Simplification - Notes | Study Quantitative Techniques for CLAT - CLAT

Simplification

VBODMAS Rule

• When an expression involves the + and – signs, the order of operation is from left to right.
• Thus 20 – 15 + 2 – 4 means that 15 is to be subtracted from 20 then 2 is to be added to the result and then 4 is to be subtracted from the result.
• When an expression involves x and ÷ signs, the order of operation is from left to right. Thus 55 ÷ 5 x 3 means that 55 is to be divided by 5 and the result is to be multiplied by 3.
• 45 x 5 ÷ 3 means that 45 is to be multiplied by 5/3.
• When an expression involves all the signs (i.e. +, –, x, ÷) then division and multiplication must be performed before addition and subtraction.

• The expressions within brackets are to be treated as a single identity thus 6 x (3 – 2) means that 6 is to be multiplied by the difference of 3 and 2.
• So 6 x (3 – 2) = 6 x 1 = 6

Note: Operations within the bracket are to be carried first.

Brackets are of 4 types. 

• Bar bracket or Vinculum
• Circular bracket ( )
• Curly bracket { }
• Square bracket [ ]

Removal of a bracket

• 5 + (9 + 2) is equal to 5 + 11 = 16
• 5 + (9 – 2) is equal to 5 + 7 = 12
• The rule is that when a bracket is preceded by a + sign, the bracket may be removed without making any change in the expression.
• Now consider the expressions when the bracket is preceded by a ‘–’ sign
•  20 – (8 + 3) = 20 – 11 = 9 also 20 – (8 + 3) = 20 – 8 – 3 = 9
• 20 – (8 – 3) = 20 – 5 = 15 also 20 – (8 – 3) = 20 – 8 + 3 = 12 + 3 = 15
• The rule is that if a bracket is preceded by a negative (–) sign, the bracket can be removed by changing the sign of every term within the bracket.

Bracket within a bracket

• Let us take an example
• Supposing we want to simplify

• We will first remove the bar bracket then circular bracket, then curly bracket and at the last stage we remove the square bracket.
• So the expression given above is

= 24 – [12 – {8 – (9 – 2 – 3)}]

= 24 – [12 – {8 – 4}]

= 24 – [12 – 4]

= 24 – 8

= 16

• The rule of VBODMAS gives us the arrangement according to which an expression is to be simplified. VBODMAS stands for vinculum, brackets, of, division, multiplication, addition and subtraction respectively.
• Algebraic Identities: The algebraic identities noted below are used as expression formulae:

General Rules For Solving Problems In Arithmetic

• (a+b) (a-b) = a² - b² 
• (a+b)² = a² + b² + 2ab 
• (a – b) ² = a² + b² – 2ab 
• (a + b)³ = a³ + b³ + 3ab (a+b) 
• (a – b)³ = a³ – b³ – 3ab (a–b)
• a² + b²  =  (a + b)²   – 2ab
• a³ + b³ = (a+b) (a² + b² – ab) = (a+b)³ – 3ab (a+b) 
• a³ - b³  =  (a – b) ( a² + b² + ab) =  (a–b)³ + 3ab (a-b) 
• a³ + b³ + c³ = (a + b + c) (a² + b² + c² – ab – bc – ca) + 3abc If a + b + c = 0, then a³ + b³ + c³ = 3abc 
• (a+b+c)² = a²+b²+c²+2ab+2bc+2ca 
• (a+b+c)³ = a³+b³+c³+3a² (b+c) + 3b²(a+b)+3c²(a+b)+6abc [remember there are ten terms] 
• To find the sum of a and b, given their difference and product:
  a + b =    
• To find the difference of a and b, given their sum and product: a – b =

 Questions on Simplification 

• Generally the types of questions which can be asked for simplification are of the following types.

 Type 1 

• Questions involving expressions in which the items / numbers are either to be added or subtracted. The numbers may be integers or decimals.

 Type – II 

• The numbers to be added or subtracted are fractions. The fractions may be pure or mixed fractions.

 Type – III 

• The numbers or items in the expression are connected with the signs of multiplication or division e.g.

i) 1.6 x 20 x 12 + 30 x 2

ii) 2.21 ÷ 0.7 = ? + 5.5

iii) 4545 ÷ 50 ÷ 5 = ?

Type – IV 

• These questions contain signs of multiplication, division, percentage, of, etc and also fraction e.g.

i) 35% of 495 + ? = 250

ii)

iii) 30% 150 + ? of 300 = 40 % of 450

 Type V

• Square – roots and cube roots along with fractions etc. are there in this type of questions on simplification like.

Type VI 

• This type of questions may relate to surds and / or indices like

i) (19.7)⁵ ÷ (19.7)⁴ =?  

ii) 23^10.5 x 23^5.1 ÷ 23^1.2 = 23^?

• Type VII may relate to inequalities or equalities with signs of (<, =, >)