Advanced Operations – Distributive Property and Expressions Simplification

Theory: The distributive property explains how multiplication distributes over addition or subtraction. That is, multiplying a number by a sum/difference gives the same result as multiplying it by each term separately and then adding/subtracting the results.
Example:
Question: Simplify 3 × (5 + 2)
Solution:
3 × (5 + 2) = 3 × 5 + 3 × 2 = 15 + 6 = 21

Theory: When using the distributive property, multiply the outside term with each term inside the bracket, then perform the addition or subtraction.
Example:
Question: Simplify 4 × (7 – 3)
Solution:
4 × 7 – 4 × 3 = 28 – 12 = 16

Theory: Distributive property helps break complex expressions into smaller, simpler parts for easier calculations.
Example:
Question: Simplify 6 × (8 + 5)
Solution:
6 × 8 + 6 × 5 = 48 + 30 = 78

Theory: Multiplying a number by a sum means multiplying it with each term in the sum separately, then adding.
Example:
Question: Evaluate 2 × (3 + 6)
Solution:
2 × 3 + 2 × 6 = 6 + 12 = 18

Theory: Multiplying a number by a difference means multiplying it with each term in the subtraction and then subtracting.
Example:
Question: Simplify 5 × (9 – 4)
Solution:
5 × 9 – 5 × 4 = 45 – 20 = 25

Theory: Breaking expressions into parts using distributive, associative, and commutative properties helps simplify evaluation and comparison.
Example:
Question: Simplify: 3 × (4 + 2 – 1)
Solution:
First evaluate inside: 4 + 2 – 1 = 5 → 3 × 5 = 15

Theory: Distributive property is useful in daily situations like calculating cost of repeated items or budgeting combined expenses.
Example:
Question: A pencil costs ₹5 and a pen costs ₹10. What is the cost of 3 sets each having 2 pencils and 1 pen?
Solution:
Each set: 2 × 5 + 1 × 10 = 10 + 10 = ₹20
Total = 3 × 20 = ₹60

Theory: Commutative property allows reordering of terms; associative property allows regrouping. Both don’t affect the sum.
Example:
Question: Check: (6 + 2) + 3 = 6 + (2 + 3)
Solution:
Left: 8 + 3 = 11, Right: 6 + 5 = 11 → True

Theory: Like terms are terms with the same variable or operation type and can be added or subtracted directly to simplify.
Example:
Question: Simplify: 2 + 3 + 5 – 4
Solution:
2 + 3 = 5 → 5 + 5 = 10 → 10 – 4 = 6

Theory: To remove brackets, apply the distributive property by multiplying the term outside with each term inside.
Example:
Question: Remove brackets and simplify: 2 × (6 – 1)
Solution:
2 × 6 – 2 × 1 = 12 – 2 = 10