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Digital Sum Balance (D.S.)


A single digit is obtained by adding all the digits of a number is called Digital Sum Balance (D.S.)
 
Ex : 12 → 1 + 2 → 3

24 → 2 + 4 → 6

21 → 2 + 1  → 3

9876 → 9 + 8 + 7 + 6 → 30 → 3 + 0 → 3

3452 → 3 + 4 + 5 + 2 → 14 → 1 + 4 → 5

Or,  On Dividing the digits by 9, the obtained Remainder is Digital Sum Balance. 


Ex :    12

  •    ━━ = 3

               9

  •      24

━━━ = 6

     9

  •    3452

━━━━ =  5

     9

KEY HIGHLIGHTS : The range of D.S. always falls from 0 to 9.

When we add 9 to a number, its Digital Sum will not change therefore in order to calculate Digital Sum we need to cancel out  all the 9’s. 


9 + 1 → 10 → 1

9 + 2 → 11 → 2

9 + 3 → 12 → 3

9 + 4 → 13 → 4

9 + 5 → 14 → 5

9 + 6 → 15 → 6

9 + 7 → 16 → 7

9 + 8 → 17 → 8

9 + 9 → 18 → 9


Example : 12345 → 6 → Digital Sum

367215 → 6 → Digital Sum

                 49219 → 7 → Digital Sum  

                 59991 → 6 → Digital Sum

                 637218 → 9 → Digital Sum

63271894503 → 3 → Digital Sum


When the Digital Sum on the Digits turns out to be 0, consider it to be 9 (During Addition and Subtraction). But this does not follow for multiplication.


Decimal and percentage do not have any impact on Digital Sum of a number.

Example : 14 → 5 → Digital Sum

                 1.4 → 5 → Digital Sum

                 140 → 5 → Digital Sum

                 1400 → 5 → Digital Sum


Example  : 62% → 8 → Digital Sum 

                  6.2 % → 8 → Digital Sum

                  620 % → 8 → Digital Sum

Properties of Digital Sum 

  •  D.S. (a + b) = D.S. {D.S.(a) + D.S.(b)}

  • D.S. (a - b) = D.S. {D.S.(a) - D.S.(b)}

  • D.S. (a x b) = D.S. {D.S.(a) x D.S.(b)}

Example of  Property 1: D.S. (a + b) = D.S. {D.S.(a) + D.S.(b)}

  •                  786        +    152    =    938

                             ↓                  ↓              ↓

                       D.S. (21)  +   D.S.(8)  =   D.S. (11)

                             ↓                  ↓              ↓

                             3         +       8      =      2

                                11

                                  ↓

                                  2 → Digital Sum

                                  L.H.S.            =  R.H.S.

  •                46      =      34        +     12

                           ↓                 ↓                 ↓

                     D.S. (10)  =  D.S. (7 )  +   D.S. (3)

                           ↓                 ↓                 ↓

                            1                7        +       3    = 10

                                                                            ↓

                                                 Digital Sum ←  1     

               L.H.S.    =  R.H.S.            

  

Example of  Property 2: D.S. (a - b) = D.S. {D.S.(a) -  D.S.(b)}

  •                 962        -    151     =     811

                             ↓                  ↓              ↓

                        D.S. (8)   -   D.S.(7)  =   D.S. (10)

                             ↓                  ↓              ↓

                             8        -        7       =      1

                                  1 → Digital Sum

                                 L.H.S.              =  R.H.S.

  •                 729        -    591     =    138

                             ↓                  ↓              ↓

                        D.S. (9)   -   D.S.(6)  =   D.S. (3)

                             ↓                  ↓              ↓

                             9        -        6       =      3

                                  3 → Digital Sum

                                 L.H.S.              =  R.H.S.


Example of  Property 3: D.S. (a x b) = D.S. {D.S.(a) x  D.S.(b)}

  •                  35        x     16     =     560

                             ↓                  ↓              ↓

                        D.S. (8)   x   D.S.(7)  =   D.S. (11)

                             ↓                  ↓              ↓

                             8        x       7       =      2

                         56 → 11 

                                    ↓  

                                    2 → Digital Sum

                                 L.H.S.              =  R.H.S.

  •                  54        x      76     =    4104

                             ↓                  ↓              ↓

                        D.S. (9)   x   D.S.(13)  =   D.S. (9)

                             ↓                  ↓              ↓

                             9        x       4       =      9

                          36 → 9  

                                    ↓  

                                    9 → Digital Sum

                                 L.H.S.              =  R.H.S.

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