Quantitative Aptitude :: Simplication Formulas

Simplication Questions and Answers

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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