)) ÷ 12 = (5x + 2) ÷ 12.
Example 3 (Division): Simplify (x² − 9) ÷ (x + 2) ÷ (x − 3) ÷ (x + 2).
= (x − 3)(x + 3) ÷ (x + 2) × (x + 2) ÷ (x − 3) = x + 3.
Example 4 (Reducible equation): Solve 2x⁴ − 11x² + 12 = 0.
Let y = x². Then 2y² − 11y + 12 = 0, so (2y − 3)(y − 4) = 0, giving y = 3/2 or y = 4.
So x² = 3/2 gives x = ±√(3/2), and x² = 4 gives x = ±2.
Example 5 (Work rate): If A can finish a job in 6 days and B in 12 days, how long do they take together?
Combined rate = 1/6 + 1/12 = 2/12 + 1/12 = 3/12 = 1/4. So together they take 4 days.
Important Formulas at a Glance
| Concept | Rule |
|---|---|
| Simplify | Factorize both parts, cancel common factors |
| Add/subtract | Use the LCM of the denominators |
| Multiply | Cancel first, then multiply remaining factors |
| Divide | Multiply by the reciprocal |
| ax²ⁿ + bxⁿ + c = 0 | Substitute y = xⁿ |
| Combined work rate | 1/(time A) + 1/(time B) = 1/(time together) |
Exercises in Unit 5
| Exercise | Main Focus |
|---|---|
| 5.1 | Reducing rational expressions to lowest form |
| 5.2 | Addition and subtraction of rational expressions |
| 5.3 | Multiplication and division of rational expressions |
| 5.4 | Equations reducible to quadratic form |
| 5.5 | Real-world problems with rational equations |
| Review Exercise 5 | MCQs, short and long questions |
Exam Tips
- Factorize fully before cancelling, since a partial factorization hides common factors.
- Always find the LCM before adding or subtracting unlike fractions.
- Choose the substitution carefully in reducible equations, and remember to substitute back.
- In work-rate problems, add the rates (1/time), never the times themselves.
About HSA Notes
HSA Notes is a free educational platform for secondary and higher secondary students in Pakistan. It publishes solved notes, model papers, and study material aligned with the NBF, PECTAA, PCTAA, and FBISE curricula, covering Mathematics, Physics, Chemistry, Biology, Computer Science, and English. Our solutions are written by our own team, giving students clear, original, step-by-step help.
Online Classes
Students who want guided learning can join our online classes run by Hamdard Science Academy. The classes follow the new PECTAA book unit by unit, so Unit 5 methods, examples, and exercises are taught in a clear order alongside these notes. To ask about the schedule, fees, or admission, email hsanotes48@gmail.com.
Conclusion
Algebraic Fractions rewards students who are comfortable with factorization. With careful cancelling, correct LCM work, and steady practice on reducible equations, this unit can become a reliable source of marks. Free solved exercises and PDF notes are available on HSA Notes for all Punjab boards, and online classes are there if you need extra support.
1 thought on “Unit 5: Class 10 Math PECTAA Notes New Book”