Simplification: Formulas, Concepts, Tricks & Examples

For competitive exams, the Simplification section tests numerical ability, speed, and accuracy.

Basic Concepts to Revise

Candidates must be fluent with:

• Converting Fractions to Decimals and vice versa
• Comparing fractions
• Percentage equivalences (½ = 50%, ⅓ = 33.33%, ⅕ = 20%, etc.)
• Reciprocal values (1/2 = 0.5, 1/3 ≈ 0.333, 1/4 = 0.25, etc.)
• BODMAS rule (Bracket → Of → Division → Multiplication → Addition → Subtraction)
• Other simplifcation techniques

Percentage Decimal Fraction Conversion Table

Common Squares, Cubes & Roots to memorize

Square roots till 20 and cubes till 10

Square & Cube Roots (1 – 3)

Common Powers

Remember common powers will help you speed up & be more confident in the examination. You can refer the list of common powers you must memorize for the exam here

BODMAS

BODMAS is the standard order used to solve mathematical expressions with multiple operations.

BODMAS stands for:

• B — Brackets
• O — Orders (powers, roots, exponents)
• D — Division
• M — Multiplication
• A — Addition
• S — Subtraction

The order of solving is:

$\text{Brackets} \rightarrow \text{Orders} \rightarrow \text{Division/Multiplication} \rightarrow \text{Addition/Subtraction}$

Important: Division and multiplication have equal priority, so solve them from left to right. The same applies to addition and subtraction.

Basic Example

$6 + 2(5^2 – 3) \div 11$

Step 1: Orders

$5^2 = 25$

Step 2: Brackets

$25 – 3 = 22$

Expression becomes:

$6 + 2(22) \div 11$

Step 3: Multiplication and Division

$2 \times 22 = 44$

$44 \div 11 = 4$

Step 4: Addition

$6 + 4 = 10$

Additional Rules

• Fractions behave like brackets
• Absolute values act like grouping symbols
• Functions like $\sin(x)$ and $\log(x)$ are evaluated before addition/subtraction
• Implied multiplication: $2(3+4)$ means $2 \times (3+4)$
• Exponents are usually evaluated right to left

Example:

$2^{3^2} = 2^{(3^2)} = 2^9 = 512$

Negative Number Rule

$-3^2$ means:

$-(3^2) = -9$

But:

$(-3)^2 = 9$

Common Mistakes

• Doing addition before multiplication
• Ignoring left-to-right evaluation
• Confusing $-3^2$ with $(-3)^2$
• Skipping brackets

Simplification Techniques

Approximations

Use quick rounding to simplify calculations mentally.

• Round decimals and fractions to the nearest whole number.
• Approximate surds for faster estimation.

Example: √50 ≈ 7 instead of 7.07

Working with Fractions

Simplify fraction operations using faster comparison and conversion methods.

• Use cross multiplication to compare fractions.
• Convert mixed fractions into improper fractions before calculations.

Example: Compare 3/4 and 5/6 → 3×6 = 18, 5×4 = 20 → 5/6 is larger.

Surds & Indices

Apply standard laws to simplify powers and roots quickly.

• a^m × aⁿ = a^m+n
• (a^m)ⁿ = a^mn
• Simplify square roots into compact forms.

Example: √72 = √(36×2) = 6√2

Digital Sum & Unit Digit Techniques

Use digit patterns for quick verification and shortcuts.

• Find the last digit of large powers.
• Check divisibility and calculation accuracy rapidly.

Example: Unit digit of 7⁴ = 1

Algebraic Manipulations

Use algebraic identities and base methods to speed up multiplication.

• Multiply numbers near a base like 100 or 1000.
• Apply identities for faster mental calculations.

Examples:

• 99 × 97 = (100−1)(100−3) = 9603
• 101 × 99 = 100² − 1² = 9999

Refer Aptitude Concepts

Practice Aptitude Questions on Simplification

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