Simplification Formulas & Shortcuts

🔢 BODMAS Rule

Order of Operations

B - Brackets
O - Of (Orders: powers and roots)
D - Division
M - Multiplication
A - Addition
S - Subtraction

Remember: BODMAS

1. Brackets (Parentheses, Curly, Square)
2. Orders (Powers, Roots)
3. Division and Multiplication (Left to Right)
4. Addition and Subtraction (Left to Right)

📚 Bracket Rules

Types of Brackets (from inside to outside)

( ) - Parentheses or Small Brackets
{ } - Curly Brackets or Middle Brackets
[ ] - Square Brackets or Big Brackets

Solving Brackets

Solve innermost bracket first
Work your way outward
Example: [{(3 + 2) × 4} - 5] = [{5 × 4} - 5] = [20 - 5] = 15

✖️ Multiplication Shortcuts

Multiplying by 5

Number × 5 = (Number × 10)/2
Example: 47 × 5 = 470/2 = 235

Multiplying by 9

Number × 9 = (Number × 10) - Number
Example: 67 × 9 = 670 - 67 = 603

Multiplying by 11

Two-digit method: (a b) × 11 = a | a+b | b
Example: 45 × 11 = 4 | 4+5 | 5 = 495

If middle digit > 9: Carry over
Example: 87 × 11 = 8 | 8+7 | 7 = 8 | 15 | 7 = 957

Multiplying by 99

Number × 99 = (Number × 100) - Number
Example: 234 × 99 = 23400 - 234 = 23166

Multiplying by 25

Number × 25 = (Number × 100)/4
Example: 68 × 25 = 6800/4 = 1700

Multiplying by 125

Number × 125 = (Number × 1000)/8
Example: 56 × 125 = 56000/8 = 7000

➗ Division Shortcuts

Dividing by 5

Number ÷ 5 = (Number × 2)/10
Example: 125 ÷ 5 = 250/10 = 25

Dividing by 25

Number ÷ 25 = (Number × 4)/100
Example: 300 ÷ 25 = 1200/100 = 12

Dividing by 125

Number ÷ 125 = (Number × 8)/1000
Example: 1000 ÷ 125 = 8000/1000 = 8

🧮 Fraction Operations

Addition of Fractions

a/b + c/d = (ad + bc)/bd

Subtraction 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) = ad/bc

Simplifying Complex Fractions

(a/b)/(c/d) = (a/b) × (d/c) = ad/bc

🎯 Square and Cube Operations

Square Numbers

(a + b)² = a² + 2ab + b²
(a - b)² = a² - 2ab + b²

Quick Squares:

• Numbers ending with 5: (a5)² = a(a+1) | 25 Example: 35² = 3×4 | 25 = 1225

Cube Numbers

(a + b)³ = a³ + 3a²b + 3ab² + b³
(a - b)³ = a³ - 3a²b + 3ab² - b³

Difference of Squares

a² - b² = (a + b)(a - b)

🔢 Decimal Operations

Addition and Subtraction

• Align decimal points
• Add or subtract as usual
• Place decimal point in result

Multiplication

• Multiply without decimals
• Count total decimal places
• Place decimal point accordingly

Division

• Make divisor a whole number
• Move decimal point in dividend same places
• Divide normally

📈 Percentage Simplification

Percentage to Fraction

x% = x/100

Common Percentages

10% = 1/10
20% = 1/5
25% = 1/4
50% = 1/2
75% = 3/4
33⅓% = 1/3
66⅔% = 2/3

Successive Percentage

a% and b% successive = a + b + (ab/100)

🧮 Surds and Indices

Laws of Indices

aᵐ × aⁿ = aᵐ⁺ⁿ
aᵐ ÷ aⁿ = aᵐ⁻ⁿ
(aᵐ)ⁿ = aᵐⁿ
a⁰ = 1 (a ≠ 0)
a⁻ⁿ = 1/aⁿ

Fractional Indices

a¹/ⁿ = ⁿ√a
aᵐ/ⁿ = (ⁿ√a)ᵐ

Simplifying Surds

√(ab) = √a × √b
√(a/b) = √a/√b

🎪 Mixed Operations

Order of Mixed Operations

1. Brackets (inside out)
2. Powers and Roots
3. Division and Multiplication (left to right)
4. Addition and Subtraction (left to right)

Example with Mixed Operations

Simplify: 15 + 3 × 4 - 8 ÷ 2
= 15 + 12 - 4
= 27 - 4
= 23

📊 Unit Digit Method

Finding Unit Digit

Focus only on the unit digit of each number
Perform operations on unit digits only

Example: Find unit digit of 34 × 27 + 19

• Unit digits: 4 × 7 + 9
• 28 + 9 = 37
• Unit digit = 7

Cyclicity of Unit Digits

2, 3, 7, 8: Cycle of 4
4, 9: Cycle of 2
0, 1, 5, 6: Always same

🔢 Approximation Techniques

Rounding Method

Round to nearest 10, 100, 1000
Calculate with rounded numbers
Choose closest option

Range Method

Find approximate range
Eliminate options outside range

📝 Practice Examples

Example 1: Mixed Operations

Simplify: 25 + 5 × (16 - 8) ÷ 4
= 25 + 5 × 8 ÷ 4
= 25 + 40 ÷ 4
= 25 + 10
= 35

Example 2: Fraction Operations

Simplify: (3/4 + 2/3) ÷ (5/6)
= (9/12 + 8/12) ÷ (5/6)
= (17/12) ÷ (5/6)
= (17/12) × (6/5)
= 102/60 = 17/10

Example 3: Percentage Simplification

Simplify: 30% of 250 + 20% of 150
= (30/100) × 250 + (20/100) × 150
= 75 + 30
= 105

🎯 Time-Saving Tips

1. Look for Patterns

Identify common patterns
Use memorized results

2. Work Backwards

From answer options
Test which one works

3. Use Estimation

Estimate the range
Eliminate impossible options

4. Digital Sum Method

For checking addition/subtraction
Sum of digits should match

🔗 Related Topics

• Percentage - Percentage calculations
• Ratio and Proportion - Ratio problems
• Average - Average calculations
• Number System - Number system basics

📚 Continue Learning