NUMBER SYSTEM CHAPTER 4

                             MULTIPLICATION IN SPACIAL CASES

 Multiplication by 9:

       Let us take any number ‘a’ which is to be multiplied by ‘9’.

                      a*9= a*(10-1) ===.> a*10-a

Example : 7253*9=?

·         Put a zero on the right side of the given number.

·         Subtract the given number from the number obtained.

 72530-7253=65277

Example:   1165*9=?

                         11650-1165= 10485.

Example : 5678*9=?

                         56780-5678=51102.

Now we are extend this method to multiplication by number having any number of 9’s

        Like as          9=10-1

                            99=100-1

                            999=1000-1.

    We can use same logic ,we can write several other rules.

            Example :   352*98=?

                              35200-(352*2)

                              35200-704=34496.

Multiplication by 11:

                     a is any number which is to multiplied by 11.

                             (a*11) = a*(10+1)

                                 (a*10)+a

Example : 235*11=?

·         Put a zero on the right side of the given number.

·         Add the given number from the number obtained.

                  235*10=2350,  2350+235=2585.

Example : 7295*11=?

                 7295*10=72950, 72950+7295= 80245.

Example : 3652*11=?

                 3652*10=36520,36520+3652= 40172.

Multiplication by 12:

                        Put a zero on the right side of the given number.Add to it twice the given number.

Example :  312*12=?

                 312*10=3120

                 3120+312*2=3744.

Example : 5427*12=?

               5427*10=54270

               54270+5427*2=65124.

Some more spacial cases:

 When the sum of the digits of units places of the multiplicand and multiplier is 10 and other figure are the same in the both numbers.

     Example : 99*91= ?

       Ans: 9*1=09,   9*10=90

              99*91=9009.

Let us explain in detailed

Step 1: let us take the unit digits first then multiplied.

            9*1 = 09

Step 2: then take the rest of the digits with a number one greater then it.

         9*10 =90.

Step 3: then combine the tenth place and unit place.

               99*91 = 9009

Example : 84*86 =?

               4*6 = 24

               8*9 = 72

              84*86= 7224.

Example : 992*998 =?

                 2*8=16

                 99*100= 9900

                 992*998 = 990016.

If the digits of the units place is the same and sum of the tenth’s place is 10.

  Rule: (product of the ten’s figure+units figure)(product of units figure)

Examples:

        86*26 = (8*2+6)(6*6) = (22)(36) =2236.

        52*52 = (5*5+2)(2*2) = (27)(04) =2704. 

        64*44 = (6*4+4)(4*4) = (28)(16) =2816.

The units figure are ‘5’ and difference between ten’s figure is 1 then

     Rule: [(larger ten’s figure+1)(smaller ten’s figure)][(75)].

Example :

           35*45 = (4+1)(3) [75] =1575.

           65*55 = (6+1)(5) [75] =3575.

          105*95 = (10+1)(9) [75] = 9975.

Multiplication by repeated digits

      Multiplication by 11,111,1111,………… etc

 Example :   1245*11

                  1 (2+1) (4+2) (4+5) 5

                  1245*11= 13695

Explanation:

Ø   We right the extreme right digit of given number.

Ø  Sum of the two right digits of the given number (up to 9 only then add to next number)

Ø  Leave the right digit and add the next  two digits.

Ø  Last step write down the extreme left digit.

Example :

                  2893*11 = ?

                  2 (8+2) (9+8) (9+3) 3

                  2893*11 =31823.

Example :

                 4591*11 =?

                 4 (5+4) (9+5) (1+9) 1

                 4591*11 =50501.

Same as the multiplication by 111

    Example : 5423*111 = ?

                   5 (5+4) (5+4+2) (4+2+3) (2+3) 3

                   5423*111 = 601953.

    Example :  456723*111

                    4 (4+5) (4+5+6) (5+6+7) (6+7+2) (7+2+3) (2+3) 3

                    456723*111 = 50696253.