Important Formulas

Simplification

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Simplification

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Important Formulas & Concepts

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Simplification

Simplification is one of the most important and high-scoring topics in Quantitative Aptitude. Questions from simplification are frequently asked in SSC, Banking, Railway, CDS, NDA, CAT, UPSC, and various competitive examinations.

This chapter focuses on reducing complex mathematical expressions into simpler forms by applying arithmetic operations in the correct sequence. Strong command over simplification techniques improves calculation speed, accuracy, and overall aptitude performance.

Why Simplification is Important?

  • High-weightage topic in competitive exams.
  • Improves calculation speed significantly.
  • Foundation for arithmetic and algebra problems.
  • Useful in approximation and numerical ability tests.
  • Helps solve lengthy calculations quickly and accurately.

What is Simplification?

Simplification is the process of converting a complex mathematical expression into a simpler form by performing arithmetic operations according to a specific order.

A mathematical expression may contain:

  • Addition (+)
  • Subtraction (−)
  • Multiplication (×)
  • Division (÷)
  • Brackets
  • Fractions
  • Roots and powers

To solve such expressions correctly, we follow the VBODMAS Rule.

VBODMAS Rule

VBODMAS defines the correct order of operations in arithmetic expressions.

VBODMAS Stands For

Letter Meaning Operation
V Vinculum Bar or Line
B Bracket () {} []
O Of Multiplication with Fractions/Percentages
D Division ÷
M Multiplication ×
A Addition +
S Subtraction −

✔ Operations are always performed from left to right.

✔ Follow the exact sequence of VBODMAS to avoid mistakes.

Order of Solving Brackets

If multiple brackets are present, solve them in the following order:

  1. Vinculum (Bar)
  2. Small Bracket ()
  3. Curly Bracket {}
  4. Square Bracket []

Example:

[12 + {8 − (3 × 2)}]

First solve:

(3 × 2) = 6

Then:

{8 − 6} = 2

Finally:

12 + 2 = 14

Vinculum (Bar) Concept

A vinculum is a horizontal bar placed over numbers or expressions.

Operations under the vinculum are solved first.

Example:

8 + 3 + 2 × 4

First solve the vinculum:

3 + 2 = 5

Then:

8 + 5 × 4

= 8 + 20

= 28

Concept of "Of"

The word "of" means multiplication.

According to VBODMAS, "of" is solved before division and multiplication.

Example:

2/3 of 18

= (2/3) × 18

= 12

Basic Arithmetic Operations

1. Addition

Combining two or more numbers to get a total value.

Example:

25 + 18 = 43

2. Subtraction

Finding the difference between numbers.

Example:

56 − 19 = 37

3. Multiplication

Repeated addition of a number.

Example:

12 × 5 = 60

4. Division

Splitting a quantity into equal parts.

Example:

84 ÷ 7 = 12

Fractions in Simplification

Fractions are solved by:

  • Finding LCM of denominators
  • Reducing into simplest form
  • Applying VBODMAS sequence

Example:

1/2 + 1/3

LCM of 2 and 3 = 6

= 3/6 + 2/6

= 5/6

Decimal Operations

While solving decimal expressions:

  • Align decimal points correctly.
  • Count decimal places carefully.
  • Apply VBODMAS sequence properly.
Expression Answer
12.5 + 7.25 19.75
5.2 × 0.5 2.6
18.6 ÷ 3 6.2

Square and Cube Concepts

Squares

Square of a number = Number × Number

Number Square
12 144
25 625
30 900

Cubes

Cube of a number = Number × Number × Number

Number Cube
2 8
5 125
10 1000

Important Algebraic Identities

Identity Formula
Square of Sum (a + b)² = a² + b² + 2ab
Square of Difference (a − b)² = a² + b² − 2ab
Difference of Squares (a + b)(a − b) = a² − b²
Cube Sum Identity a³ + b³ = (a + b)(a² − ab + b²)
Cube Difference Identity a³ − b³ = (a − b)(a² + ab + b²)

Absolute Value of a Real Number

The absolute value of a real number is always non-negative.

It represents the distance of the number from zero.

|m| = m, if m > 0

|m| = −m, if m < 0

Expression Value
|5| 5
|−5| 5
|12| 12

Important Simplification Formulae

Concept Formula
Square of Number a² = a × a
Cube of Number a³ = a × a × a
Difference of Squares a² − b² = (a + b)(a − b)
Average Formula Average = Sum / Number of Terms
Fraction Addition a/b + c/d = (ad + bc)/bd

Important Exam Tips

  • Always follow VBODMAS sequence strictly.
  • Solve brackets carefully from inner to outer.
  • Practice multiplication tables, squares, and cubes regularly.
  • Use algebraic identities for fast calculations.
  • Be careful while handling negative signs.
  • Simplify fractions before multiplication whenever possible.
  • Use approximation techniques in lengthy calculations.

Simplification is one of the most scoring chapters in Quantitative Aptitude. Strong command over VBODMAS, arithmetic operations, fractions, decimals, and algebraic identities helps candidates solve aptitude questions quickly and accurately in competitive examinations.

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