NUMBER SYSTEM CHAPTER 4
MULTIPLICATION IN
SPACIAL CASES
Multiplication by 9:
Let us take any number ‘a’ which is to
be multiplied by ‘9’.
a*9= a*(10-1) ===.> a*10-a
Example
: 7253*9=?
·
Put
a zero on the right side of the given number.
·
Subtract
the given number from the number obtained.
72530-7253=65277
Example: 1165*9=?
11650-1165= 10485.
Example
: 5678*9=?
56780-5678=51102.
Now we are
extend this method to multiplication by number having any number of 9’s
Like as 9=10-1
99=100-1
999=1000-1.
We can use same logic ,we can write several other rules.
Example : 352*98=?
35200-(352*2)
35200-704=34496.
Multiplication by 11:
a is any number which is
to multiplied by 11.
(a*11) = a*(10+1)
(a*10)+a
Example
: 235*11=?
·
Put
a zero on the right side of the given number.
·
Add
the given number from the number obtained.
235*10=2350, 2350+235=2585.
Example
: 7295*11=?
7295*10=72950, 72950+7295= 80245.
Example : 3652*11=?
3652*10=36520,36520+3652= 40172.
Multiplication
by 12:
Put a zero on the right
side of the given number.Add to it twice the given number.
Example
: 312*12=?
312*10=3120
3120+312*2=3744.
Example : 5427*12=?
5427*10=54270
54270+5427*2=65124.
Some more spacial cases:
When the sum of the digits of units
places of the multiplicand and multiplier is 10 and other figure are the same in
the both numbers.
Example : 99*91= ?
Ans: 9*1=09, 9*10=90
99*91=9009.
Let us
explain in detailed
Step
1: let us take the unit
digits first then multiplied.
9*1
= 09
Step
2: then take the rest
of the digits with a number one greater then it.
9*10 =90.
Step
3: then combine the
tenth place and unit place.
99*91 = 9009
Example
: 84*86 =?
4*6 = 24
8*9 = 72
84*86= 7224.
Example
: 992*998 =?
2*8=16
99*100= 9900
992*998 = 990016.
If the
digits of the units place is the same and sum of the tenth’s place is 10.
Rule: (product of the ten’s figure+units
figure)(product of units figure)
Examples:
86*26 = (8*2+6)(6*6) = (22)(36) =2236.
52*52 = (5*5+2)(2*2) = (27)(04) =2704.
64*44 = (6*4+4)(4*4) = (28)(16) =2816.
The units
figure are ‘5’ and difference between ten’s figure is 1 then
Rule: [(larger ten’s figure+1)(smaller
ten’s figure)][(75)].
Example :
35*45 = (4+1)(3) [75] =1575.
65*55 = (6+1)(5) [75] =3575.
105*95 = (10+1)(9) [75] = 9975.
Multiplication by repeated digits
Multiplication by 11,111,1111,………… etc
Example : 1245*11
1 (2+1) (4+2) (4+5) 5
1245*11= 13695
Explanation:
Ă˜
We right the extreme right digit of given
number.
Ă˜
Sum
of the two right digits of the given number (up to 9 only then add to next
number)
Ă˜
Leave
the right digit and add the next two
digits.
Ă˜
Last
step write down the extreme left digit.
Example
:
2893*11 = ?
2 (8+2) (9+8) (9+3) 3
2893*11 =31823.
Example
:
4591*11 =?
4 (5+4) (9+5) (1+9) 1
4591*11 =50501.
Same as the multiplication by 111
Example :
5423*111 = ?
5
(5+4) (5+4+2) (4+2+3) (2+3) 3
5423*111 = 601953.
Example : 456723*111
4 (4+5) (4+5+6) (5+6+7) (6+7+2) (7+2+3) (2+3) 3
456723*111 = 50696253.
NUMBER SYSTEM CHAPTER 4
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