Simplification
Overview
Simplification forms the backbone of the Quantitative Ability section in HSSC CET. Almost every arithmetic problem—whether it involves percentages, profit-loss, or time-work—requires you to simplify expressions quickly and accurately. Examiners use simplification questions to test your command over basic operations, order of precedence, and mental math speed.
Expect 3–5 direct simplification questions, plus many indirect applications across the paper. Mastery here means faster solving everywhere else. Students must be fluent in BODMAS rules, fraction-decimal conversions, and square/cube root calculations. A single sign error or misplaced bracket can cost you the entire question, so precision matters as much as speed.
Key Concepts
- BODMAS Rule: The golden rule for order of operations—Brackets, Orders (powers/roots), Division, Multiplication, Addition, Subtraction. Always work left to right within the same priority level.
- Brackets Hierarchy: Solve innermost brackets first. Order: Vinculum (bar) → Parentheses ( ) → Curly braces { } → Square brackets [ ].
- Fractions: A fraction a/b represents a parts out of b equal parts. To add/subtract fractions, make denominators equal (LCM). To multiply, multiply numerators and denominators directly. To divide, multiply by the reciprocal.
- Decimals: Decimals are base-10 fractions. Align decimal points for addition/subtraction. Count total decimal places for multiplication. Move decimal point for division.
- Mixed Numbers: Convert mixed numbers (like 3½) to improper fractions (7/2) before calculating. Convert back only at the end.
- Square Roots: √a × √b = √(ab). √a / √b = √(a/b). To rationalize, multiply by conjugate.
- Cube Roots: ∛(a × b) = ∛a × ∛b. Useful cube roots to memorize: ∛8 = 2, ∛27 = 3, ∛64 = 4, ∛125 = 5, ∛216 = 6, ∛343 = 7, ∛512 = 8, ∛729 = 9, ∛1000 = 10.
- Surds Simplification: √50 = √(25×2) = 5√2. Always extract perfect square factors.
Formulas / Key Facts
| Concept | Formula / Fact |
|---|---|
| BODMAS order | B → O → D → M → A → S (same level: left to right) |
| Fraction addition | a/b + c/d = (ad + bc) / bd |
| Fraction multiplication | a/b × c/d = ac / bd |
| Fraction division | a/b ÷ c/d = a/b × d/c |
| Decimal to fraction | 0.25 = 25/100 = 1/4 |
| Square root property | √(a²) = a, √a × √a = a |
| Cube root property | ∛(a³) = a |
| Rationalizing denominator | 1/√a = √a / a |
| Perfect squares (1–15) | 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225 |
| Perfect cubes (1–10) | 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000 |
Worked Examples
Example 1: BODMAS with brackets
Simplify: 18 ÷ 3 × 2 + 4 × (5 – 2)
Step 1: Solve bracket → (5 – 2) = 3 Step 2: Expression becomes → 18 ÷ 3 × 2 + 4 × 3 Step 3: Division/Multiplication left to right → 18 ÷ 3 = 6 → 6 × 2 = 12 → 4 × 3 = 12 Step 4: Addition → 12 + 12 = 24
Example 2: Fraction simplification
Simplify: 2/3 + 3/4 – 1/6
Step 1: Find LCM of 3, 4, 6 → LCM = 12 Step 2: Convert fractions → 2/3 = 8/12, 3/4 = 9/12, 1/6 = 2/12 Step 3: Calculate → 8/12 + 9/12 – 2/12 = 15/12 = 5/4 or 1¼
Example 3: Square and cube roots
Simplify: √144 + ∛216 – √81
Step 1: √144 = 12 Step 2: ∛216 = 6 Step 3: √81 = 9 Step 4: Calculate → 12 + 6 – 9 = 9
Example 4: Decimal multiplication
Simplify: 2.5 × 0.4 × 3
Step 1: 2.5 × 0.4 = 1.00 = 1 Step 2: 1 × 3 = 3
Alternative: Count decimal places. 2.5 (1 place) × 0.4 (1 place) = 100 → place decimal 2 places from right = 1.00
Example 5: Mixed expression
Simplify: [48 ÷ {12 – (6 – 2)}] + 5²
Step 1: Innermost bracket → (6 – 2) = 4 Step 2: Curly brace → {12 – 4} = 8 Step 3: Square bracket → [48 ÷ 8] = 6 Step 4: Orders → 5² = 25 Step 5: Addition → 6 + 25 = 31
Common Mistakes
- Wrong BODMAS order: Students often add before multiplying. Fix: Remember "DM before AS"—division and multiplication come before addition and subtraction.
- Ignoring bracket hierarchy: Solving [ ] before ( ) leads to wrong answers. Fix: Always start from the innermost bracket and work outward.
- Forgetting to convert mixed numbers: Calculating 2½ + 1¼ directly without converting to improper fractions. Fix: Convert to 5/2 + 5/4 = 15/4 = 3¾.
- Decimal point misplacement: In 0.3 × 0.3, students write 0.9 instead of 0.09. Fix: Total decimal places in product = sum of decimal places in factors (1 + 1 = 2).
- Confusing √ and ∛: Taking cube root when square root is needed or vice versa. Fix: Read the symbol carefully—√ (two) vs ∛ (three).
- Sign errors in subtraction: Especially when subtracting negative numbers or multiple terms. Fix: Use brackets explicitly: 5 – (–3) = 5 + 3 = 8.
Quick Reference
- BODMAS: Brackets → Orders → Division/Multiplication (L→R) → Addition/Subtraction (L→R)
- LCM trick: For 2–3 small denominators, multiply them and simplify at the end if time is short
- 0.5 = 1/2, 0.25 = 1/4, 0.125 = 1/8, 0.2 = 1/5 — memorize these conversions
- √2 ≈ 1.41, √3 ≈ 1.73, √5 ≈ 2.24 — useful for approximation questions
- To simplify √72: Factor as √(36 × 2) = 6√2
- When in doubt, convert everything to fractions — decimals introduce rounding errors in multi-step problems