MULTIPLICATION TECHNIQUES

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HTML clipboa There is no change when any number is multiplied by 1.When we multiply one number by another then it is increased and becomes further away from one. When 4 is multiplies by 5 it becomes 20 which is further away from 4 and 5. Using our multiplication techniques, we relate each number very close to another number called base. The difference between the number and the base is termed as deviation. Deviation may be positive or negative. Positive deviation is written without the positive sign and the negative deviation, is written using a bar or negative sign on the number
HTML clipboard A-Multiplication by using base 10
HTML clipboard Let N1 and N2 be two numbers near to a given base in powers of 10, and D1 and D2 are their respective deviations from the base. Then N1 X N2 can be represented as
HTML clipboardThis is the formula = (N1+D2) or (N2+D1) / D1xD2
  Example 1:      Multiply 7 by 8.    :     base number as 10. HTML clipboard Here we compare this multiplication with the formula N1 = 7 , N2 = 8 and Deviation D1 = -3 & D2 = -2 
HTML clipboard 7 -3
HTML clipboard8 -2
HTML clipboard(7+(-2)) or (8+(-3)) / 3x2 =56
HTML clipboard B - Multiplication by using base 100
HTML clipboardFormula is same like part or section A but base are changed  
HTML clipboardExample 2:    Multiply 98 by 97
HTML clipboardCompare this multiplication with the formula N1 = 98 , N2 = 97 and Deviation D1 = -02 & D2 = -03 Since the base is 100 than D1xD2 = 06
HTML clipboard 98 -02 
HTML clipboard 97 -03 
HTML clipboard(98+(-03)) or (97+(-02)) / 03x02 = 9506
 HTML clipboard Example3 :    Multiply 75 by 95
HTML clipboard Compare this multiplication with the formula N1 = 75 , N2 = 95 and Deviation D1 = -25 & D2 = -05. Since the base is 100 than D1xD2 = 125 written 25 & 1 carry to left
HTML clipboard 75 -25
 95 -05
(75+(-05)) or (95+(-25)) / 25x05 = 70/125
 HTML clipboard C -Multiplication by using base 1000
 HTML clipboard Example 3:    Multiply 786 by 998
Compare multiplication with the formula N1 = 786 , N2 = 998 & Deviation D1 = -214 & D2 = -002
786 -214
998 -002
(786+(-002)) or (998+(-214)) / 214x002 = 784/428 = 784428

HTML clipboard D- Multiplication whose first figure are same And whose last figure add up to 10, 100 ,1000 etc.
 Example.1 : Find 43 X 47
Check for  R.H.S :  3 + 7 = 10,  L.H.S. portion remains the same i.e.,, 4.

Multiply 4 ( the same figure in both the numbers ) by the next number up 5. This gives 4 X 5 = 20 as the first part of the answer and the last part is 3 X 7 = 21 so the answer is 2021
4 3
4 7
4X5 / 3X7  2021
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Example.3 : Find 127 X 123    
Check for : 7 + 3 = 10, L.H.S. portion remains the same
12 7
12 3
12X13 / 7X3 = 15621
Assignments :    Q1.  23 X 27     Q2.  34 X 36          Q3.  62 X 68     Q4.  136 X 134

 HTML clipboard E. Numbers of which the last 2 or 3 or 4 digits added up gives 100, 1000, 10000 etc.
HTML clipboard The same rule works when the sum of the last 2, last 3, last 4 digits added respectively equal to 100, 1000, 10000
 HTML clipboard Example.1 : Find 292 X 208 
HTML clipboard Here 92 + 08 = 100, L.H.S portion is same i.e. 2
 2 92
2 08
2X3 / 92X08 = 60/736
HTML clipboardExample.3 : Find 693 X 607
HTML clipboard Check for : 93 + 07 = 100, L.H.S. portion remains the same i.e., 6
HTML clipboard HTML clipboard 6 93
6 07
HTML clipboard 420651
HTML clipboard Now R.H.S product 93 X 07 can be obtained mentally. 693 x 607 = 6 x 7 / 93 x 07 = 420 / 651 (for 100 raise the L.H.S. product by 0 i.e. 42X10) = 420651.
HTML clipboard Assignments :     Q1.  393 X 307   Q2.  696 X 604    Q3.  873 X 827      Q4.  188 X 112
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F- Multiplication by using other base
HTML clipboard Example.1 : Find 213 X 203     [Base is 200 ]
HTML clipboard Complement of 213 is 13  Complement of 203 is 03.
HTML clipboard 213 / 13
HTML clipboard 203 / 03
HTML clipboard 213+3 / 39 = 216 x 2 / 39 
 since the base is 200 i e 2X100.  The numbers are close to 200 which is 100X 2 we multiply only the left hand part of the answer by 2 to get 43239.HTML clipboard
HTML clipboard G- Numbers are close to different bases 10,000, 100 etc..
HTML clipboard Example.6 : Find 9998 X 94
Numbers are close to different bases 10,000 and 100
9998 / 02  [Base 10000]
HTML clipboard 94    / - 06[Base 100]   
HTML clipboard 9398 /   12   =  939812
HTML clipboard Note that 6 is not subtracted from 8, but from the 9 above the 4 in 94  Second column from left. So 9998 becomes 9398.Right hand side will as lower base 100