Simplification with solved examples and shortcut methods

1.Simplification

Rule: BODMAS

V=  Vinculum  means bar as  ￣

B = Bracket (),{],[]

O = of

D = Division 

M = Multiplication

A = Addition

S = Subtraction

i. Addition and Subtraction of Whole Number

Ex. 6874+9632+9841+8974+21147

Sol. 56,468

Ex. 6589+4125-7841+2125+4998

Sol. 9996

ii.  Addition and subtraction of Decimal:

For the solution, we put zeros after the decimal equal to the maximum digit after decimal present in the total numbers and then we operate addition and subtraction.

Ex. 43.632 + 3.05 + 437.102 + 232.56

Sol. +  43.632

+    3.050

+437.102

-232.590

251.224 Ans.

Ex. 268.31+834.8+2.061 = x +5.728

Sol. X = 268.31 + 834.80 - 2.061 - 5.728 = 1095.321 Ans.

iii. Addition and Subtraction of Mixed Fractions:

In these types of questions, whole numbers are added and subtracted together.

Ex. 27½+15¾-12⅖+18⅘

Sol. (27+15-12+18) +1/2 + 3/4 - 2/5 +4/5

       48+ 33/20

       48+1+13/20

2.Some multiplication Techniques:

Case i. When both numbers are greater than 100.

Ex. 106*103

Sol. 106 is 6 distances away from 100 and 103 is 3 distances away from 100.

        106 * +6

        103 * +3

        109    18

So the solution is 10918

Case ii. When both numbers are less than 100.

Ex. 85 * 95 

Sol. 88 is -12 distance away from 100 and 95 is -5 distance away from 100.

       88   -12

       95    -5

       83   60

       8360 is solution    

Add 88 with -5 or 95 with -5 we get 83 and multiply -12 * -5 = 60

Case iii. When one is greater than 100 and the other is less than 100.

Ex. 105 * 93

Sol. 105 is 5 distances away from 100 and 93 is -7 distance away from 100

         105   +5

          93    -7

       9800   -35

9800-35 = 9765 is the solution 

Add 105 with -7 or 93 with +5 we get 98

Multiply +5 * -7 = -35

3. Find the square of the numbers:

i. Formula method

(a/b)² = a²/2ab/b²

Ex.1 42²

Sol. 4² / 2*4*2 / 2²

         16/16/4

Pass 1 to 16 it becomes 17

then the solution is  1764 Ans.

Ex 2. Find a square of 112

Sol. Break it into two parts 11/2

Now using formula

11²/2*11*2/2²

121/44/4

Pass  and add 4 to 121 it becomes 121+4 = 125

the equation becomes 125/4/4  =12544 Ans.

ii. Squares of the numbers of having 5 in the unit's place can be calculated easily.

(n5)² = [n*(n+1)]25

Ex.1 25² = [2*(2+1)]25 = (2 * 3)25 = 625 

Ex.1 45² = [4*(4+1)]25 = (4 * 5)25 = 2025 

iii. When the number is between 41 and 50.

Ex. Find the square of 44.

Sol. 44 is 6 distances away from 50

       5² = 25, So we add -6 to 25 we got 19.

      And -6² = 36

Thus the square of 11 is 1936.

iv. When the number is between 50 and 60.

Ex. find the square of 58.

Sol. 58 is a +8 distance away from 50.

then,

25+8 = 33

8² = 64

3364 is the square of 58.

v. When the number is less than 100.

Ex. find the square of 96.

Sol. .96 is 4 distances away from 100.

        Subtract 4 from 96 

        (96-4)/4² = 9216 Ans

vi. When the Number is greater than 100

Since the base is 100 the right part is always in 2 digits.

Ex. Find the square of 108

Sol. 108 is 8 distances away from 100.

      (108+8)/8² = 11664 Ans.

Ex. Find the square of 114.

Sol. 114 is 14 distance away from 100

        (114+14)/14² = 12996 Ans.

vii. When the Number is less than 200.

Ex. Find the Square of 194

Sol. 194 is 6 distances away from 200.

       (194-6)/6² = 188 *2/36 = 37636 Ans.

imp. For Base 200 we multiply the left-hand side result by 2

viii. When the number is near 1000

Ex. Find the square of 996

Sol. (996-4)/4² = 99216

    

imp. For base 1000 right-hand side part has 3 digits.

Ex. find the square of  1016.

Sol. 1016+16/16² = 1032256

4. Square Roots

Last digit of squares of :

1 = 1

2 = 4

3 = 9

4 = 6

5 = 5

6 = 6

7 = 9

8 = 4

9 = 1

So the last digit of,

1 is the last digit of squares of 1 and 9.

4 is the last digit of squares of 2 and 8.

9 is the last digit of squares of 3 and 7.

6 is the last digit of squares of 4 and 6.

5 is only when square 5.

i. perfect squares:

 If the last digit of the number is 1, 4, 9, 6, 5 then it may be a perfect square.

ii. Non-Perfect squares:

If the last digit of the number is 2, 3, 7, 8. Then it is called a Non-perfect square.

Ex. The square root of 9216.

Sol. See the last 2 digits of 9216.

       Then solution may have 4 or 6 as its last digit.

       Now,

       9² is less than 92. The first digit is 9

       The solution may be 94 or 96.

       Now,

       9 *(9+1) = 90

       90 is less than 92

       thus the solution is 96.

Ex. The square root of 17424

Sol. The last digit is 4.

       Thus the last digit may be 2 or 8.

       And 13² < 174 < 14²

       First digit is 13

       Solution may be 132 or 138

       13 * 14 = 182

       174 is less than 182 thus solution is 132.

5. Cube Roots

Ex. Cube root of 300763.

Sol. The last three digits are 763

        Now the cube root of the last digit is 27. 

        Thus 7 is the last digit of the answer

        6³ < 300 < 7³

        6 is the first digit

       So, 67 is the cube roots

Ex. Cube root of 175616.

Sol. See the last three-digit is 616.

       The cube of the last digit is 6 is 216.

       Thus last digit is 6

       5³ < 175 < 6³ 

       First digit is 5.

       Thus the solution is 53