Simplification
Overview
Simplification forms the backbone of the Aptitude and Mental Ability section in TNPSC Group II/IIA. Nearly every quantitative problem—whether on percentages, profit-loss, or time-work—ultimately requires you to simplify an expression quickly and accurately. Examiners test your command of BODMAS, ability to handle fractions and decimals under time pressure, and comfort with square/cube roots.
Mastering simplification delivers two benefits: direct marks from standalone simplification questions and faster solving across all numerical problems. Your goal is error-free mental calculation within 30–45 seconds per question.
Key Concepts
- BODMAS Rule: The universal order of operations—Brackets, Orders (powers/roots), Division, Multiplication, Addition, Subtraction. Division and multiplication share equal priority (left to right); same for addition and subtraction.
- Bracket Hierarchy: Solve innermost bracket first. Order: Vinculum (bar) → Parentheses ( ) → Braces { } → Square brackets [ ].
- Fraction Basics: Numerator ÷ Denominator. To add/subtract fractions, find LCM of denominators. To multiply, multiply numerators together and denominators together. To divide, multiply by reciprocal.
- Decimal-Fraction Conversion: Move decimal point to convert; 0.25 = 25/100 = 1/4. Memorize common conversions (1/8 = 0.125, 1/6 ≈ 0.167, etc.).
- Square Roots: √(a × b) = √a × √b. Memorize squares 1–30 and cubes 1–15. For non-perfect squares, use approximation or factorization.
- Cube Roots: ∛(a × b) = ∛a × ∛b. Recognize perfect cubes: 8, 27, 64, 125, 216, 343, 512, 729, 1000.
- Simplification of Surds: Rationalize denominators when needed; √a + √b and √a − √b are conjugates.
- Recurring Decimals: 0.333... = 1/3; 0.666... = 2/3; 0.142857... = 1/7. Recognize patterns to convert quickly.
Formulas / Key Facts
| Concept | Formula / Fact |
|---|---|
| BODMAS sequence | B → O → D/M (left to right) → A/S (left to right) |
| Addition of fractions | a/b + c/d = (ad + bc) / bd |
| Multiplication of fractions | a/b × c/d = ac / bd |
| Division of fractions | a/b ÷ c/d = a/b × d/c |
| Decimal to percent | Multiply by 100 (0.45 = 45%) |
| Square root product | √(ab) = √a × √b |
| Square root quotient | √(a/b) = √a / √b |
| Rationalizing denominator | 1/(√a + √b) = (√a − √b) / (a − b) |
| 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 Mixed Operations
Problem: Simplify 18 + 12 ÷ 3 × 2 − 4
Solution:
- No brackets or orders. Start with D/M left to right.
- 12 ÷ 3 = 4
- 4 × 2 = 8
- Now A/S left to right: 18 + 8 − 4 = 22
Answer: 22
Example 2: Fraction Simplification
Problem: Simplify 3/4 + 2/5 − 1/2
Solution:
- Find LCM of 4, 5, 2 → LCM = 20
- Convert: 3/4 = 15/20; 2/5 = 8/20; 1/2 = 10/20
- Compute: 15/20 + 8/20 − 10/20 = 13/20
Answer: 13/20
Example 3: Nested Brackets with Decimals
Problem: Simplify [2.5 + {3.2 − (1.7 + 0.8)}]
Solution:
- Innermost bracket: 1.7 + 0.8 = 2.5
- Braces: 3.2 − 2.5 = 0.7
- Square bracket: 2.5 + 0.7 = 3.2
Answer: 3.2
Example 4: Square Root Simplification
Problem: Simplify √(144/25) + √49
Solution:
- √(144/25) = √144 / √25 = 12/5 = 2.4
- √49 = 7
- Sum: 2.4 + 7 = 9.4
Answer: 9.4 or 47/5
Common Mistakes
| Wrong Thinking | Correct Fix |
|---|---|
| Performing addition before division in 8 + 6 ÷ 2, getting 7 | Division comes before addition; correct answer is 8 + 3 = 11 |
| Adding fractions by adding numerators and denominators (2/3 + 1/4 = 3/7) | Find common denominator first; 2/3 + 1/4 = 8/12 + 3/12 = 11/12 |
| Writing √9 + √16 = √25 = 5 | Roots don't distribute over addition; √9 + √16 = 3 + 4 = 7 |
| Ignoring the vinculum (bar) in expressions like 8 ÷ 4̅+̅2̅ | Vinculum acts as bracket; 4 + 2 = 6 first, then 8 ÷ 6 = 4/3 |
| Converting 0.125 incorrectly as 1/12 | 0.125 = 125/1000 = 1/8 (memorize standard conversions) |
| Rushing decimal point placement in multiplication | Count total decimal places in factors; place same count in product |
Quick Reference
- BODMAS order: Brackets → Orders → Division/Multiplication (L→R) → Addition/Subtraction (L→R)
- Fraction addition: Common denominator mandatory; never add across.
- Fraction division: Flip the second fraction, then multiply.
- √(a×b) = √a × √b but √(a+b) ≠ √a + √b
- Memorize: Squares up to 30, cubes up to 15, common fraction-decimal pairs.
- Verify: Plug approximate values to check if answer is reasonable.