Simplification & Approximation – Complete Short Notes

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1. Basics of Simplification

• Simplification involves solving arithmetic expressions using BODMAS / PEMDAS rules:
  B – Brackets → (), {}, []
  O – Orders → powers, roots
  D – Division
  M – Multiplication
  A – Addition
  S – Subtraction

• Order of operations matters:
  6+2×(32−5)÷2=?

  Step 1: Brackets → $3^2 -5 =4$
  Step 2: Division/Multiplication → $4 ÷ 2 = 2$, $2 × 2 = 4$
  Step 3: Addition → $6+4=10$

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2. Important Rules / Shortcuts

A. Fractions

• Combine fractions before operations:

  $$\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$$​

B. Decimal Approximation

• Round decimals for quick calculation:

  • 3.141 ≈ 3.14

  • 6.278 ≈ 6.28

• Use 2 decimal places for exam-level speed unless specified

C. Square Roots / Cube Roots

• Approximate square roots between nearest perfect squares

  • Example: ple: $\sqrt{50} ≈ 7.07$≈ 7.07) 

• Cube roots similarly between nearest perfect cubes

D. Percentage / Fraction Shortcuts

• Convert % → decimal → approximate: 18% ≈ 0.18

• Fractions → nearest decimal: (7/8 ≈ 0.875 ≈ 0.88)

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3. Common Simplification Techniques

1. Eliminate brackets:
   (a+b) + (c-d) = a+b+c-d

2. Use squares/cubes:$(a+b)^2 = a^2 + 2ab + b^2$

3. Factorization:

• Example: (12×25 = 3×4×25 = 3×100 = 300)

4. Use reciprocal / inversion for division:
   
   $\frac{a}{b} ÷ \frac{c}{d} = \frac{a}{b} × \frac{d}{c}$

5. Combine like terms in addition/subtraction

6. Round-off for approximation:

• Example: (198 × 502 ≈ 200 × 500 = 100,000)

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4. Approximation Methods

A. Rounding Off

• Round numbers to nearest 10, 100, or decimal place depending on question

B. Estimation by Fraction

• Replace fractions with nearest decimals or simple fractions for faster calculation

• Example: (7/9 ≈ 0.78 ≈ 0.8)

C. Significant Figures

• Keep 2 or 3 significant figures for estimation questions

D. Compatible Numbers

• Adjust numbers to nearest compatible numbers for easy mental calculation:

  • Example: (398 ÷ 19 ≈ 400 ÷ 20 = 20)

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5. Tricks for Competitive Exams

1. Multiplication:

• Use factor pairs: 98×102=(100−2)(100+2)=10000−4=9996

2. Division:

• Convert divisors close to 10, 100, 1000 for easy mental math

3. Addition / Subtraction:

• Group numbers to nearest tens/ hundreds

• Example: 297+416 → 300+413=713

4. Square Roots Approximation

• • $\sqrt{105} ≈ 10 + (105-100)/(2×10) = 10 + 5/20 = 10.25$

5. Cube Roots Approximation:

• $\sqrt[3]{216}=\mathrm{<span\; class="base"><span\; class="strut"></span><span\; class="mord">6 </span>(perfect\; cube)</span>}$

• $\sqrt[3]{220} ≈ 6 + (220-216)/(3×6^2) = 6 + 4/108 ≈ 6.037$

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6. Examples for Practice

1. Simplify: $25 × 32 ÷ 8 + 12^2 ÷ 6$

• Step 1: $25×32 ÷ 8 = 25×4=100$

• Step 2: $12^2 ÷6=144÷6=24$

• Step 3: $100+24=124$

2. Approximate: $198 × 49 ÷ 7$

• Step 1: Round → $200 × 50 ÷ 7 ≈ 10,000 ÷7 ≈ 1428.57 ≈ 1430$

3. Simplify: $(75×16)-(24×50)$

• $75×16 = 1200$, $24×50 =1200$ → Result = 0

4. Approximate: $\sqrt{1025} ≈ ?$

• Nearest perfect square: $32^2=1024$ → $\sqrt{1025}\approx \mathrm{32.015\; =32}$

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7. One-Liner Formulas / Tricks

• BODMAS: Brackets → Orders → Division → Multiplication → Addition → Subtraction

$(a+b{)}^2=a^2+2ab+b^2$
$(a-b{)}^2=a^2-2ab+b^2$

$(a+b)(a-b)=a^2-b^2$

• Square root ≈ nearest perfect square + fractional adjustment

• Cube root ≈ nearest cube + adjustment using formula

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8. Exam Day Tips

• Always follow BODMAS

• Use rounding / estimation for quick calculation

• Convert fractions to decimals if easier

• Memorize square & cube tables for simplification

• Use factorization for multiplication/division shortcuts

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✅ Quick Revision Checklist:

• BODMAS rules

• Square & cube formulas

• Fraction → decimal conversion

• Rounding and estimation

• Mental math tricks for multiplication/division

• Approximation for roots & powers