In the previous article, we learnt multiplication by taking base. We discussed following three cases:
A. Numbers are below the base number
B. Numbers are above the base number
C. One number is above the base and the other number is below it
And now, we shall discuss rest of the two cases here:
D. Numbers are not near the base number, but are near a multiple of the base number, like 20, 30, 50 , 250 , 600 etc.
E. Numbers near different bases like multiplier is near to different base and multiplicand is near to different base.
Before going further, let us take few examples of larger numbers with the same method, which we have discussed in previous article.
Example: 6848 x 9997
6848 -3152
9997 -0003
----------------
6845 | 9456 (Refer to previous article for details of approach)
----------------
Example: 87654 x 99995
Let us discuss the fourth case:
D. Numbers are not near the base number, but are near a multiple of the base number, like 20, 30, 50 , 250 , 600 etc.
As you know that this method is applicable in all the cases, but works best when numbers are close to the base. Here, we shall apply the same sutras “All from 9 and the last from 10” and "Vertically and Crosswise" discussed earlier and also the sub sutra i.e. "Proportionately".
Take example as 207 x 213 . Here the numbers are not near to any of the bases that we used before: 10, 100, 1000 etc. But these are close to '200'. So, when neither the multiplicand nor the multiplier is near to the convenient power of 10 then we can take a convenient multiple or sub-multiple of a suitable base (as our 'Working Base'). And then perform the necessary operation by multiply or divide the result proportionately. Like in this example, we take 100 as a 'theoretical base' and take multiple of 100 i.e. 200 (100 x 2) as our 'working base'.
207 x 213
207 +007
213 +013
------------
= 220 | 091
------------
= (220 x 200) | 91
= 44000 + 91
= 44091
As they are close to 200, therefore deviations are 7 and 13 as shown above. From the usual procedure (refer to previous article), we get, 220 | 91. Now since our base is 200, we multiply the left-hand part of the answer by 200 and add it to the right-hand part. That is, (220 x200) + 91 {(Left side x base) + right side}. The final answer is '44091'.
Let us take one more example: 406 x 417
Here Working base is 400 ,which is (100 x 4)
406 +06
417 +17
------------
= 423 | 102
------------
= 423 x 400 | 102
= 169200 | 102
= 169200 + 102
= 169302
Another example is, 46 x 43. We can solve it in two ways:
First, we take 10 as a theoretical base and take multiple of 10 i.e. 50 (10 x 5) as our working base.
Second, we take 100 as a theoretical base and take sub-multiple of 100 i.e. 50 (100/2) as our working base.
46 -4
43 -7
-----------
39 | 28
-----------
From our usual procedure, we get 39 | 28. Now,
First way (10 x 5 = 50):
1. Multiply the left-hand part of the answer by 50 (39 x 50)and get 1950 as the first part.
2. Now add 1950 to 28 and get 1978.
3. Final answer is '1978'
Second way (100/2 = 50):
1. Divide the left-hand part of the answer by 2 and get 19.5
Note:-Here 39 being odd, its division by 2 gives us a fractional quotient or decimal number.So, number after decimal(i.e. 5) of the left hand side is carried over to the right hand (as 50).
2. Left-hand part is now '19' and right-hand part is 78(i.e. 50 + 28)
3. Final answer is: 1978
In the above discussed two examples, we can also apply the second method. Like in 207 x 213 , we can also take working base: 1000/4 = 250 or 1000/5 = 200. And in 406 x 417 , we can also take working base: 1000/2 = 500. We have to choose the best convenient multiple or sub-multiple.
Following an example in which you solve it in many ways:
62 x 48
(1) Working Base 10 x 4 = 40
62 +22
48 + 8
-----------
70 | 176
x40
-----------
2800 | 176
-----------
= 2800 + 176
= 2976
(2) Working Base 10 x 6 = 60
62 + 2
48 -12
----------
50 | -24
x60
-----------
3000 | -24
-----------
= 3000 - 24
= 2976
(3) Working Base 10 x 5 =50
62 +12
48 - 2
----------
60 | -24
x50
-----------
3000 | -24
-----------
= 3000 - 24
= 2976
(4) Working Base 100/2 = 50
62 +12
48 - 2
----------
60 | -24
/2
-----------
30 | -24
-----------
= (30 x 100) - 24
= 2976
Example: 43 x 47
In this, the base is 100/2 = 50 , so divide the left side by 2 to get the answer.
43 -7 (as above we subtract 50 from our numbers)
47 -3
----------
40 | 21
/2 (calculate the left side, then divide the answer by 2)
-----------
20 | 21
-----------
=(20 x 100) + 21
=2021
This gives: 43 x 47= 2021
E. Numbers near different bases i.e. 'multiplier' is near to one base and multiplicand is near to other base.
Many of times we need to multiply numbers that are not near to same base but are near to different bases.
In the example given below, you find that one number is close to 1000 and the other is close to 100.
Example: 996 × 97
= 996 × (97 x 10) / 10 (can be rewritten like this)
= 966 x 970 / 10 (which can futher be written like this)
Let us first solve 966 x 970
996 –004
970 –030
------------
966 | 120
------------
= 966 x 1000 + 120 (multiply left side result by base)
= 966000 + 120
= 966120
So the answer for 966 x 970 is 966120
Now we shall solve the complete equation i.e. will divide it by 10,
= 966120 / 10
= 96612
So 996 x 97 = 96612
Example: 10005 x 1002
Here the numbers are close to different bases: 10000 and 1000
So equation can be written as
= 10005 x (1002 x 10) / 10
Now let us solve 10005 X 10020 first
10005 + 005
10020 + 020
-----------------
10025 | 0100
= 10025 x 10000 + 100 (multiply left side result by base)
= 100250000+ 100
= 100250100
So 10005 X 10020 = 100250100
Now let us solve the original equation i.e. will divide the answer by 10
= 100250100/10
= 10025010
So 10005 x 1002 = 10025010
Example: 212 x 104
Rewrite as : 2 x (106 x 104)
= 2 x 11024
= 22048
Example: 192 × 44.
Here you can halve 192 and double 44.
= 192 x 44
= (192 / 2) x (44 x 2) (divide by 2 and multiple by 2)
= 96 x 88
Now solve this equation by procedure defined above
= 96 × 88
= 8448.
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A. Numbers are below the base number
B. Numbers are above the base number
C. One number is above the base and the other number is below it
And now, we shall discuss rest of the two cases here:
D. Numbers are not near the base number, but are near a multiple of the base number, like 20, 30, 50 , 250 , 600 etc.
E. Numbers near different bases like multiplier is near to different base and multiplicand is near to different base.
Before going further, let us take few examples of larger numbers with the same method, which we have discussed in previous article.
Example: 6848 x 9997
6848 -3152
9997 -0003
----------------
6845 | 9456 (Refer to previous article for details of approach)
----------------
Example: 87654 x 99995
Let us discuss the fourth case:
D. Numbers are not near the base number, but are near a multiple of the base number, like 20, 30, 50 , 250 , 600 etc.
As you know that this method is applicable in all the cases, but works best when numbers are close to the base. Here, we shall apply the same sutras “All from 9 and the last from 10” and "Vertically and Crosswise" discussed earlier and also the sub sutra i.e. "Proportionately".
Take example as 207 x 213 . Here the numbers are not near to any of the bases that we used before: 10, 100, 1000 etc. But these are close to '200'. So, when neither the multiplicand nor the multiplier is near to the convenient power of 10 then we can take a convenient multiple or sub-multiple of a suitable base (as our 'Working Base'). And then perform the necessary operation by multiply or divide the result proportionately. Like in this example, we take 100 as a 'theoretical base' and take multiple of 100 i.e. 200 (100 x 2) as our 'working base'.
207 x 213
207 +007
213 +013
------------
= 220 | 091
------------
= (220 x 200) | 91
= 44000 + 91
= 44091
As they are close to 200, therefore deviations are 7 and 13 as shown above. From the usual procedure (refer to previous article), we get, 220 | 91. Now since our base is 200, we multiply the left-hand part of the answer by 200 and add it to the right-hand part. That is, (220 x200) + 91 {(Left side x base) + right side}. The final answer is '44091'.
Let us take one more example: 406 x 417
Here Working base is 400 ,which is (100 x 4)
406 +06
417 +17
------------
= 423 | 102
------------
= 423 x 400 | 102
= 169200 | 102
= 169200 + 102
= 169302
Another example is, 46 x 43. We can solve it in two ways:
First, we take 10 as a theoretical base and take multiple of 10 i.e. 50 (10 x 5) as our working base.
Second, we take 100 as a theoretical base and take sub-multiple of 100 i.e. 50 (100/2) as our working base.
46 -4
43 -7
-----------
39 | 28
-----------
From our usual procedure, we get 39 | 28. Now,
First way (10 x 5 = 50):
1. Multiply the left-hand part of the answer by 50 (39 x 50)and get 1950 as the first part.
2. Now add 1950 to 28 and get 1978.
3. Final answer is '1978'
Second way (100/2 = 50):
1. Divide the left-hand part of the answer by 2 and get 19.5
Note:-Here 39 being odd, its division by 2 gives us a fractional quotient or decimal number.So, number after decimal(i.e. 5) of the left hand side is carried over to the right hand (as 50).
2. Left-hand part is now '19' and right-hand part is 78(i.e. 50 + 28)
3. Final answer is: 1978
In the above discussed two examples, we can also apply the second method. Like in 207 x 213 , we can also take working base: 1000/4 = 250 or 1000/5 = 200. And in 406 x 417 , we can also take working base: 1000/2 = 500. We have to choose the best convenient multiple or sub-multiple.
Following an example in which you solve it in many ways:
62 x 48
(1) Working Base 10 x 4 = 40
62 +22
48 + 8
-----------
70 | 176
x40
-----------
2800 | 176
-----------
= 2800 + 176
= 2976
(2) Working Base 10 x 6 = 60
62 + 2
48 -12
----------
50 | -24
x60
-----------
3000 | -24
-----------
= 3000 - 24
= 2976
(3) Working Base 10 x 5 =50
62 +12
48 - 2
----------
60 | -24
x50
-----------
3000 | -24
-----------
= 3000 - 24
= 2976
(4) Working Base 100/2 = 50
62 +12
48 - 2
----------
60 | -24
/2
-----------
30 | -24
-----------
= (30 x 100) - 24
= 2976
Example: 43 x 47
In this, the base is 100/2 = 50 , so divide the left side by 2 to get the answer.
43 -7 (as above we subtract 50 from our numbers)
47 -3
----------
40 | 21
/2 (calculate the left side, then divide the answer by 2)
-----------
20 | 21
-----------
=(20 x 100) + 21
=2021
This gives: 43 x 47= 2021
E. Numbers near different bases i.e. 'multiplier' is near to one base and multiplicand is near to other base.
Many of times we need to multiply numbers that are not near to same base but are near to different bases.
In the example given below, you find that one number is close to 1000 and the other is close to 100.
Example: 996 × 97
= 996 × (97 x 10) / 10 (can be rewritten like this)
= 966 x 970 / 10 (which can futher be written like this)
Let us first solve 966 x 970
996 –004
970 –030
------------
966 | 120
------------
= 966 x 1000 + 120 (multiply left side result by base)
= 966000 + 120
= 966120
So the answer for 966 x 970 is 966120
Now we shall solve the complete equation i.e. will divide it by 10,
= 966120 / 10
= 96612
So 996 x 97 = 96612
Example: 10005 x 1002
Here the numbers are close to different bases: 10000 and 1000
So equation can be written as
= 10005 x (1002 x 10) / 10
Now let us solve 10005 X 10020 first
10005 + 005
10020 + 020
-----------------
10025 | 0100
= 10025 x 10000 + 100 (multiply left side result by base)
= 100250000+ 100
= 100250100
So 10005 X 10020 = 100250100
Now let us solve the original equation i.e. will divide the answer by 10
= 100250100/10
= 10025010
So 10005 x 1002 = 10025010
Example: 212 x 104
Rewrite as : 2 x (106 x 104)
= 2 x 11024
= 22048
Example: 192 × 44.
Here you can halve 192 and double 44.
= 192 x 44
= (192 / 2) x (44 x 2) (divide by 2 and multiple by 2)
= 96 x 88
Now solve this equation by procedure defined above
= 96 × 88
= 8448.
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