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The Base Method is a technique used for multiplication
Suppose I ask you how long will you take to multiply9999998 by 9999999 ?
Perhaps you would sink at the sheer thought of multiplying
these numbers
However, using the Base Method you can get the answer in
less than 5 seconds !
Yes, it is true !
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In the base method, we use certain bases.
Normally, we use numbers like 10, 100, 1000 etc. as
bases,.
However this is not a rule.
You can use any other number toobut generally we prefer
powers of 10
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We select a particular base depending on the numbers
given in the question
Suppose we are asked to multiply 996 by 998, in this case
we take 1000 as the base as both the numbers are closer to
1000
Suppose we are asked to multiply 105 by 112, in this case
we take 100 as the base as both the numbers are closer to
100..
In the questions that follow, we will solve the answer in two
parts
The left hand part will be called LHS and the right hand partwill be called RHS
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Let us have a look at the steps involved in the procedure..
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STEPS
Find the Base and the Difference
Number of digits in RHS = Number of zeros in Base
Multiply the differences in RHS
Put the cross answer in LHS
These are the four basic rules of the system. However,
they will only become clear when we solve some
examples..We will solve three examples simultaneously..
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(Q) Find the answers to the following questions.
(A) (B) (C)
9 6 9 9 8 8 9 9 9 9 9 9 9
x 9 8 x 9 9 9 6 x 9 9 9 9 9 9 9
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STEP AFind the Base and the Difference
The first part says find the base and the difference.
Let us have a look at example A. The numbers are 96 and
98. Since, both the numbers are closer to 100 our base in
this case will be 100.
Similarly, in example B both the numbers are closer to
10,000 so our base will be 10,000
In Example C, both the numbers are closer to 1,00,00,000
and hence we will take 10 million (1 crore) as our base
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So, our bases will be
(A) (B) (C)
100 1 0 0 0 0 1 0 0 0 0 0 0 0
9 6 9 9 8 8 9 9 9 9 9 9 9
x 9 8 x 9 9 9 6 x 9 9 9 9 9 9 9
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We are still on Step A.
Next, we have to find the differences
In Example A, the difference between 100 and 96 is 4 and the
difference between 100 and 98 is 2
In Example B, the difference of 9988 from 10,000 is 12 and thedifference of 9996 from 10,000 is 4
In Example C, the difference of both the numbers from ten million is 1
(A) (B) (C)
100 1 0 0 0 0 1 0 0 0 0 0 0 0
9 6 9 9 8 8 9 9 9 9 9 9 9
x 9 8 x 9 9 9 6 x 9 9 9 9 9 9 9
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Step A is complete. The question looks as given below :
(A) (B) (C)
100 1 0 0 0 0 1 0 0 0 0 0 0 0
9 6 - 4 9 9 8 8 - 12 9 9 9 9 9 9 9 - 1
x 9 8 - 2 x 9 9 9 6 - 4 x 9 9 9 9 9 9 9 - 1
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Let us make a provision for it..
(A) (B) (C)
100 1 0 0 0 0 1 0 0 0 0 0 0 0
9 6 - 4 9 9 8 8 - 12 9 9 9 9 9 9 9 - 1
x 9 8 - 2 x 9 9 9 6 - 4 x 9 9 9 9 9 9 91
_ _ _ _ _ _ _ _ _ _ _ _ _
We have put equivalent empty blanks in the RHS to
accommodate the answer
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We come to Step C which says, multiply the differences and
put it as the RHS of the answer
(A) (B) (C)
100 1 0 0 0 0 1 0 0 0 0 0 0 0
9 6 - 4 9 9 8 8 - 12 9 9 9 9 9 9 9 - 1
x 9 8 - 2 x 9 9 9 6 - 4 x 9 9 9 9 9 9 91
_ _ _ _ _ _ _ _ _ _ _ _ _
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(A) (B) (C)
100 1 0 0 0 0 1 0 0 0 0 0 0 0
9 6 - 4 9 9 8 8 - 12 9 9 9 9 9 9 9 - 1
x 9 8 - 2 x 9 9 9 6 - 4 x 9 9 9 9 9 9 91
0 8
In Example A, we multiply the differences -4 by -2and get the
answer 8. But, we need a two-digit answer. So, we express thenumber 8 as 08 and put it in the RHS.
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(A) (B) (C)
100 1 0 0 0 0 1 0 0 0 0 0 0 0
9 6 - 4 9 9 8 8 - 12 9 9 9 9 9 9 9 - 1
x 9 8 - 2 x 9 9 9 6 - 4 x 9 9 9 9 9 9 91
0 8 0 0 4 8
In Example B, we multiply the differences -12 by -4and get the
answer 48. But, we need a four-digit answer. So, we express the
number 48 as 0048 and put it in the RHS.
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(A) (B) (C)
100 1 0 0 0 0 1 0 0 0 0 0 0 0
9 6 - 4 9 9 8 8 - 12 9 9 9 9 9 9 9 - 1
x 9 8 - 2 x 9 9 9 6 - 4 x 9 9 9 9 9 9 91
0 8 0 0 4 8 0 0 0 0 0 0 1
In Example C, we multiply the differences -1 by -1and get the
answer 1. But, we need a seven-digit answer. So, we express the
number 1 as 0000001 and put it in the RHS.
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(A) (B) (C)
100 1 0 0 0 0 1 0 0 0 0 0 0 0
9 6 - 4 9 9 8 8 - 12 9 9 9 9 9 9 9 - 1
x 9 8 - 2 x 9 9 9 6 - 4 x 9 9 9 9 9 9 91
9 40 8 0 0 4 8 0 0 0 0 0 0 1
In Example A, the cross answer can be obtained by doing (962)or (984) in a cross manner. In either case, the answer is 94.
Thus, the complete answer of 96 x 98 is 9408
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(A) (B) (C)
100 1 0 0 0 0 1 0 0 0 0 0 0 0
9 6 - 4 9 9 8 8 - 12 9 9 9 9 9 9 9 - 1
x 9 8 - 2 x 9 9 9 6 - 4 x 9 9 9 9 9 9 91
9 4 0 8 9 9 8 40 0 4 8 0 0 0 0 0 0 1
In Example B, the cross answer can be obtained by doing (998812) or (99964) in a cross manner. In either case, the answer is
9984.
Thus, the complete answer of 9998 x 9996 is 99840048
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(A) (B) (C)
100 1 0 0 0 0 1 0 0 0 0 0 0 0
9 6 - 4 9 9 8 8 - 12 9 9 9 9 9 9 9 - 1
x 9 8 - 2 x 9 9 9 6 - 4 x 9 9 9 9 9 9 91
9 4 0 8 9 9 8 40 0 4 8 9 9 9 9 9 9 80 0 0 0 0 0 1
In Example C, the cross answer can be obtained by doing(99999991) in a cross manner. In either case, the answer is
9999998.
Thus, the complete answer of 99999980000001
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Let us do one complete example.
(Q) Multiply 99980 by 99980
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9 9 9 8 0
x 9 9 9 8 0
Both the numbers are closer to 1,00,000so the base is 1, 00,000The difference between the base and 99,980 is 20 each
The base has 5 zeros and so RHS will be a 5 digit answer
Multiplying the differences will yield 20 x 20 = 400. We write the
number as 00400
The cross answer is 99,98020 = 99,960.
The final answer is 9996000400
1,00,000
- 20
- 20
_ _ _ _ _0 0 4 0 09 9 9 6 0
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(Q) Multiply 999 by 850
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9 9 9
x 8 5 0
Both the numbers are closer to 1,000so the base is 1, 000The difference between the base and the numbers is (-1) and (-
150) respectively
The base has 4 zeros and so RHS will be a 4 digit answer
Multiplying the differences will yield 1 x 150 = 150. We write thenumber as 0150
The cross answer is 8501 = 849
1,000
- 1
- 150
_ _ _ _0 1 5 08 4 9
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Let us take an example where both the numbers are above
the base
(Q) Multiply 1002 by 1003
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1 0 0 2
x 1 0 0 3
Both the numbers are closer to 1,000so the base is 1, 000The difference between the base and the numbers is (+2) and (+3)
respectively
The base has 3 zeros and so RHS will be a 3 digit answer
Multiplying the differences will yield 2 x 3 = 6. We write the numberas 006
The cross answer is 1002 + 3 = 1005. The complete answer is
1005006.
1,000
+ 2
_ _ _0 0 61 0 0 5
+ 3
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Let us take an example where both the numbers are above
the base
(Q) Multiply 10010 by 10010
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1 0 0 1 0
x 1 0 0 1 0
Both the numbers are closer to 10,000so the base is 10,000The difference between the base and the numbers is (+10) and
(+10) respectively
The base has 4 zeros and so RHS will be a 4 digit answer
Multiplying the differences will yield 10 x 10 = 100. We write thenumber as 0100
The cross answer is 10010 + 10 = 10020. The complete answer is
100200100.
10,000
+ 10
_ _ _ _0 1 0 01 0 0 2 0
+ 10
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Sometimes you might have a number where there is a
carry-over involved. Let us see how to deal with such
cases.
Suppose you have to multiply 950 by 950
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9 5 0
x 9 5 0
Both the numbers are closer to 1,000so the base is 1,000
The difference is (-50) and (-50) respectively
The base has 3 zeros and so RHS will be a 3 digit answer
Multiplying the differences will yield (-50) x (-50) = 2500. The number
2500 is a four-digit number but we can fit only 3 digits. So, we write downthe last three digits (500) and carry over 2.
The cross answer is 95050 = 900 (+2 carry over) is 902.
The final answer is 902500
1,000
- 5 0
_ _ _9 0 0
- 5 0
5 0 02
9 0 2 5 0 0
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(Q) Multiply 1200 by 1020
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1 2 0 0
x 1 0 2 0
Both the numbers are closer to 1,000so the base is 1,000
The difference between the base and the numbers is (+200) and(+20) respectively
The base has 3 zeros and so RHS will be a 3 digit answer
Multiplying the differences will yield (+200) x (+20) = 4000. The
number 4000 is a four-digit number but we can fit only 3 digits. So,
we write down the last three digits (000) and carry over 4.
The cross answer is 1200 + 20 (+4 carry over) is 1224.
1,000
+ 2 0 0
_ _ _1 2 2 4
+ 2 0
0 0 0
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We come to the last part of this topic..
How to multiply one number above the base and one
number below the baseSuppose you have to multiply 95 by 115.
In this case, 95 is below the base (100) and 115 is above
the base (100)
In such examples, where one number is above the base
and one number is below the base. we use the same
technique with a different last step.
Let us have a look at an example :
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9 5
x 1 1 5
The base is 100 and the differences are (-5) and (+15)
respectively. RHS will be a 2 digit answer
Multiplying the differences will yield (-5) x (+15) = (-75). The LHS
will be 110. Our final answer is 110(-75). However, minus sign isnot permitted in the final answer.
In such cases, we multiply LHS with Base and subtract RHS to get
the final answer.The final answer will be 110 x 10075 = 10925.
100
- 5
_ _1 1 0
+ 1 5
(- 7 5 )
1 0 9 2 5
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(Q) Multiply 1042 by 998
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1 0 4 2
x 9 9 8
The base is 1000 and the differences are (+42) and (-2)
respectively. RHS will be a 3 digit answer
Multiplying the differences will yield (+42) x (-2) = (-084). The LHS
will be 1040. Our final answer is 1040(-084).We multiply LHS with Base and subtract RHS to get the final
answer.The final answer will be 1040 x 1000(084) = 1039916.
1000
+ 4 2
_ _ _1 0 4 0
- 2
(- 0 8 4)
1 0 3 9 9 1 6
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Thus, we have seen many variations with the Base Method.
You can use this method for a variety of multiplication problems
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Dh l B thi