November 16, 2020
Multiplication of numbers with a series of 9's
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Multiplication of numbers with a series of 9's
In my seminars , I often have
an audience challenge round . In this round , the audience members ask me to
perform various mental calculations and give them the correct answer . They
generally ask me to multiply numbers which involve a lot of 9's in them . The
general perception is that the higher the number of 9's the tougher it will be
for me to calculate . However , the truth is exactly the opposite - the higher
the number of 9's in the question , the easier it is for me to calculate the
correct answer . I use two methods for this . The first method is given below
and the second method is explained in the chapter ' Base Method of
Multiplication ' .
Using
the method given below , we can multiply any given number with a series of
nines . In other words , we can instantly multiply any number with 99 ,
999,9999 , 99999 , etc.
The technique is divided
into three cases . In the first case , we will be multiplying a given number
with an equal number of nines . In the second case we will be multiplying a
number with a higher number of nines . In the third case , we will be
multiplying a number with a lower number of nines ,
Case 1 ( Multiplying a number with an equal number of nines )
(
Q ) Multiply 654 by 999
• 654
x 999
-------
=653 346
• We subtract 1 from 654 and write half the answer as 653 . Answer at
this stage is 653 . .
• Now we will be dealing with 653. Subtract each of the digits six ,
five and three from nine and write them in the answer one by one ,
• Nine minus six is 3. Nine minus five is 4. Nine minus three is 6
.
• The answer already obtained was 653 and now we suffix to it the digits
3 , 4 and 6. The complete answer is 653346
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( Q ) Multiply 9994 by 9999
• 9094
X9099
-----------
=9993 0006
We subtract one from 9994
and write it as 9993. This becomes our left half of the answer . Next , we . ,
447 x 900 subtract each of the digits of 9993 from 9 and write the answer as
0006. This becomes the right half of the answer . The complete answer is
99930006
(Q) Multiply 456789 by 999999
• 156789
x 99999
------------------
=456789
543211
We subtract 1 from
456789 and get the answer 456788. We write this down on the left hand side .
Next , we subtract each of the digits of 456788 ( left hand side ) from 9 and
get 543211 which becomes the right hand part of our answer . The complete
answer is 456788543211
More examples :
• 7777
X 9999
----------
=77762223
X99999
-----------
=6543134568
X999
----------
=446553
X99999
-----------
=9090809091
The simplicity of this method can be vouched from the examples given above . Now we move toCase 2. In this case , we will multiply a given number with a higher number of nines .
Case 2
( Multiplying a number with a higher number of nines )
( Q )
Multiply 45 with 999
45 → 045X999=044955
X 999
--------
There are three nines in the multiplier . However , the
multiplicand 45 has only two digits . So we add a zero and convert 45 to 045
and make it a three digit number . After having done so , we can carry on with
the procedure explained in Case 1
. First we subtract 1 from 045 and write it down as 044. Next ,
we subtract each of the digits of 044 from 9 and write the answer as 955. The
complete answer is 044955 ,
( Q ) Multiply
888 with 9999
888 0888X9999 = 8879112
X9999
We convert 888 to 0888 and
make the digits equal to the number of nines in the multiplier . Next , we
subtract 1 from 0888 and write the answer as 0887. Finally , we subtract each
digit of 0887 from 9 and write the answer as 9112. The final answer is 08879112
which is 8879112
( Q )
Multiply 123 by 99999
123 → 00123 X99999 =
x 99999 00122 / 99877
The multiplicand is a three - digit number and the multiplier is a five
- digit number . Therefore we
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add two zeros in the
multiplicand so that the digits are equal in the multiplicand and the
multiplier .
We now subtract 1 from
00123 and write the left hand part of the answer as 00122. Next , we subtract
each of the digits of the left hand part of the answer from 9 and write it down
as 99877 as the right hand part of the answer . The complete answer is
12299877
Other examples :
• 162
X9999
-----------
0161/9838
X99999
-----------
05554/94445
• 363
X999999
-------------
000362/999637
We can see that this technique is not only simple and easy to follow , but it also enables one to calculate the answer in the mind itself . This is the uniqueness of these systems . As you read the chapters of this book , you will realize how simple and easy it is to find the answer to virtually any problem of mathematics that one encounters in daily life and especially in the exams . And the approach is so different from the traditional methods of calculation that it makes the whole process enjoyable
( Q ) Multiply 654 by 99 : In this case the number of digits are more than the number of nines in the multiplier . Instead of multiplying the number 654 with 99 we will multiply it with ( 100-1 ) . First we will multiply 654 with 100 and then we will subtract from it 654 multiplied by
1.
654
X 99
65400
- 654
--------------
= 64746
( Q )
Multiply 80020 by 999
We will multiply 80020 with (
1000 - 1 ) .
80020000
- 800203
---------------
79939980
This method is so obvious that it needs no further
elaboration .
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