Multiplying numbers by 9, 99, 999 and so on (Part-2)

In Part-1, we saw how to multiply numbers by 9, 99, 999, 9999 and so on in the fastest way for multiplier having equal or more digits than the multiplicand. Here in part-2; we will see how to multiply numbers by 9, 99, 999, 9999 and so on in the faster way irrespective of digits the multiplicand have. That is; this method is applicable for multiplying any number with any-number of digit composed of 9.

Write as many ‘0’ after the multiplicand as there are ‘9’ in the multiplier (the number composed of 9) and then subtract the original number from this result.

Let’s illustrate with example:

                      59876

          X    999       

               ?

There are three 9’s in the number composed of 9. So, place three ‘0’ after 59876 getting 59876000 and then subtract 59876 from 59876000.

Now, 59876000 – 59876 = 59816124.

So, 59876 x 999 = 59816124 

More examples,

9×9                       = 90 – 9                          = 81(there is one 9, so put one 0)

46×9                     = 460−46                        = 414 (there is one 9, so place one 0)

46×99                   = 4600 – 46                    = 4554 (two 9’s, so place 00)

38×999                  = 38000-38                     = 37962 (three 9’s, put 000)

59876 × 9999        = 598760000 – 59876      = 598700124 (four 9’s)

6789543 × 99999   = 678954300000 – 6789543 = 678947510457

Math behind this: This works because of the Distributive law of multiplication

9        = 10-1

99      = 100-1

999    = 1000-1

9999  = 10000-1

And so on…

Hence,

46 × 99        = 46 × (100-1)       = 4600 – 46                     = 4554

38 × 999      = 38 × (1000-1)     = 38000 – 38                   = 37962

59876×9999 = 59876×(10000-1)=598760000–59876         = 598700124