NCERT Solutions for Class 6 Maths Chapter 11 Algebra Exercise 11.2

Welcome to Chapter 11, Exercise 11.2 of CBSE Class 6 Mathematics! In this section, we take our first big step into real algebraic thinking. Up until now, you have used specific numbers to find the perimeters of shapes or to solve arithmetic problems. Now, you will learn the magic of generalization. By using simple letter variables like 'l', 'r', 'a', and 'b', you can write mathematical rules that apply to every single case, no matter how large or small the numbers are! This lesson will guide you through expressing geometric perimeters, circle measurements, and properties of arithmetic (like commutativity and distributivity) in a neat, algebraic language. Master these principles, and you will unlock a fundamental skill used in higher maths, physics, and computer coding.

Generalizing Math: The Power of Variables in Formulas

In arithmetic, if you want to find the perimeter of an equilateral triangle with a side of 5 cm, you add 5 + 5 + 5 = 15 cm. If the side is 8 cm, you calculate 8 + 8 + 8 = 24 cm. Doing this for every single number is exhausting!

Algebra offers a smarter way. Since an equilateral triangle always has three equal sides, we can represent the length of any side with a variable, say 'l'. Instead of writing separate additions for every triangle, we write a single general rule: Perimeter = 3 times l (or 3l).

Here, 'l' is a variable because its value can change depending on which triangle we are looking at, while '3' is a constant because a triangle always has three sides. This approach helps us write short, elegant, and universal formulas for shapes and mathematical operations.

Step-by-Step Process: Writing a Geometric Rule

  1. Step 1: Identify the Constant Properties — Determine what stays the same. For example, if you are looking at a regular pentagon, it will always have exactly 5 equal sides. The number 5 is your constant.
  2. Step 2: Define your Variable — Choose a letter to represent the changing quantity. Let 's' represent the length of one side of the pentagon. Clearly state: 'Let the side of the regular pentagon be s'.
  3. Step 3: Write the General Rule — Combine the constant and variable. Perimeter is the total boundary length, which is the sum of all sides. Since there are 5 equal sides of length 's', the Rule for Perimeter = 5 × s = 5s.

Expressing Rules of Arithmetic Using Variables

  • Commutative Property of Addition: If we add two numbers, the order doesn't matter. (e.g., 4 + 5 = 5 + 4). Using variables 'a' and 'b', we express this universal rule as: a + b = b + a.
  • Commutative Property of Multiplication: When multiplying two numbers, changing the order gives the same result. (e.g., 3 × 8 = 8 × 3). Written algebraically: a × b = b × a.
  • Distributive Property of Multiplication over Addition: To multiply a number by the sum of two numbers, we can multiply it by each addend individually and then add. (e.g., 2 × (3 + 4) = (2 × 3) + (2 × 4)). Algebraically, we express this as: a × (b + c) = a × b + a × c.

Exam Tip: Avoiding Common Algebraic Notation Mistakes

When writing algebraic expressions for formulas, keep these golden rules in mind to score full marks:

  1. Do not use the multiplication sign unnecessarily: In algebra, '3 × l' is written simply as '3l'. Writing '3 × l' is not wrong, but '3l' is the standard, cleaner format.
  2. Always define your variables: If a question asks you to express a rule, always start by declaring what your letter stands for. For example, write: 'Let the radius of the circle be r'.
  3. Do not mix up multiplication with addition: 'l + l + l' is '3l' (multiplication by 3). It is NOT 'l³' (which means l multiplied by itself three times). Keep your addition rules distinct from multiplication!

Practice Questions with Solutions

  • Q: The side of a square is denoted by 's'. Express the perimeter of the square using 's'. A: Step 1: Identify the geometric property of a square. A square has 4 equal sides. Step 2: Use the given variable. Let the length of one side of the square be 's'. Step 3: Write the formula. Perimeter is the sum of the lengths of all sides. Thus, Perimeter = s + s + s + s = 4 × s. Final answer: Perimeter of the square = 4s.
  • Q: A regular octagon has eight equal sides. If the length of each side is represented by 'x', write the rule for the perimeter of the octagon. A: Step 1: Note that a regular octagon always has 8 equal sides. Step 2: The length of each side is given as 'x'. Step 3: Express the perimeter. Perimeter = sum of all 8 equal sides = 8 × x. Final answer: Perimeter of the regular octagon = 8x.
  • Q: The diameter of a circle (d) is twice its radius (r). Write this relationship using variables and algebra. A: Step 1: Identify the given terms. Let 'd' be the diameter and 'r' be the radius. Step 2: Set up the algebraic relationship. The diameter is twice (2 times) the radius. Step 3: Form the expression. d = 2 × r. Final answer: d = 2r.
  • Q: If we have three numbers 5, 10, and 15, show how the distributive property works using variables a, b, and c. A: Step 1: State the general formula for the distributive property: a × (b + c) = a × b + a × c. Step 2: Let a = 5, b = 10, and c = 15. Step 3: Solve the Left-Hand Side (LHS): 5 × (10 + 15) = 5 × 25 = 125. Step 4: Solve the Right-Hand Side (RHS): (5 × 10) + (5 × 15) = 50 + 75 = 125. Step 5: Since LHS = RHS, the property is verified. Final answer: The generalized algebraic formula is a × (b + c) = a × b + a × c.

Frequently Asked Questions

What is a variable in Class 6 Algebra?

A variable is a letter or symbol (like x, y, l, or r) used to represent a number whose value is not fixed and can change depending on the situation.

How do we write the perimeter of a regular shape using algebra?

To write the perimeter, multiply the number of equal sides of the shape by the variable representing the length of a single side. For example, a regular hexagon with side 's' has a perimeter of 6s.

Why do we use variables instead of regular numbers in formulas?

Variables allow us to write a single general rule that works for all possible values. This is much more efficient than writing a new calculation for every different number.