Simplefications

COMPETITIVE EXAMINATION HANDBOOK

SIMPLIFICATION

PAGE 1 — BASIC CONCEPT & BODMAS

Simplification means reducing a mathematical expression to its simplest or final value by following the correct order of operations. It is one of the most important topics in arithmetic and is frequently used in competitive examinations.

SIMPLIFICATION = Calculation according to the correct order of operations

The BODMAS Rule

The most important rule used in simplification is BODMAS.

Letter	Meaning	Order
B	Brackets	1st
O	Orders / Of	2nd
D	Division	3rd
M	Multiplication	4th
A	Addition	5th
S	Subtraction	6th

Important: Division and multiplication have the same priority. Calculate them from left to right. Addition and subtraction also have the same priority and are calculated from left to right.

Example 1

8 + 6 × 2

First multiply:
6 × 2 = 12

Then add:
8 + 12 = 20

Example 2

25 − 12 ÷ 3

Division first:
12 ÷ 3 = 4

25 − 4 = 21

Example 3

5 + 4 × 3 − 6

4 × 3 = 12
5 + 12 − 6
= 17 − 6
= 11

Why BODMAS is Important

It prevents calculation mistakes.
It gives a fixed order for solving expressions.
It saves time in competitive examinations.
It is useful for fractions, decimals and algebraic expressions.

Quick Practice

7 + 5 × 2 = ?
30 − 18 ÷ 3 = ?
12 + 4 × 5 = ?
50 ÷ 5 + 8 = ?

SIMPLIFICATION HANDBOOK

BRACKETS & ORDERS

PAGE 2 — BRACKETS, OF AND POWERS

Types of Brackets

Brackets are solved before other operations according to the BODMAS rule.

Bracket	Name	Example
( )	Small / Round Bracket	(5 + 3)
{ }	Curly Bracket	{8 − 2}
[ ]	Square Bracket	[10 + 4]

Rule for Multiple Brackets

When brackets occur inside other brackets, solve the innermost bracket first.

20 − [8 + {6 − (3 + 1)}]

First:
(3 + 1) = 4

Then:
{6 − 4} = 2

Then:
[8 + 2] = 10

Finally:
20 − 10 = 10

The Meaning of "Of"

In many arithmetic questions, the word of means multiplication. It is especially common when fractions or percentages are involved.

1/2 of 20
= 1/2 × 20
= 10

25% of 80
= 25/100 × 80
= 20

Orders / Powers

Orders include powers, squares, cubes and roots. They are normally calculated before multiplication, division, addition and subtraction.

Squares

5² = 25

10² = 100

12² = 144

Cubes

2³ = 8

3³ = 27

5³ = 125

Example with Power

5 + 3² × 2

3² = 9
9 × 2 = 18
5 + 18 = 23

Example with Brackets and Power

(4 + 2)² ÷ 3

(4 + 2) = 6
6² = 36
36 ÷ 3 = 12

Exam Tip: Never solve an expression simply from left to right. First identify brackets, orders, division/multiplication and then addition/subtraction.

Practice

10 + (5 × 2) = ?
20 − [8 + 2] = ?
2³ + 5 = ?
4² + 3 × 2 = ?
1/4 of 40 = ?
50% of 90 = ?

SIMPLIFICATION HANDBOOK

FRACTIONS & DECIMALS

PAGE 3 — SIMPLIFICATION WITH FRACTIONS

Basic Fraction Rule

When adding or subtracting fractions with different denominators, first find their LCM and convert them into equivalent fractions.

1/2 + 1/3

LCM of 2 and 3 = 6

1/2 = 3/6
1/3 = 2/6

3/6 + 2/6 = 5/6

Fraction Multiplication

2/3 × 5/7

= (2 × 5)/(3 × 7)
= 10/21

Answer = 10/21

Fraction Division

For division, multiply by the reciprocal of the second fraction.

3/4 ÷ 2/5

= 3/4 × 5/2
= 15/8
= 1 7/8

Simplification of Complex Fraction

(1/2 + 1/4) ÷ 3/4

1/2 + 1/4 = 2/4 + 1/4 = 3/4

3/4 ÷ 3/4
= 1

Decimal Simplification

Decimal calculations follow the same BODMAS rule.

2.5 + 3.5 × 2

3.5 × 2 = 7
2.5 + 7 = 9.5

10.5 − 4.5 ÷ 3

4.5 ÷ 3 = 1.5
10.5 − 1.5 = 9

Fraction + Decimal

1/2 + 0.25

1/2 = 0.50

0.50 + 0.25 = 0.75

Useful Fraction–Decimal Values

Fraction	Decimal
1/2	0.5
1/4	0.25
3/4	0.75
1/5	0.2
1/8	0.125
1/10	0.1

Shortcut: Before multiplying fractions, cancel common factors between numerator and denominator whenever possible. This reduces calculation time and avoids large numbers.

Practice

1/2 + 1/4 = ?
3/4 − 1/2 = ?
2/5 × 3/4 = ?
3/5 ÷ 2/3 = ?
2.5 + 1.5 × 2 = ?
10.5 − 2.5 ÷ 2 = ?

COMPETITIVE EXAMINATION HANDBOOK

SHORTCUTS & PRACTICE

PAGE 4 — EXAM SHORTCUTS AND MIXED SIMPLIFICATION

Shortcut 1 — Multiplication by 5

To multiply a number by 5, multiply it by 10 and divide by 2.

48 × 5
= 48 × 10 ÷ 2
= 480 ÷ 2
= 240

Shortcut 2 — Multiplication by 25

To multiply by 25, multiply by 100 and divide by 4.

36 × 25
= 36 × 100 ÷ 4
= 3600 ÷ 4
= 900

Shortcut 3 — Multiplication by 125

Multiply by 1000 and divide by 8.

32 × 125
= 32 × 1000 ÷ 8
= 32000 ÷ 8
= 4000

Shortcut 4 — Percentage Simplification

20% of 250
= 20/100 × 250
= 1/5 × 250
= 50

Mixed BODMAS Example

50 − {12 + 6 × (8 − 5)} ÷ 3

Step 1:
(8 − 5) = 3

Step 2:
6 × 3 = 18

Step 3:
12 + 18 = 30

Step 4:
30 ÷ 3 = 10

Step 5:
50 − 10 = 40

Another Mixed Example

100 − [25 + {15 − (8 + 2)}] × 2

(8 + 2) = 10
15 − 10 = 5
25 + 5 = 30
30 × 2 = 60
100 − 60 = 40

Golden Rules for Exams

Always check the brackets first.
Calculate powers and roots before multiplication.
Division and multiplication are solved from left to right.
Addition and subtraction are solved from left to right.
Cancel fractions before multiplying whenever possible.
Convert percentages into fractions when convenient.
Keep decimal points aligned during addition and subtraction.
Do not perform unnecessary calculations.

Final Practice Set

15 + 6 × 3 = ?
48 ÷ 6 + 7 = ?
25 − 5 × 3 = ?
10 + [20 − (5 × 2)] = ?
50 − {10 + 5 × 4} = ?
2/3 + 1/6 = ?
5/6 − 1/3 = ?
3/4 × 8/9 = ?
5/8 ÷ 10/16 = ?
2.5 + 4.5 × 2 = ?
20.5 − 5.5 ÷ 5 = ?
25% of 240 = ?
35 × 25 = ?
64 × 125 = ?
100 − [20 + {30 ÷ 5}] = ?

MASTER FORMULA

BRACKETS → ORDERS → DIVISION / MULTIPLICATION → ADDITION / SUBTRACTION

UtkalPress Exam Tip:
Read the complete expression once before starting. Mark the brackets, identify fractions or percentages, and then solve step-by-step. Accuracy is more important than speed; with regular practice, both accuracy and speed improve.