Simplification

This summary explains the Simplification chapter from your PDF for SSC and other competitive exams. You’ll find key concepts, formulas, solved examples, a cheatsheet, practice questions, and visual aids for better understanding¹.

1. Number Basics

• Digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
• Types of Numbers:
  • Natural numbers: 1, 2, 3, …
  • Whole numbers: 0, 1, 2, 3, …
  • Integers: …, -3, -2, -1, 0, 1, 2, 3, …
  • Real numbers: All rational and irrational numbers (e.g., 2.8, -10, 3.13)
  • Even numbers: 2, 4, 6, …
  • Odd numbers: 1, 3, 5, …
  • Prime numbers: 2, 3, 5, 7, 11, …

2. Algebraic Identities (Cheatsheet)

3. Addition & Subtraction Tricks

Example 1

Q: $8 + 88 + 888 + 8888 + 88888 = ?$

Trick: Factor out 8: $8 \times (1 + 11 + 111 + 1111 + 11111) = 8 \times 12345 = 98760$

Example 2

Q: $0.8 + 0.88 + 0.888 + 0.8888 + 0.88888 = ?$

Trick: $8 \times (0.1 + 0.11 + 0.111 + 0.1111 + 0.11111) = 8 \times 0.54321 = 4.34568$

Example 3

Q: $8.8 + 8.88 + 8.888 + 8.8888 + 8.88888 = ?$

Trick: Add the whole numbers: $8 \times 5 = 40$ Add decimals from previous: $4.34568$ Total: $44.34568$

4. Multiplication Shortcuts

Type 1: Unit Digits Add to 10, Other Digits Same

Q: $43 \times 47$

• $4 \times (4+1) = 20$
• $3 \times 7 = 21$
• Answer: 2021

Type 2: Tens Digits Add to 10, Units Same

Q: $46 \times 66$

• $(4 \times 6) + 6 = 30$
• $6 \times 6 = 36$
• Answer: 3036

Type 3: Both Units 5, Numbers Differ by 10

Q: $75 \times 65$

• $6 \times (7+1) = 48$
• Write 75
• Answer: 4875

5. Squares and Cubes

• Squares of 1–50: Memorize for quick calculation.
• Example: $56^2$
  • Split as 5 and 6.
  • $(5/6)^2 = 5^2 / 2 \times 5 \times 6 / 6^2 = 25 / 60 / 36$
  • Combine: $3136$

6. Fractions

• Proper Fraction: Numerator < Denominator (e.g., $\frac{3}{5}$)
• Improper Fraction: Numerator > Denominator (e.g., $\frac{7}{5}$)
• Mixed Fraction: Whole number + proper fraction (e.g., $8 \frac{7}{2}$)

7. VBODMAS Rule

Order of Operations:

• Vinculum (bar)
• Brackets: (), {}, []
• Orders (powers, roots)
• Division
• Multiplication
• Addition
• Subtraction

Table:

8. VBODMAS Examples

Example 1

$35 \div 7 \times 5 = ?$

• Divide first: $35 \div 7 = 5$
• Multiply: $5 \times 5 = 25$

Example 2

$35 \div 5$ of $7 = ?$

• ‘Of’ first: $5 \times 7 = 35$
• Then divide: $35 \div 35 = 1$

Example 3

$48 \div 12$ of $2 + [3 + 17 \times 2]$

• Solve inside brackets: $17 \times 2 = 34$, $3 + 34 = 37$
• ‘Of’: $12 \times 2 = 24$
• Divide: $48 \div 24 = 2$
• Add: $2 + 37 = 39$

Example 4

$2 \div 2 \div 2 \div 2 \div 2 \div 2 = ?$

• Left to right: $2 \div 2 = 1$, $1 \div 2 = 0.5$, $0.5 \div 2 = 0.25$, etc.
• Final answer: $1/16$

9. Practice Questions

1. $7 + 77 + 777 + 7777 = ?$
2. $0.9 + 0.99 + 0.999 + 0.9999 = ?$
3. $53 \times 57$ (use special multiplication)
4. $85 \times 75$
5. $(a+b)^2$ when $a = 12, b = 8$
6. $(a-b)^3$ when $a = 10, b = 4$
7. $25 + 75 \div 5 \times 2$
8. $16 \div 2$ of $4 + 2$
9. $3.5 + 4.5 + 5.5 + 6.5$
10. $2 \div 2 \div 2$

10. Visuals & Graphics

A. VBODMAS Pyramid

    Vinculum
     Brackets
      Orders
      Divide
      Multiply
      Add
      Subtract

B. Multiplication Pattern

For numbers like 43 × 47:

|--4--|3| × |--4--|7|
Same tens, units add to 10

11. Quick Reference Table

12. Tips & Tricks

• Memorize squares and cubes up to at least 25 for speed.
• Use multiplication patterns for numbers close to each other.
• Always follow VBODMAS for order of operations.
• For decimals, align digits for addition/subtraction.

With these explanations, formulas, examples, cheatsheet, practice questions, and visuals, you’ll be ready to master Simplification for any exam!