Unit 5 Class 9 Math Solutions Sindh Board

Unit 5 Class 9 Math Solutions Sindh Board are available below for Exercises 5.1 to 5.3 and the Unit 5 Review Exercise. Students can view and download each PDF to study complete step-by-step solutions.

This unit covers HCF and LCM of algebraic expressions, operations on algebraic fractions, and square roots of algebraic expressions. For other units, visit our Class 9 Math Notes Sindh Board page.

Unit 5 Class 9 Math Exercise Solutions

Select an exercise below to view or download its complete PDF solutions.

Unit 5 Quick Overview

unit 5 class 9 math solutions sindh board

What Is Algebraic Manipulation?

Algebraic manipulation means changing an algebraic expression into a simpler or more useful form without changing its value.

In this unit, students learn how to find the HCF and LCM of algebraic expressions, simplify algebraic fractions, perform operations on fractions and calculate square roots of algebraic expressions.

For example:\[ 6x^2+12x \]

can be simplified by taking the common factor \(6x\):\[ 6x^2+12x=6x(x+2) \]

Similarly, consider the algebraic fraction:\[ \frac{x^2-9}{x^2+5x+6} \]

It can be simplified by factorizing the numerator and denominator:\[ \frac{x^2-9}{x^2+5x+6} = \frac{(x-3)(x+3)}{(x+2)(x+3)} \]

Cancel the common factor \(x+3\):\[ \frac{x^2-9}{x^2+5x+6} = \frac{x-3}{x+2} \]

Factorization is therefore an important part of algebraic manipulation.

Exercise 5.1 – HCF and LCM of Algebraic Expressions

Exercise 5.1 focuses on finding the highest common factor and least common multiple of algebraic expressions.

Students learn how to find HCF and LCM by factorization and division methods.

Highest Common Factor

The highest common factor, or HCF, is the greatest expression that divides all the given expressions exactly.

For example, consider:\[ 12x^2y \]

and:\[ 18xy^2 \]

Factorize the numerical coefficients:\[ 12=2^2\times3 \] \[ 18=2\times3^2 \]

The common numerical factor is:\[ 2\times3=6 \]

The common algebraic factors are \(x\) and \(y\). Therefore:\[ \operatorname{HCF}=6xy \]

When finding the HCF, students should select only the common factors with their smallest powers.

HCF by Factorization

Consider the expressions:\[ x^2-9 \]

and:\[ x^2+5x+6 \]

Factorize the first expression:\[ x^2-9=(x-3)(x+3) \]

Factorize the second expression:\[ x^2+5x+6=(x+2)(x+3) \]

The common factor is:\[ x+3 \]

Therefore:\[ \operatorname{HCF}=x+3 \]

HCF by Division Method

The division method may be used when algebraic expressions are difficult to factorize directly.

Arrange both expressions in descending powers.
Divide the larger expression by the smaller expression.
Divide the previous divisor by the remainder.
Continue the process until the remainder becomes zero.
The last non-zero divisor is the HCF.

Students should include zero coefficients when a power of the variable is missing.

Least Common Multiple

The least common multiple, or LCM, is the smallest expression that is exactly divisible by all the given expressions.

For example, consider:\[ 6x^2y \]

and:\[ 8xy^3 \]

Factorize the coefficients:\[ 6=2\times3 \] \[ 8=2^3 \]

For the LCM, take every factor with its greatest power:\[ \operatorname{LCM} = 2^3\times3\times x^2\times y^3 \]

Therefore:\[ \operatorname{LCM}=24x^2y^3 \]

Relationship Between HCF and LCM

For two algebraic expressions \(p(x)\) and \(q(x)\), the relationship is:\[ \operatorname{HCF}\times\operatorname{LCM} = p(x)\times q(x) \]

This relationship can be used to calculate an unknown HCF or LCM.\[ \operatorname{LCM} = \frac{p(x)q(x)}{\operatorname{HCF}} \]

Applications of HCF and LCM

HCF is generally used when a quantity must be divided into the largest equal groups or pieces.

LCM is generally used when repeated events must occur together again.

Finding the longest equal pieces uses HCF.
Finding when repeating events occur together uses LCM.

Students should read each question carefully before deciding whether HCF or LCM is required.

Exercise 5.2 – Operations on Algebraic Fractions

Exercise 5.2 covers simplification, addition, subtraction, multiplication and division of algebraic fractions.

An algebraic fraction has the form:\[ \frac{p(x)}{q(x)} \]

where \(p(x)\) and \(q(x)\) are algebraic expressions and:\[ q(x)\neq0 \]

For example:\[ \frac{x+2}{x-3} \]

is an algebraic fraction, provided:\[ x\neq3 \]

Simplification of Algebraic Fractions

An algebraic fraction is simplified by factorizing the numerator and denominator and cancelling common factors.

For example:\[ \frac{x^2-4}{x^2+x-6} \]

Factorize the numerator:\[ x^2-4=(x-2)(x+2) \]

Factorize the denominator:\[ x^2+x-6=(x-2)(x+3) \]

Therefore:\[ \frac{x^2-4}{x^2+x-6} = \frac{(x-2)(x+2)}{(x-2)(x+3)} \]

Cancel the common factor \(x-2\):\[ \frac{x^2-4}{x^2+x-6} = \frac{x+2}{x+3} \]

Only common factors can be cancelled. Terms joined by addition or subtraction cannot be cancelled individually.

Addition of Algebraic Fractions

Fractions with the same denominator are added by combining their numerators:\[ \frac{a}{m}+\frac{b}{m} = \frac{a+b}{m} \]

When the denominators are different, first find their LCM.

For example:\[ \frac{1}{x}+\frac{1}{x+1} \]

The common denominator is:\[ x(x+1) \]

Therefore:\[ \frac{1}{x}+\frac{1}{x+1} = \frac{x+1+x}{x(x+1)} \] \[ = \frac{2x+1}{x(x+1)} \]

Subtraction of Algebraic Fractions

The common-denominator method is also used for subtraction.

For example:\[ \frac{3}{x}-\frac{2}{x+1} \]

Using the common denominator \(x(x+1)\):\[ \frac{3}{x}-\frac{2}{x+1} = \frac{3(x+1)-2x}{x(x+1)} \] \[ = \frac{3x+3-2x}{x(x+1)} \] \[ = \frac{x+3}{x(x+1)} \]

Students should use brackets around the numerator of the fraction being subtracted.

Multiplication of Algebraic Fractions

Algebraic fractions are multiplied by multiplying their numerators and denominators:\[ \frac{a}{b}\times\frac{c}{d} = \frac{ac}{bd} \]

Factorization should be completed before multiplication so that common factors can be cancelled.

For example:\[ \frac{x^2-4}{x^2-1} \times \frac{x+1}{x+2} \]

Factorize:\[ = \frac{(x-2)(x+2)}{(x-1)(x+1)} \times \frac{x+1}{x+2} \]

Cancel the common factors:\[ = \frac{x-2}{x-1} \]

Division of Algebraic Fractions

To divide by an algebraic fraction, multiply by its reciprocal:\[ \frac{a}{b}\div\frac{c}{d} = \frac{a}{b}\times\frac{d}{c} \]

For example:\[ \frac{x^2-9}{x^2-4} \div \frac{x+3}{x+2} \]

Change division into multiplication:\[ = \frac{x^2-9}{x^2-4} \times \frac{x+2}{x+3} \]

Factorize:\[ = \frac{(x-3)(x+3)}{(x-2)(x+2)} \times \frac{x+2}{x+3} \]

Cancel the common factors:\[ = \frac{x-3}{x-2} \]

Students should reverse only the second fraction when changing division into multiplication.

Exercise 5.3 – Square Roots of Algebraic Expressions

Exercise 5.3 teaches students how to find the square root of an algebraic expression using factorization and division methods.

Students also learn how to complete an expression so that it becomes a perfect square.

Square Root by Factorization

An algebraic expression is first written as the square of another expression.

For example:\[ x^2+6xy+9y^2 \]

Use the identity:\[ a^2+2ab+b^2=(a+b)^2 \]

Therefore:\[ x^2+6xy+9y^2=(x+3y)^2 \]

Hence:\[ \sqrt{x^2+6xy+9y^2}=x+3y \]

Similarly:\[ x^2-10xy+25y^2=(x-5y)^2 \]

Therefore:\[ \sqrt{x^2-10xy+25y^2}=x-5y \]

Square Root of a Product

If an expression is written as a product of perfect-square factors, take the square root of each factor.

For example:\[ (4x^2-4x+1)(9x^2-54x+81) \]

Factorize the first expression:\[ 4x^2-4x+1=(2x-1)^2 \]

Factorize the second expression:\[ 9x^2-54x+81=(3x-9)^2 \]

Therefore:\[ \sqrt{(4x^2-4x+1)(9x^2-54x+81)} = (2x-1)(3x-9) \]

Square Root by Division Method

The division method is useful for lengthy algebraic expressions that cannot be recognized easily as perfect squares.

Arrange the expression in descending powers.
Find the square root of the first term.
Write it as the first term of the required root.
Square it and subtract.
Double the root obtained so far.
Find the next term of the root.
Continue until the remainder becomes zero.

For example:\[ 4x^4+12x^3-19x^2-42x+49 \]

has the square root:\[ 2x^2+3x-7 \]

because:\[ (2x^2+3x-7)^2 = 4x^4+12x^3-19x^2-42x+49 \]

Students should include zero coefficients for any missing powers.

Making an Expression a Perfect Square

Some questions ask what must be added to or subtracted from an expression to make it a perfect square.

For example:\[ x^2+6x \]

Half the coefficient of \(x\) is:\[ \frac{6}{2}=3 \]

Square the result:\[ 3^2=9 \]

Therefore:\[ x^2+6x+9=(x+3)^2 \]

Thus, \(9\) must be added.

Students may also compare an expression with:\[ (a+b+c)^2 = a^2+b^2+c^2+2ab+2bc+2ca \]

to find missing terms or unknown coefficients.

Unit 5 Review Exercise

The Unit 5 Review Exercise combines the important concepts from all three exercises.

HCF by factorization
HCF by division
LCM by factorization
LCM by division
Relationship between HCF and LCM
Applications of HCF and LCM
Simplification of algebraic fractions
Addition and subtraction of algebraic fractions
Multiplication and division of algebraic fractions
Square roots by factorization
Square roots by division
Completing algebraic expressions as perfect squares
Multiple-choice and short questions

Students should attempt the review exercise after completing Exercises 5.1 to 5.3.

Important Formulas from Unit 5

The relationship between two algebraic expressions and their HCF and LCM is:\[ \operatorname{HCF}\times\operatorname{LCM} = \text{Product of the two expressions} \]

For algebraic fractions:\[ \frac{a}{b}+\frac{c}{d} = \frac{ad+bc}{bd} \] \[ \frac{a}{b}-\frac{c}{d} = \frac{ad-bc}{bd} \] \[ \frac{a}{b}\times\frac{c}{d} = \frac{ac}{bd} \] \[ \frac{a}{b}\div\frac{c}{d} = \frac{a}{b}\times\frac{d}{c} \]

Useful perfect-square identities include:\[ a^2+2ab+b^2=(a+b)^2 \] \[ a^2-2ab+b^2=(a-b)^2 \] \[ (a+b+c)^2 = a^2+b^2+c^2+2ab+2bc+2ca \]

Common Mistakes to Avoid

Selecting the greatest powers when finding the HCF
Selecting the smallest powers when finding the LCM
Forgetting to factorize expressions completely
Confusing HCF with LCM in application questions
Cancelling individual terms instead of common factors
Adding fractions without finding a common denominator
Forgetting brackets during subtraction
Failing to factorize before multiplication
Reversing the first fraction instead of the second during division
Ignoring restrictions on denominator values
Assuming an expression is a perfect square without checking it
Omitting missing powers during the square-root division method
Stopping before the remainder becomes zero

Students should verify their simplified expressions whenever possible.

How to Prepare Unit 5 for Exams

Begin by revising factorization because it is used throughout this unit.

For HCF questions, select common factors with their smallest powers. For LCM questions, include every factor with its greatest power.

Practise identifying whether an application question requires HCF or LCM. Questions involving the largest equal groups generally require HCF, while questions involving repeating events generally require LCM.

For algebraic fractions, factorize before cancelling any factors. Find the LCM of the denominators before addition or subtraction.

For square roots, first check whether the expression matches a known perfect-square identity. Use the division method when the expression is lengthy or cannot be recognized easily.

After attempting the textbook questions, use the Unit 5 Class 9 Math Solutions Sindh Board PDFs to check each step and correct mistakes.

Why These Unit 5 Solutions Are Helpful

Understand HCF and LCM methods
Solve application questions involving HCF and LCM
Simplify algebraic fractions correctly
Perform operations on algebraic fractions
Find square roots by factorization
Understand the square-root division method
Check homework answers
Prepare for class tests and board examinations
View or download each exercise separately

Students should use the solutions to understand the complete working rather than copy final answers only.

Frequently Asked Questions

What is the name of Unit 5 in Class 9 Sindh Board Mathematics?

The name of Unit 5 is Algebraic Manipulation.

How many exercises are included in Unit 5?

Unit 5 contains three exercises, from Exercise 5.1 to Exercise 5.3, followed by a Review Exercise.

What topics are covered in Exercise 5.1?

Exercise 5.1 covers HCF and LCM of algebraic expressions using factorization and division methods.

What topics are covered in Exercise 5.2?

Exercise 5.2 covers simplification, addition, subtraction, multiplication and division of algebraic fractions.

What topics are covered in Exercise 5.3?

Exercise 5.3 covers square roots of algebraic expressions using factorization and division methods.

Are all Unit 5 exercises solved on this page?

Yes. Separate PDFs are provided for Exercises 5.1, 5.2, 5.3 and the Unit 5 Review Exercise.

Can students download the solution PDFs?

Yes. Students can view and download each exercise-wise PDF on mobile phones, tablets and computers.

Related Class 9 Math Resources

Disclaimer

These solutions are prepared for educational support. Students should also consult their official Sindh Textbook Board Mathematics book and follow the instructions provided by their teachers.

Final Words

The Unit 5 Class 9 Math Solutions Sindh Board provide complete exercise-wise help with HCF, LCM, algebraic fractions and square roots of algebraic expressions.

Students should first attempt each question independently and then use the PDFs to check their working, understand missing steps and correct calculation errors.

For solutions to all other units, visit the Class 9 Math Notes Sindh Board page.