🧮 Simplification - Complete Theory

Master calculation techniques - speed is everything in IBPS!

🎯 BODMAS Rule

The Order of Operations:

B - Brackets (solve innermost first)
O - Of (multiplication)
D - Division
M - Multiplication
A - Addition
S - Subtraction

Important: Division and Multiplication have EQUAL priority (left to right) Important: Addition and Subtraction have EQUAL priority (left to right)

Bracket Types (Innermost to Outermost):

() - Parentheses (round brackets)
{} - Braces (curly brackets)
[] - Square brackets
— - Vinculum (bar over numbers)

💡 Solved Examples - BODMAS

Example 1: Basic BODMAS

Q: Simplify: 12 + 18 ÷ 6 × 2 - 3

Solution:

Step 1: Division first: 18 ÷ 6 = 3
= 12 + 3 × 2 - 3

Step 2: Multiplication: 3 × 2 = 6
= 12 + 6 - 3

Step 3: Addition: 12 + 6 = 18
= 18 - 3

Step 4: Subtraction: 18 - 3 = 15

Answer: 15

Example 2: With Brackets

Q: Simplify: 5 + [(24 - 12) ÷ 3] × 2

Solution:

Step 1: Innermost bracket: 24 - 12 = 12
= 5 + [12 ÷ 3] × 2

Step 2: Division inside bracket: 12 ÷ 3 = 4
= 5 + 4 × 2

Step 3: Multiplication: 4 × 2 = 8
= 5 + 8

Step 4: Addition: 5 + 8 = 13

Answer: 13

Example 3: Complex Brackets

Q: Simplify: 100 - [50 - {25 - (15 - 5)}]

Solution:

Step 1: Round brackets: 15 - 5 = 10
= 100 - [50 - {25 - 10}]

Step 2: Curly brackets: 25 - 10 = 15
= 100 - [50 - 15]

Step 3: Square brackets: 50 - 15 = 35
= 100 - 35

Step 4: Final: 100 - 35 = 65

Answer: 65

🔢 Fraction Simplification

Basic Operations

Addition/Subtraction:

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

Example: 2/3 + 3/4 = (8 + 9) / 12 = 17/12

Multiplication:

a/b × c/d = (a × c) / (b × d)

Example: 2/3 × 3/5 = 6/15 = 2/5

Division:

a/b ÷ c/d = a/b × d/c

Example: 2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6

Example 4: Mixed Fractions

Q: Simplify: 2(1/3) + 1(2/5)

Solution:

Convert to improper fractions:
2(1/3) = 7/3
1(2/5) = 7/5

7/3 + 7/5 = (35 + 21) / 15 = 56/15 = 3(11/15)

Answer: 3(11/15) or 56/15

🔷 Decimal Simplification

Operations

Addition:

12.5 + 3.75 = 16.25
(Align decimal points)

Multiplication:

2.5 × 1.2 = 3.0
(Count total decimal places: 1 + 1 = 2)

Division:

12.5 ÷ 2.5 = 125 ÷ 25 = 5
(Remove decimals by multiplying both by 10)

Example 5: Decimal Operations

Q: (0.5 × 0.5 × 0.5) ÷ (0.05 × 0.05 × 0.05)

Solution:

= (5 × 5 × 5) / (10 × 10 × 10) ÷ (5 × 5 × 5) / (100 × 100 × 100)
= 125/1000 ÷ 125/1,000,000
= 125/1000 × 1,000,000/125
= 1,000

Shortcut:

= (0.5/0.05)³ = (10)³ = 1,000 ✓

Answer: 1,000

√ Surds & Indices

Square Roots

Key Values (Memorize!):

√1 = 1          √36 = 6
√4 = 2          √49 = 7
√9 = 3          √64 = 8
√16 = 4         √81 = 9
√25 = 5         √100 = 10
                √121 = 11
                √144 = 12
                √169 = 13
                √196 = 14
                √225 = 15

Surd Rules

√(a × b) = √a × √b
√(a/b) = √a / √b
(√a)² = a
√a × √a = a

Example 6: Surds

Q: Simplify: √50 + √32

Solution:

√50 = √(25 × 2) = 5√2
√32 = √(16 × 2) = 4√2

√50 + √32 = 5√2 + 4√2 = 9√2

Answer: 9√2

🔺 Indices/Powers

Basic Laws

a^m × a^n = a^(m+n)
a^m ÷ a^n = a^(m-n)
(a^m)^n = a^(mn)
a^0 = 1
a^(-n) = 1/a^n
a^(1/n) = ⁿ√a

Example 7: Indices

Q: Simplify: (2³ × 2²) ÷ 2⁴

Solution:

= 2^(3+2) ÷ 2⁴
= 2⁵ ÷ 2⁴
= 2^(5-4)
= 2¹
= 2

Answer: 2

⚡ Approximation Techniques

When to Approximate

When question asks for "approximate value"
When options are far apart
When exact calculation takes too long

Techniques

1. Rounding Off:

23.7 ≈ 24
98.2 ≈ 98 or 100
4.91 ≈ 5

2. Using Standard Values:

√2 ≈ 1.414
√3 ≈ 1.732
π ≈ 3.14 or 22/7

3. Fraction Approximation:

1/3 ≈ 0.33
2/3 ≈ 0.67
1/7 ≈ 0.14

Example 8: Approximation

Q: Approximate: √101 + √99

Solution:

√101 ≈ √100 = 10
√99 ≈ √100 = 10

√101 + √99 ≈ 10 + 10 = 20

Answer: ≈ 20

🎯 Special Techniques

Technique 1: Simplifying Big Numbers

12,345 × 8 = 12,345 × 2 × 4
           = 24,690 × 4
           = 98,760

Or use: 12,345 × 8 = (12,000 + 345) × 8
                    = 96,000 + 2,760
                    = 98,760

Technique 2: Division by 5

Divide by 5 = Multiply by 2 and divide by 10

345 ÷ 5 = (345 × 2) ÷ 10 = 690 ÷ 10 = 69

Technique 3: Multiplication by 11

For 2-digit numbers:
Add the digits and put sum in middle

34 × 11 = 3_(3+4)_4 = 374

Technique 4: Squaring Numbers Ending in 5

(10a + 5)² = a(a+1) × 100 + 25

25² = 2 × 3 × 100 + 25 = 625
35² = 3 × 4 × 100 + 25 = 1,225
45² = 4 × 5 × 100 + 25 = 2,025

💡 More Examples

Example 9: Finding Unknown

Q: If 5x - 3(x - 2) = 14, find x.

Solution:

5x - 3x + 6 = 14
2x + 6 = 14
2x = 8
x = 4

Answer: x = 4

Example 10: Percentage Simplification

Q: What is 15% of 40% of 500?

Solution:

= 15/100 × 40/100 × 500
= 0.15 × 0.4 × 500
= 0.06 × 500
= 30

Answer: 30

Example 11: Mixed Operations

Q: Simplify: (64)^(2/3) + (125)^(1/3)

Solution:

(64)^(2/3) = (∛64)² = 4² = 16
(125)^(1/3) = ∛125 = 5

Total = 16 + 5 = 21

Answer: 21

⚡ Quick Calculation Tricks

Trick 1: Multiplying by 9, 99, 999

45 × 9 = 45 × (10 - 1) = 450 - 45 = 405
45 × 99 = 45 × (100 - 1) = 4,500 - 45 = 4,455

Trick 2: Division by 9

To check if divisible by 9: Sum of digits divisible by 9
234 → 2+3+4 = 9 → divisible ✓

Trick 3: Square Near 50

48² = (50-2)² = 2,500 - 200 + 4 = 2,304
52² = (50+2)² = 2,500 + 200 + 4 = 2,704

Trick 4: Percentage of Numbers

10% of x = x/10
5% of x = x/20
25% of x = x/4
20% of x = x/5

⚠️ Common Mistakes

❌ Mistake 1: Wrong BODMAS Order

Wrong: 10 - 5 + 3 = 10 - 8 = 2 ✗
Right: 10 - 5 + 3 = 5 + 3 = 8 ✓
(Addition and subtraction: left to right!)

❌ Mistake 2: Bracket Removal

Wrong: 5 - (3 - 2) = 5 - 3 - 2 = 0 ✗
Right: 5 - (3 - 2) = 5 - 1 = 4 ✓
(Negative sign before bracket changes signs inside!)

❌ Mistake 3: Decimal Multiplication

Wrong: 0.5 × 0.5 = 0.25 counted as "2.5" ✗
Right: Count decimal places carefully ✓

❌ Mistake 4: Zero Power

Wrong: 5⁰ = 0 ✗
Right: 5⁰ = 1 (any number to power 0 = 1) ✓

❌ Mistake 5: Negative Indices

Wrong: 2⁻³ = -8 ✗
Right: 2⁻³ = 1/2³ = 1/8 ✓

📊 Important Values to Memorize

Squares (1-30)

1² = 1      11² = 121    21² = 441
2² = 4      12² = 144    22² = 484
3² = 9      13² = 169    23² = 529
4² = 16     14² = 196    24² = 576
5² = 25     15² = 225    25² = 625
6² = 36     16² = 256    26² = 676
7² = 49     17² = 289    27² = 729
8² = 64     18² = 324    28² = 784
9² = 81     19² = 361    29² = 841
10² = 100   20² = 400    30² = 900

Cubes (1-15)

1³ = 1      6³ = 216     11³ = 1,331
2³ = 8      7³ = 343     12³ = 1,728
3³ = 27     8³ = 512     13³ = 2,197
4³ = 64     9³ = 729     14³ = 2,744
5³ = 125    10³ = 1,000  15³ = 3,375

📝 Practice Problems

Level 1:

1. Simplify: 24 ÷ 6 + 3 × 2 - 1
2. Simplify: 2/5 + 3/10
3. What is √169?

Level 2:

4. Simplify: 100 - [80 - {60 - (40 - 20)}]
5. (0.8 × 0.8 × 0.8) ÷ (0.2 × 0.2 × 0.2)
6. Simplify: 3² × 3³ ÷ 3⁴

Level 3:

7. √48 + √75 - √12
8. If 3x - 2(x - 5) = 20, find x
9. (216)^(2/3) + (64)^(1/3)

🔗 Related Topics

Prerequisites:

• Number System - Foundation for calculations

Uses Simplification:

• Percentage - Simplifying percentage calculations
• Average - Simplifying sums
• Profit & Loss - Complex calculations

Practice:

• Simplification Questions
• Shortcuts & Tricks

Master Simplification - Speed and accuracy win IBPS exams! 🧮