Simplification — Study Notes for SSC CHSL
Overview
Simplification is the foundational arithmetic skill tested in every SSC CHSL exam. It forms 3–5 direct questions in Tier 1 and also appears embedded within word problems across other topics. The core challenge is computing complex arithmetic expressions accurately under time pressure — typically 30–40 seconds per question.
Mastery of simplification requires two skills: **rule knowledge** (BODMAS order, fraction operations, root properties) and **calculation speed** (mental math, pattern recognition, strategic approximation). Questions range from straightforward BODMAS chains to tricky combinations of fractions, decimals, and surds. Students who automate these rules solve faster and avoid the calculation errors that cost 0.50 marks per mistake in negative marking.
The topic connects directly to Number Systems and all Arithmetic chapters. Every percentage, ratio, or interest calculation ultimately reduces to a simplification problem. Build speed here and you accelerate across 40% of the Quant section.
Key Concepts
• **BODMAS/BODMAS hierarchy** — The universal order for evaluating expressions: Brackets (parentheses first, then curly, then square) → Of (powers, roots) → Division and Multiplication (left to right) → Addition and Subtraction (left to right). Violating this order is the #1 error source.
• **Fraction arithmetic** — Addition/subtraction requires a common denominator (LCM of denominators). Multiplication is direct (numerator × numerator / denominator × denominator). Division inverts the second fraction then multiplies. Mixed numbers convert to improper fractions before operations.
• **Decimal operations** — Align decimal points for addition/subtraction. For multiplication, ignore decimals, multiply integers, then place the decimal point counting total decimal places. Division converts the divisor to a whole number by shifting decimal points equally.
• **Square and cube roots** — Simplify √(a×b) = √a × √b and ∛(a×b) = ∛a × ∛b. Rationalisation removes roots from denominators by multiplying by the conjugate (for √) or suitable factors (for ∛). Memorise squares 1–30 and cubes 1–15.
• **Approximation strategies** — When answer choices differ by >5%, round intermediate steps to one decimal place. For addition-heavy expressions, club similar magnitude terms. For product-heavy chains, cancel common factors before multiplying.
• **Simplification identities** — (a + b)² = a² + 2ab + b², (a − b)² = a² − 2ab + b², a² − b² = (a + b)(a − b). Recognising these patterns saves 3–4 calculation steps in algebraic simplifications.
• **Operator precedence with signs** — Negative signs distribute over brackets: −(a − b) = −a + b. When multiple operations chain, preserve signs with each intermediate result. Track whether you're adding or subtracting at each BODMAS layer.
• **Strategic bracket expansion** — Don't blindly expand all brackets. First check if terms cancel or if factoring is simpler. In expressions like (x + 5)(x − 5), recognise the difference-of-squares pattern rather than expanding term-by-term.
Formulas / Key Facts
1. **BODMAS order**: Brackets → Of → Division/Multiplication (L→R) → Addition/Subtraction (L→R) 2. **Fraction addition**: a/b + c/d = (ad + bc) / bd 3. **Fraction multiplication**: (a/b) × (c/d) = ac / bd 4. **Fraction division**: (a/b) ÷ (c/d) = (a/b) × (d/c) = ad / bc 5. **Square root property**: √(ab) = √a × √b; √(a/b) = √a / √b 6. **Cube root property**: ∛(ab) = ∛a × ∛b 7. **Rationalisation**: 1/√a = √a/a; 1/(√a + √b) = (√a − √b)/(a − b) 8. **Perfect squares**: 1²=1, 2²=4, 3²=9, ..., 25²=625, 30²=900 9. **Perfect cubes**: 1³=1, 2³=8, 3³=27, 4³=64, 5³=125, 10³=1000, 15³=3375 10. **(a ± b)² = a² ± 2ab + b²**; **a² − b² = (a + b)(a − b)**
Worked Examples
**Example 1: BODMAS chain** Simplify: 15 + 18 ÷ 6 × 3 − 4
*Solution*: Step 1: Division first (left to right among ×÷): 18 ÷ 6 = 3 Expression becomes: 15 + 3 × 3 − 4 Step 2: Multiplication: 3 × 3 = 9 Expression becomes: 15 + 9 − 4 Step 3: Addition and subtraction (left to right): 15 + 9 = 24, then 24 − 4 = **20**
**Example 2: Fraction mixed operations** Simplify: (3/4 + 5/6) × (2/3 ÷ 4/9)
*Solution*: Step 1: Bracket 1 — LCM of 4 and 6 is 12: 3/4 = 9/12, 5/6 = 10/12 → 9/12 + 10/12 = 19/12 Step 2: Bracket 2 — Division converts to multiplication: 2/3 ÷ 4/9 = 2/3 × 9/4 = 18/12 = 3/2 Step 3: Multiply results: (19/12) × (3/2) = 57/24 = **19/8** or **2.375**
**Example 3: Square root rationalisation** Simplify: 1/(√7 − √3)
*Solution*: Multiply numerator and denominator by the conjugate (√7 + √3): = [1 × (√7 + √3)] / [(√7 − √3)(√7 + √3)] Denominator uses (a − b)(a + b) = a² − b²: = (√7 + √3) / (7 − 3) = (√7 + √3) / 4 = **(√7 + √3)/4**
**Example 4: Algebraic identity application** Simplify: (27)² − (23)²
*Solution*: Recognise a² − b² = (a + b)(a − b): = (27 + 23)(27 − 23) = 50 × 4 = **200** (Avoid calculating 27² = 729 and 23² = 529 separately — saves time!)
Common Mistakes
**Mistake 1: BODMAS violation — performing operations left-to-right regardless of hierarchy** *Wrong*: 10 + 2 × 5 = 12 × 5 = 60 *Correct*: Multiply first → 10 + (2 × 5) = 10 + 10 = 20. Always handle ×÷ before +−.
**Mistake 2: Adding fractions without common denominator** *Wrong*: 1/3 + 1/4 = 2/7 (adding numerators and denominators directly) *Correct*: LCM(3,4)=12 → 4/12 + 3/12 = 7/12. Never add/subtract fractions without matching denominators.
**Mistake 3: Distributing square root over addition** *Wrong*: √(9 + 16) = √9 + √16 = 3 + 4 = 7 *Correct*: √(9 + 16) = √25 = 5. The property √(ab) = √a × √b does NOT work for √(a + b).
**Mistake 4: Ignoring negative signs in brackets** *Wrong*: 20 − (15 − 8) = 20 − 15 − 8 = −3 *Correct*: 20 − (15 − 8) = 20 − 7 = 13. The minus distributes: −(15 − 8) = −15 + 8.
**Mistake 5: Premature rounding in multi-step calculations** *Wrong*: [(23.7 × 4.2) ÷ 3.1] → round each step → 24 × 4 ÷ 3 = 32 *Correct*: Compute exactly or round only the final answer. Intermediate rounding compounds error — actual answer ≈ 32.1.
Quick Reference
• **BODMAS never fails**: Brackets → Of → Divide/Multiply (L→R) → Add/Subtract (L→R). • **Fractions**: Common denominator for ±; invert-and-multiply for ÷. • **√ and ∛ multiply/divide inside, not add/subtract**: √(a×b) = √a × √b ≠ √(a+b). • **Memorise squares to 30, cubes to 15** — saves 10 seconds per root question. • **Check answer choices first** — if options differ by >10%, approximate boldly. • **Use identities over expansion**: (a±b)², a²−b² patterns cut calculation time in half.