Multiplying by a number consisting only of 9’s i.e. 9, 99, 999 and so on (Part-1)

Here is the most inquisitive method to multiply numbers by 9, 99, 999, 9999 and so on in the fastest way when the multiplier (the number composed of 9) has equal or more number of digits than the multiplicand (number to be multiplied).

Step-1:  Write 1 less than the multiplicand. This is the first part of the answer.

Step-2:  The last part of the answer is the difference between the multiplier and the findings in step-1 or complement of the multiplicand (subtracting ‘all from 9 and the last from 10’).

Note:  You can carry out step-2 this way also: 

If the number of digits of the multiplier (the number composed of 9) is more than the number of digits of multiplicand; then affix 9’s for every extra digit in the multiplier i.e. as many extra digits as many 9’(s) and then Subtract each digit of the multiplicand ‘all from 9 and the last from 10’.

Let’s illustrate with example:

                       

	873
×	999
	?

Step-1:     1 less than 873 is 872. You just subtract 1 from the last (one’s) digit. This is the first part of the answer.

Step-2:     Then subtract 872 from the multiplier (999); 999 - 872 = 127. (It may be easier to calculate the difference of each digit individually with 9.)

Or, Complement of the multiplicand 873; 1000 – 873 = 127. (Subtracting ‘all from 9 and the last from 10’).

That’s it. Simple, easy and fast.

So, our final answer is, 873 X 999 = 872,127.

More examples,

3 × 9        = 1 less than first digit (3-2) i.e 2 and (9-2)= 7 i.e, 27.

35 × 99   =1 less than 35 i.e, 34 and (9-3), (10-5)= 65 i.e, 3465.

Now, 145 × 99999 =?

1 less than 145 is 144. Since, the multiplier (99999) has two more digits than the multiplicand (145), so affix two 9’s after 144 to get 14499.

The last part of the answer will come from subtracting each digit of multiplicand ‘all from 9 and the last from 10’. So, (9-1) = 8, (9-4) = 5 and (10-5) = 5 i.e, 855

So the final answer is 14,499,855.

See the pattern?

6789543 x 9999999 =?

1 less than 6789543 is 6789542

And, (9-6), (9-7), (9-8), (9-9), (9-5), (9-4), (10-3) = 3210457

So, 6789543 x 9999999 = 67895423210457 

Find 878 x 9999

• 878 - 1 = 877

• 9999 - 877 = 9122 (or Complement of 878, 10000 - 878)

So, 878 x 9999 = 8779122

It is so simple that the whole calculation can be done mentally, with no help from calculators, calculation tables or even pen and paper, even if the number is a gigantic one.

More examples,

1456 × 9999                = 14558544

1456 × 99999              =145598544

1456 × 999999            =1455998544