Simplification

Aptitude Tutorial

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A process which convert complex arithmetical expression into simple expression is called simplification.

Order of Operations
When we simplify expression and equation we used order of operation.

VBODMAS Rule

In this rule, we use operations in order
V - Vinculum or Bar
B - Brackets - Order of brackets
First- Small brackets ( )
second- Middle brackets{ }
Third- Square brackets[ ]
O - Of
D - Division
M - Multiplication A - Addition
S - Subtraction

Order of above mentioned operations is same as the order of letters in the VBODMAS from left to right

Example- Simplify 5 - [ 4 - {9 - ( 7 - 5 - 3 )}]
Solution:-
5 - [ 4 - {9 - ( 7 - 5 - 3 )}]
= 5 - [ 4 - { 9 - ( 7 - 5 + 3 )}]
= 5 - [ 4 - { 9 - ( 7 - 2 )}]
= 5 - [ 4 - { 9 - 5 }]

= 5 - [ 4 - 4 ]
= 5 - 0
= 5
Example-
Simplify 34500 ÷ 30 ÷ 10
Solution:- 34500 ÷ 30 ÷ 10
= 34500/30 ÷ 10
= 1150 ÷ 10
= 1150/10
= 115
Example

= - 5
1
 + 3
2
 + 2
3
7 7 7

Solution:-
= 5
1
 + 3
2
 + 2
3
7 7 7
= (5 + 2 + 2 ) +
1
 +
2
 +
3
7 7 7
= 10
5
7
= 10
5
7

Example- Simplify 34500 ÷ 30 ÷ 10
Solution:- 34500 ÷ 30 ÷ 10
= 34500/30 ÷ 10
= 1150 ÷ 10
= 1150/10
= 115
Example
= - 5 1 + 3 2 + 2 1
7 7 7

Solution:-
= 5 1 + 3 2 + 2 1
7 7 7
= (5 + 3 + 2) + 1 + 2 + 1
7 7 7
= 10
5
7

Basic Formulae

(a + b)² = a² + b² + 2ab


Example- Find the square of 107.
Solution:-

( 107 )2 = ( 100 + 7 )2
= 1002 + 72 + 2 × 100 × 7
= 10000 + 49 + 1400
= 11449

(a - b)² = a² + b² - 2ab


Example- Find the square 999.
Solution:-

( 999 )2 = ( 1000 - 1 )2
= 10002 + 12 - 2 × 1000 × 1
= 1000000 + 1 - 2000
= 998001

a² - b² = (a + b)(a - b)


Example- Find the product of 998 and 1002.
Solution:- ( 1000 - 2 ) ( 1000 + 2 )
= 1000² - 2²
= 1000000 - 4
= 999996
Example- Find the value of 48² - 46²
Solution:- 48² - 46²
= ( 48 + 46 ) ( 48 - 46 )
= 94 × 2
= 188

(a + b)³ = a³ + b³ + 3ab (a - b)


Example- Find the value of ( 103 )³
Solution:-

( 103 )³
= ( 100 + 3 )³
= 100³ + 3³ + 3 × 100 × 3 ( 100 + 3 )
= 1000000 + 27 + 900 × 103
= 1000000 + 27 + 92700
= 109272

(a - b)³ = a³ - b³ - 3ab (a - b)


Example- Find the value of ( 98 )³.
Solution:- ( 98 )³
= ( 100 - 2 )³
= 100³ - 2³ - 3 × 100 × 2 ( 100 - 2 )
= 1000000 - 8 - 600 × 98
= 1000000 - 8 - 58800
= 1000000 - 58808
= 941192

a³ + b³ = (a + b) - (a² - ab + b²)


Example- Find the value of ( 4.8 )³ + ( 1.2 )³ / ( 4.8 )² - 4.8 × 1.2 + ( 1.2 )²
Solution:- Let a = 4.8 and b = 1.2
a³ + b³ / (a² - ab + b² )
= {( a + b ) ( a2 - ab + b2 ) }/ (a2 - ab + b2)
= ( a + b )
= 4.8 + 1.2
=6

7.

a³ - b³ = (a - b) - (a² + ab + b²)


Example- Find the value of ( 11.3 )³ -( 1.3 )³ / ( 11.3 )³ +11.3 ×1.3 +( 1.3 )³
Solution:- Let a = 11.3 and b = 1.3
= a³ - b³ / a² + ab + b²
= ( a - b ) ( a² + ab + b² ) / a² + ab + b²
= ( a - b )
= 11.3 - 1.3
=10

8.

(a + b + c)² = a² + b² + C² +2ab + 2bc + 2ca


Example- If a + b + c = 14 and a² + b² + c² = 96, then find the value of ab + bc + ca .
Solution:- a + b + c = 14
Squaring on both sides
or, ( a + b + c )² = (14 )²
or, a² + b² + c² + 2ab + 2bc + 2ca = 196
or, 96 + 2ab + 2bc + 2ca = 196
or, 96 + 2 ( ab + bc + ca ) = 196
or, 2 ( ab + bc + ca )
= 196-96
or, 2 ( ab + bc + ca ) = 100
∴ ab + bc + ca
= 100/2 = 50

9.

(a + b)² + (a - b)² = a² + b² - 2ba


= 2a² + 2b²
= 2 ( a² + b²)
∴ ( a + b )² + ( a - b )² = 2 ( a² + b²)
Example-
Find the value of (745 + 123)² + (745 - 123)²
745 × 745 + 123 × 123

Solution:- Let a = 745 and b = 123
= (a + b)² + (a - b)²
a² + b²
= 2 (a² + b²)
a² + b²

= 2

10.

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


= a² + b² + 2ab - a² - b² + 2ab

=2ab+2ab = 4ab
∴ ( a + b )² - ( a - b )² = 4ab

11.

a³ + b³ c³ - 3abc = (a + b + c)
= (a² + b² + c² - ab - bc - ca)

= 1 (a + b + c)[( a - b )² + ( b - c )² + ( c - a )² ]
2

If a + b + c = 0, then
a3 + b3 + c3 - 3abc = 0
or, a3 + b3 + c3 = 3abc
Example- Find the value of a3 + b3 + c3 - 3abc when a = 23, b = 24 and c = 25
Solution:- a3 + b3 + c3 - 3abc = 1/2 ( a + b + c ) [ ( a - b )2 + ( b - c )2 + ( c - a )2 ]

= 1 (23 + 24 + 25)[( 23 - 24 )² + ( 24 - 25 )² + ( 25 - 23 )²]
2
= 1 × 72 [( -1 )2 + ( -1 )2 + ( 2 )2]
2

= 36 [ 1 + 1 + 4 ]
= 36 × 6
= 261
Example-
Solution:- Let a = 5.3, b = 3.5 and c = 1.2
=
(5.3)3 + (3.5)3 + (1.2)3 - 3 × 5.3 × 3.5 × 1.2
(5.3)2 + (3.5)2 + (1.2)2 - 5.3 × 3.5 - 3.5 × 1.2 - 1.2 × 5.3
=
a3 + b3 + c3 - 3abc
a2 + b2 + c2 - ab - bc - ca

= 5.3 + 3.5 + 1.2
= 10
Example- If a = 25, b = 50 and c = -75, then find the value of a3 + b3 + c3 - 3abc.
Solution:- Here, a = 25, b = 50 and c = -75
a + b + c = 25 + 50 + ( -75 ) = 75 - 75 = 0
a3 + b3 + c3 - 3abc = ( a + b + c ) ( a2 + b2 + c2 - ab - bc - ca )
or, a3 + b3 + c3 - 3abc = 0 ( a2 + b2 + c2 - ab - bc - ca )
∴ a3 + b3 + c3 - 3abc = 0

Some Important Rule

Rule- If x + 1/x = a, then x2 + 1/x2 = a2 - 2
( x + 1/x )2 = x2 + 1/x2 + 2 × x × 1/x
or, ( x + 1/x )2 = x2 + 1/x2 + 2
∴ x2 + 1/x2 = ( x + 1/x )2 - 2

Example- If x + 1/x = 3, then find the value x2 + 1/x2.
Solution:- x + 1/x = 3
∴ x2 + 1/x2 = ( x + 1/x )2 - 2 = 32 - 2
= 9 - 2
= 7

Example- If x + 1/x = 3, then find the value x4 + 1/x4.
Solution:- x + 1/x = 3
∴ x2 + 1/x2 = ( x + 1/x )2 - 2
= 32 - 2
= 9 - 2
= 7
∴ x4 + 1/x4 = ( x2 + 1/x2 )2 - 2
= 72 - 2
= 49 - 2
= 47

Example- If x + 1/x = 3, then find the value x8 + 1/x8.
Solution:- x + 1/x = 3
∴ x2 + 1/x2 = ( x + 1/x )2 - 2
= 32 - 2
= 9 - 2
= 7
∴ x4 + 1/x4 = ( x2 + 1/x2 )2 - 2
= 72 - 2
= 49 - 2
= 47
∴ x8 + 1/x8 = ( x4 + 1/x4 )2 - 2
= 472 - 2
= 2209 - 2
= 2207

Rule- If x + 1/x = a, then x3 + 1/x3 = a3 - 3a
( x + 1/x )3 = x3 + 1/x3 + 3 × x × 1/x ( x + 1/x )
or, ( x + 1/x )3 = x3 + 1/x3 + 3( x + 1/x )
∴ x3 + 1/x3 = ( x + 1/x )3 - 3( x + 1/x )

Example- If x + 1/x = 3, then find the value of x3 + 1/x3.
Solution:- x + 1/x = 3
∴ x3 + 1/x3 = ( x + 1/x )3 - 3( x + 1/x )
= 33 - 3( x + 1/x )
= 27 - 3 × 3
= 27 - 9
= 18

Example- If x + 1/x = 3, then find the value of x9 + 1/x9.
Solution:- x + 1/x = 3
∴ x3 + 1/x3 = ( x + 1/x )3 - 3( x + 1/x )
= 33 - 3( x + 1/x )
= 27 - 3 × 3
= 27 - 9
= 18
∴ x9 + 1/x9 = ( x3 + 1/x3 )3 - 3( x + 1/x )
= 183 - 3( x3 + 1/x3 )
= 5832 - 3 × 18
= 5832 - 54 = 5778

Rule- If x + 1/x = a, then x5 + 1/x5 = [{ ( a3 - 3a ) ( a2 - 2 ) } - a ]
x5 + 1/x5 = ( x3 + 1/x3 ) ( x2 + 1/x2 ) - ( x + 1/x )

Example- If x + 1/x = 3, then find the value of x5 + 1/x5.
Solution:- x + 1/x = 3
∴ x3 + 1/x3 = ( x + 1/x )3 - 3( x + 1/x )
= 33 - 3( x + 1/x )
= 27 - 3 × 3
= 27 - 9
= 18
∴ x2 + 1/x2 = ( x + 1/x )2 - 2
= 32 - 2
= 9 - 2
= 7
∴ x5 + 1/x5 = ( x3 + 1/x3 ) ( x2 + 1/x2 ) - ( x + 1/x )
= 18 × 7 - 3
= 126 - 3
= 123

Rule- If x + 1/x = a, then x - 1/x = √a2 - 4

Example- If x + 1/x = 5, then find the value of x - 1/x.
Solution:- x + 1/x = 5
∴ x - 1/x = √( x + 1/x )2 - 4
= √( 5 )2 - 4
= √21


Example- If x3 + 1/x3 = 10, then find the value of x3 - 1/x3.
Solution:- x3 + 1/x3 = 10
∴ x3 - 1/x3 = √( x3 + 1/x3 )2 - 4
= √( 10 )2 - 4
= √100 - 4
= √96
= 4 √6

Rule- If x - 1/x = a, then x + 1/x = √a2 + 4

Example- If x - 1/x = 7, then find the value of x + 1/x.

Solution:- x - 1/x =7

∴ x + 1/x = √( x - 1/x )2 + 4

= √( 7 )2 + 4 = √49 + 4 = √53



Example- If x = 17, then find the value of x5 - 18x4 + 18x3 - 18x2 + 18x - 19.
Solution:-
x5 - 18x4 + 18x3 - 18x2 + 18x - 19.
= x5 - 17x4 - x4 + 17x3 + x3 - 17x2 - x2 + 17x + x - 19
= 175 - 17 ( 17 )4 - 174 + 17 ( 17 )3 + 173 - 17 ( 17 )2 - 172 + 17 ( 17 ) + 17 - 19
= 175 - 175 - 174 + 174 + 173 - 173 - 172 + 172+ 17 - 19
= 17 - 19
= -2