Operations on Numbers

This section covers the basic building blocks. A strong foundation here is non-negotiable for speed and accuracy.

Basic Operations: Addition, Subtraction, Multiplication, and Division are the pillars.
Properties of Numbers:
- Commutative Property: \( a + b = b + a \) ; \( a \times b = b \times a \)
- Associative Property: \( (a + b) + c = a + (b + c) \) ; \( (a \times b) \times c = a \times (b \times c) \)
- Distributive Property: \( a \times (b + c) = a \times b + a \times c \)
Order of Operations (BODMAS): This rule dictates the sequence for solving expressions: Brackets, Orders (powers/roots), Division, Multiplication, Addition, Subtraction.

Memorizing these formulas will save you crucial time during exams.

1. Algebraic Formulas:

\( (a+b)^2 = a^2 + 2ab + b^2 \)
\( (a-b)^2 = a^2 - 2ab + b^2 \)
\( a^2 - b^2 = (a+b)(a-b) \)
\( a^3 + b^3 = (a+b)(a^2 - ab + b^2) \)
\( a^3 - b^3 = (a-b)(a^2 + ab + b^2) \)

2. Summation Formulas:

Squaring Numbers Ending in 5: For a number like 'N5' (e.g., 45), the square will end in 25. The preceding digits are \( N \times (N+1) \). So, \( 45^2 = (4 \times 5) \) followed by 25 = 2025.
Multiplication Shortcuts:
- To multiply by 25, multiply by 100 and then divide by 4.
- To multiply by 125, multiply by 1000 and then divide by 8.
Unit Digit Method: In multiple-choice questions, often you can find the correct answer simply by calculating the unit digit of the expression and matching it with the options.

Master the Tables: Have multiplication tables up to at least 20 and squares up to 30 at your fingertips.
Practice Mental Math: Spend 10-15 minutes daily on mental calculations to boost your speed.
Approximate: When exact values are not needed, round off numbers to simplify calculations and get to the answer's vicinity quickly.
Don't Get Stuck: If a question involves overly complex calculations, mark it for review and move on. Return only if you have time.