In Simplification of an expression, there are certain laws which should be strictly adhered to these laws
are as follows.
VBODMAS Rule
This rule gives the correct sequence in which the mathematical operations are to be executed so as to find out
the value of a given expression.
(i) Here, 'V' stands for 'Vernaculum' (or Bar), 'B' stands for 'Bracket', 'O' stands for 'Of', 'D'
stands for 'Division', 'M' stands for 'Multiplication', 'A' stands for 'Addition' and 'S' stands for
'Subtraction'.
Here, 'VBODMAS' gives the order of simplication. Thus, the order of performing the mathematical
operations in a given expression are:
First Vernaculum or Line bracket or Bar
Second Bracket
Third Division
Fourth Of
Fifth Multiplication
Sixth Addition
Seventh Subtraction
The above order should strictly be followed.
(ii) There are four types of brackets
(a) Square bracket[]
(b) Curly bracket{}
(c) Circular bracket()
(d) Bar or Vernaculum __
Thus, in simplifying an expression, all the brackets must be removed in the order '-', '()', '{}',
and '[]'.
Basic Formulae
(i) (a + b)2 = a2 + 2ab + b2
(ii) (a - b)2 = a2 - 2ab - b2
(iii) (a + b)2 - (a - b)2 = 4ab
(iv) (a + b)2 + (a - b)2 = 2(a2 + b2)
(v) (a2 - b2) = (a + b)(a - b)
(vi) (a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca)
(vii) (a3 + b3) = (a + b)(a2 - ab + b2)
(viii)(a3 + b3) = (a - b)(a2 + ab + b2)
(ix) (a3 + b3 + c3 - 3abc) = (a + b + c)(a2 + b2 + c2 - ab - bc - ca)
(x) a3 + b3 + c3 = 3abc if a + b + c = 0
Solved Examples
Type 1
To Find the Value of (?)
Example 1 : 1005 + (500 - 10) - 80 = ?
Solution : 1005 + (500 - 10) - 80
= 1005 + 490 - 80 = 1415
Example 2 1(4/7) + 7(1/3) + 3(3/5) = ?
Solution ? = 1(4/7) + 7(1/3) + 3(3/5)
= (1 + 7 + 3) + (4/7 + 1/3 + 3/5)
= 11 + [(60 + 35 + 63)/(7 × 3 × 5)]
= 11 + (158/75)
= 11 + 2(8/75)
= 13(8/75)
Example 3 (9321 + 5406 + 1001) ÷ (498 + 929 + 660) = ?
Solution. ? = (9321 + 5406 + 1001) ÷ (498 + 929 + 660)
? = 15728/2087 = 7.54
Type 2
Words Problems
Example 4 : If one-third of a tank holds 45 L of water, then the quantity of water that half of the tank hold is
Solution : Let the capacity of the tank be x L.
Then, (1/3) x = 45
=> x = 135
=> x/2 = 67.5L
Example 5 : A boy read (2/7)th of a book on one day and (4/5)th of the remainder on another day.
If there were 12 pages unread, how many pages did the book contain?
Solution. Part read on first day = 2/7
Remaining part = (1 - (2/7)) = 5/7
Part read on second day = 4/5 × 5/7
= 4/7
Unread part = 1 - [(2/7) + (4/7)]
= 1 - 6/7 = 1/7
Then, let the total number of pages be x.
=> 1/7 x = 12
=> x = 84 pages
Example : 7 A certain number of books were purchased for 300 Rupees. Five more books could have been purchased
in the same amount, If each book was cheaper by 10 rupees. The number of books purchased was
Solution : Let the number of books purchased be x.
Then, (300/x) - (300/(x + 5)) = 10
1/x - 1/(x + 5) = 1/30
x = 10