Simplification
Simplification - Methods to simplify basic mathematical operations, BODMAS rule, decimals, fractions, percentages
Learning Content
🔢 Compression - The Basics
Simplify Mathematical Functions and Find Answers!
📚 What is compression?
SimplificationIt is the process of simplifying a mathematical expression to find the final answer.
⭐ BODMAS RULE - VERY IMPORTANT!
| Writing | English | Tamil | Code | priority |
|---|---|---|---|---|
| B | Brackets | Parentheses | ( ), { }, [ ] | 1️⃣ First |
| O | Orders (Powers/Roots) | layers/sources | ², ³, √ | 2️⃣ Second |
| D | Division | Division | ÷, / | 3️⃣ Third |
| M | Multiplication | multiplication | ×, * | 3️⃣ Third |
| A | Addition | addition | + | 4️⃣ Fourth |
| S | Subtraction | deduction | - | 4️⃣ Fourth |
📐 Brackets sort
| sequence | Code | Name (English) | Name (Tamil) |
|---|---|---|---|
| 1️⃣ | —(Vinculum/Bar) | Vinculum | Line/stripe |
| 2️⃣ | ( ) | Parentheses / Round Brackets | Small blockage |
| 3️⃣ | { } | Curly Brackets / Braces | Coil enclosure |
| 4️⃣ | [ ] | Square Brackets | Square enclosure |
🔢 Basic number types
📊 Integers
- Even numbers:+1, +2, +3...
- Opposite Numbers:-1, -2, -3...
- Zero:0
📊 Fractions
- Proper fraction:Volume < Area (2/5)
- Equivalent Fraction:Volume > Area (7/3)
- Mixed Number:2⅓
➕➖ Coding rules
| function | fate | example |
|---|---|---|
| (+) × (+) | = + | 3 × 2 = 6 |
| (-) × (-) | = + | (-3) × (-2) = 6 |
| (+) × (-) | = - | 3 × (-2) = -6 |
| (-) × (+) | = - | (-3) × 2 = -6 |
| (+) ÷ (+) | = + | 6 ÷ 2 = 3 |
| (-) ÷ (-) | = + | (-6) ÷ (-2) = 3 |
| (+) ÷ (-) | = - | 6 ÷ (-2) = -3 |
| (-) ÷ (+) | = - | (-6) ÷ 2 = -3 |
📐 Condensation - Key Formulas
Essential Formulas for TNPSC Exam!
🔢 Fractions Formulas
| function | formula | example |
|---|---|---|
| addition | a/b + c/d = (ad + bc) / bd | 1/2 + 1/3 = (3+2)/6 = 5/6 |
| deduction | a/b - c/d = (ad - bc) / bd | 3/4 - 1/2 = (6-4)/8 = 2/8 = 1/4 |
| multiplication | a/b × c/d = ac / bd | 2/3 × 3/4 = 6/12 = 1/2 |
| Division | a/b ÷ c/d = a/b × d/c = ad / bc | 2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6 |
📊 Decimals Formulas
| Change | method | example |
|---|---|---|
| Fraction → Decimal | Divide the volume by the area | 3/4 = 3 ÷ 4 = 0.75 |
| Decimal → Fraction | Divide by 10, 100... according to decimal places | 0.25 = 25/100 = 1/4 |
| Percent → Decimal | Divide by 100 | 25% = 25/100 = 0.25 |
| Decimal → Percent | Multiply by 100 | 0.35 = 0.35 × 100 = 35% |
⚡ Exponents Formulas (Powers/Exponents)
| fate | formula | example |
|---|---|---|
| Multiplication rule | aᵐ × aⁿ = aᵐ⁺ⁿ | 2³ × 2² = 2⁵ = 32 |
| Division rule | aᵐ ÷ aⁿ = aᵐ⁻ⁿ | 2⁵ ÷ 2² = 2³ = 8 |
| Layer upon layer | (aᵐ)ⁿ = aᵐˣⁿ | (2²)³ = 2⁶ = 64 |
| Zero layer | a⁰ = 1 | 5⁰ = 1 |
| opposite layer | a⁻ⁿ = 1/aⁿ | 2⁻³ = 1/2³ = 1/8 |
| fraction layer | a^(1/n) = ⁿ√a | 8^(1/3) = ³√8 = 2 |
🔲 Square and weight formulas
| formula | Expansion |
|---|---|
| (a + b)² | = a² + 2ab + b² |
| (a - b)² | = a² - 2ab + b² |
| a² - b² | = (a + b)(a − b) |
| (a + b)³ | = a³ + 3a²b + 3ab² + b³ |
| (a - b)³ | = a³ - 3a²b + 3ab² - b³ |
| a³ + b³ | = (a + b)(a² - ab + b²) |
| a³ - b³ | = (a - b)(a² + ab + b²) |
📈 Percentage Formulas
| Account type | formula |
|---|---|
| Y% of X | = (X × Y) / 100 |
| What % of X is Y? | = (X / Y) × 100 |
| Increase % | = [(New – Old) / Old] × 100 |
| Reduction % | = [(Old – New) / Old] × 100 |
🔄 Key Change Schedule
| Fraction | Decimal | percent |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/3 | 0.333... | 33.33% |
| 1/4 | 0.25 | 25% |
| 1/5 | 0.2 | 20% |
| 1/6 | 0.166... | 16.67% |
| 1/8 | 0.125 | 12.5% |
| 1/10 | 0.1 | 10% |
| 2/3 | 0.666... | 66.67% |
| 3/4 | 0.75 | 75% |
| 4/5 | 0.8 | 80% |
📝 Shortening - 10 Key Examples
TNPSC Model Questions with Step by Step Solutions!
📌 Example 1: BODMAS Basis
Question:25 - 5 × 3 + 10 ÷ 2 = ?
Step 1: Multiplication → 5 × 3 = 15
Step 2: Divide → 10 ÷ 2 = 5
Step 3: Coimbatore = 25 - 15 + 5
Step 4: Cross from left → 25 - 15 = 10
Step 5: 10 + 5 =15
📌 Example 2: Parentheses
Question:(15 + 5) × (12 - 4) ÷ 4 = ?
Step 1: First bracket → (15 + 5) = 20
Step 2: Second bracket → (12 - 4) = 8
Step 3: Coimbatore = 20 × 8 ÷ 4
Step 4: Cross from left → 20 × 8 = 160
Step 5: 160 ÷ 4 =40
📌 Example 3: Fractional addition
Question:2/3 + 3/4 + 1/6 = ?
Step 1: LCM(3, 4, 6) = 12
Step 2: 2/3 = 8/12
Step 3: 3/4 = 9/12
Step 4: 1/6 = 2/12
Step 5: (8 + 9 + 2)/12 = 19/12 =1 7/12
📌 Example 4: Multiplication of Decimals
Question:0.25 × 0.4 × 100 = ?
Step 1: 0.25 × 0.4 = 0.10
Step 2: 0.10 × 100 =10
📌 Example 5: Stacks
Question:2³ + 3² - 4¹ = ?
Step 1: 2³ = 2 × 2 × 2 = 8
Step 2: 3² = 3 × 3 = 9
Step 3: 4¹ = 4
Step 4: 8 + 9 - 4 =13
📌 Example 6: Percent
Question:20% of 250 + 30% of 150 = ?
Step 1: 20% of 250 = 250 × 20/100 = 50
Step 2: 30% of 150 = 150 × 30/100 = 45
Step 3: 50 + 45 =95
📌 Example 7: Mixed Fraction
Question:3½ × 2⅓ = ?
Step 1: 3½ = 7/2 (Convert compound to equivalent fraction)
Step 2: 2⅓ = 7/3
Step 3: 7/2 × 7/3 = 49/6
Step 4: 49/6 =8⅙
📌 Example 8: By (Square Root)
Question:√144 + √81 - √49 = ?
Step 1: √144 = 12
Step 2: √81 = 9
Step 3: √49 = 7
Step 4: 12 + 9 - 7 =14
📌 Example 9: Complex BODMAS
Question:[25 - {10 + (6 - 4)}] × 3 = ?
Step 1: Small bracket → (6 - 4) = 2
Step 2: Spiral enclosure → {10 + 2} = 12
Step 3: Squaring → [25 - 12] = 13
Step 4: 13 × 3 =39
📌 Example 10: TNPSC Model
Question:(1/2 + 1/3) ÷ (1/4 - 1/6) + 2 = ?
Step 1: 1/2 + 1/3 = 3/6 + 2/6 = 5/6
Step 2: 1/4 - 1/6 = 3/12 - 2/12 = 1/12
Step 3: (5/6) ÷ (1/12) = 5/6 × 12/1 = 60/6 = 10
Step 4: 10 + 2 =12
💡 Practice Tips:
- Always follow BODMAS rule
- Solve the parentheses from the inside out
- Change the fractions to common denominator
- Decimals can be converted into fractions and simplified
⚡ Shortcuts - Shortcuts & Tricks
Use These Shortcuts to Save Time in TNPSC Exam!
🚀 Shortcut 1: Multiply by 11
| Rule: | Place the sum of the two digits of a 2-digit number between them |
| Example: | 45 × 11 = ? 4__5 → in the middle (4+5) = 9 →495 67 × 11 = ? 6__7 → middle (6+7) = 13 → 6(1+6)(7-10+13) →737 |
🚀 Shortcut 2: Divide by 5
| Rule: | Multiply the number by 2 and divide by 10 (or change the decimal point) |
| Example: | 235 ÷ 5 = ? 235 × 2 = 470 470 ÷ 10 =47 |
🚀 Shortcut 3: Multiply by 25
| Rule: | Multiply the number by 100 and divide by 4 |
| Example: | 48 × 25 = ? 48 × 100 = 4800 4800 ÷ 4 =1200 |
🚀 Shortcut 4: Square (Nearest Ten)
| Rule: | Use the formula (a+b)² or (a-b)² |
| Example: | 47² = ? 47 = (50 - 3) 47² = 50² - 2×50×3 + 3² = 2500 - 300 + 9 =2209 |
🚀 Shortcut 5: Multiply by 9
| Rule: | Multiply the number by 10 and subtract the same number |
| Example: | 67 × 9 = ? 67 × 10 = 670 670 - 67 =603 |
🚀 Shortcut 6: Percentage Quick Calculation
| percent | Quick method | example |
|---|---|---|
| 10% | Divide by 10 | 10% of 450 = 45 |
| 5% | Find 10% and divide by 2 | 5% of 450 = 22.5 |
| 15% | 10% + 5% | 15% of 450 = 45 + 22.5 = 67.5 |
| 20% | Divide by 5 | 20% of 450 = 90 |
| 25% | Divide by 4 | 25% of 400 = 100 |
| 50% | Divide by 2 | 50% of 450 = 225 |
| 75% | 50% + 25% | 75% of 400 = 300 |
| 33⅓% | Divide by 3 | 33⅓% of 450 = 150 |
🚀 Shortcut 7: Fractional comparison
Which is bigger? a/b or c/d
| Rule: | Do cross multiplication - a×d vs b×c |
| Example: | 3/7 vs 4/9 3 × 9 = 27 7 × 4 = 28 27 < 28, therefore4/9 > 3/7 |
🚀 Shortcut 8: Multiplication of consecutive numbers
| Rule: | n × (n+1) = n² + n |
| Example: | 19 × 20 = ? = 19² + 19 = 361 + 19 =380 |
🚀 Shortcut 9: The square of a number ending in 5
| Rule: | Multiply the first digit by its next digit, then add 25 |
| Example: | 35² = ? 3 × 4 = 12 Then add 25 →1225 75² = ? 7 × 8 = 56 Then add 25 →5625 |
🚀 Shortcut 10: Multiply by 99,999
| Rule: | 99 × n = 100n - n = (n-1) Link (100-n dress number) |
| Example: | 99 × 56 = ? = 5600 - 56 =5544 (or) 55|44 = (56-1)|(100-56) 999 × 78 = ? = 78000 - 78 =77922 |
🎯 Memory Formulas:
- BODMAS- Brackets, Orders, Division, Multiplication, Addition, Subtraction
- Fraction division- Multiply by taking Reciprocal
- % → Decimal- Divide by 100
- Decimal → %- Multiply by 100
- Fraction → Decimal- Volume ÷ Area
- Class- Use (a±b)² formula
⏱️ Time Management Tips for Exam:
- Solve simple questions first
- Attempt difficult questions at the end
- Spend only 30-45 seconds on each question
- Save time by using Shortcuts
- Find answer directly from Options (Back Calculation)
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