Multiplication of Numbers near to the bases
Base is nothing but numbers like 100, 1000, 10000, 100000..etc
Case 1:
Multiplication of Numbers Below the Base
a) 99 x 98
• 99 is one less from the base 100
• 98 is two less from the base 100
Here we ll get the answer in Two Parts.
One is LHS and another is RHS.
99 – 1 x
98 – 2
——–
97/02
———
Steps:
1.Do Cross subtraction.
(99-2 or 98-1)=97
2. Multiply -1 and -2 =2. Write as 02.
LHS is 97. RHS is 02. Answer is 9702.
Always ensure that the number of digits in RHS should be equal to the number
of 0’s in the Base.
b) 96 x 92
• 96 is less than 100 by 4
• 92 is less than 100 by 8
96 – 4 x
92 – 8
——–
88/32 ans:8832
Step 1 is cross subtraction
(96-8)or(92-4)=88
step 2 is Multiplication
(-4)x(-8)=32
so, answer = 8832
c) What is 88 x 86
88 – 12 x
86 -14
——–
74/168
——–
Step 1 is cross substraction
(88-14) or (86-12)=74
step 2. is multiplication
(-12)x(-14)=168
As the base is 100, There shouild be only two digits in RHS . So carry over
the 1 to the left. It becomes 7568
d) What is 981 x 991
981 – 19 x
991 – 09
———–
972/171
———
Ans : 972171
steps:
1:Step one is cross substraction
(981-09) or (991-19)=972
2: step two is Multiplication
(-19)x(-09)=171
Ans : 972171
Multiplication of Numbers Above the Base
a) 105 x 106
105 + 5 x
106 + 6
———-
111/30
———-
ans: 11130
Steps:
1: Cross addition as it is above the base.
So, (105+6) OR (106+5) = 111
2: Multiplication
(+5)x(=6)=30
b) 112 x 118
112 + 12
118 + 18
———–
130 /216
———-
ANS:13216
Step:1 Cross addition
112+18 OR 118 + 12 = 130
Step 2: Multiplication
(+12)(+18)=216
As per the rule the number of digits in RHS should match the number
zeros of the base. So carry over the 2 to LHS
Multiplication of Numbers (Mixed Base)
This is the third and final case of Multiplication with bases.
In Case 1, we saw multiplication of numbers below to the bases (99 x 98, 997 x
996..etc)
In Case 2, we saw multiplication of numbers above to the bases (101 x 103,
1012 x 1021..etc)
In case 3, ie mixed base, we will see what to do for products like 99 x 104 or
974 x 1010..etc
a) 103 x 95
103 + 3 x
95 – 5
————
98/-15
————
ANS:9785
Step 1: cross addition or subtraction
(103-5) or (95+3)=98
Step 2: Multiplication
(+3)x(-5)=-15
so answer is 9785
NOTE:
RHS is negative , now we borrow 1 from the LHS . It becomes 98-1=97.
The 1 taken from LHS is equivalent to 100 . Now subtract 100 from 15 , we get 85
So the answer is 9785
b) 110 x 88
110 + 10
88 – 12
————–
98/-120
—————-
Step 1 is cross addition or subtraction
(110-12) or (88+10)=98
Step 2 is Multiplication
(+10)x(-12)= -120
Note: RHS Can’t be negative. RHS should have only two digits. As the base is 100.
So, take the -1 of RHS to the left. It becomes 97/-20
now borrow1 from LHS and Subtract the RHSfrom 100. It becomes 96/80
ANS is 9680
* Square of Numbers near to the Base
a) What is 972
• 97 is less than 100 by 3
So, Write like this –> (97-3) / 32
= 94/09. Answer is 9409.
Note that the number of digits in RHS should be equal to number of Zeros in
the base.
b) What is 942
• 94 is less than 100 by 6.
So, Write like this (94-6) / 62
= 88 / 36. Answer is 8836
c) What is 882
= (88 – 12) / (122
)
= (76) / (144) = 7744 (Carry Over the One to LHS. Base is 100. So it should be
two digits).
d) What is 9982
= (998 – 2) / (22
)
= (996) (004). Answer = 996004
e) 9993 x 9993
= (9993 – 7) / (72
)
= (9986) / (49) = 99860049
f) 105 x 105
= (105 + 5) / (52
) (We add as it is above the base)
= (110)/ 25. Answer = 11025.
g) 109 x 109
= (109 + 9) / (92
)
= (118)/(81) = 11881
h) 1013 x 1013
= (1013 + 13) / (132
)
= (1026) / (169)
= 1026169
i) 1025 x 1025
= (1025 + 25) / (252
)
= (1050) / (625)
= 1050625
j) 1096 x 1096
= (1096 + 96) / (962
)
= (1192) / [(96 – 4)/ (42
)]
= (1192) / (9216) = (1201) / (216)
= 1201216 (carry over 9 to LHS. Base is 1000. Digits in RHS should be three).
k) 1113 x 1113
= (1113 + 113) / (1132
)
= (1226) / [(113 + 13) / (13)2
] (Separate 1132 and do calculation for that alone)
= (1226) / [126 / 169]
= 1226 / [12769]
= 1238769 (Carryover the 12 to the left) (Understand that I write this many
steps only for your understanding)
Best Way to Calculate Percentage
” In Percentage concept, there is division as well as multiplication. It is really
time consuming for many people. In fact, it is the easiest part among all the
topics. It is easier than adding two 3 digit numbers. The Split and Merge
method can also be applied for Division also. Let’s see “
Example 1
75 25+25+25
—- x100 = ————— x 100 = 1+1+1= 300%
25 25
Points to note:
1. Never cancel the 100 while calculating percentage.
2. Always take the denominator as 100%.
3. Think about 50%,10%, and 1% of the Denominator
4. Try to come as close as possible to the Numerator
5. You can find the answer
In the above example,
What is 100% of 25? = It is 25. We just go by simple examples.
It is 3 times in the Numerator. So the answer is 300%.
Example 2
90
—— x 100
390
1. Take 390 as 100%
2. 10% of 390 is 39 ie, 3.9
3. You can split 90 into 39 +39+12
4. So it is 10% + 10% +(Nearly 3%)
5. Answer is 23%
Example 3
97
——x 100
29
1. Take 29 as 100%
2. 100% of 29 is 29 and 10% of 29 is 2.9
3. 97 can be split into 29+29+29+10
[the above three steps can be done just by using your brain with in 5 seconds]
4. It’s 300%+(10/29)=Nearly 334%
Example 4:
(132 / 7985) x 100
1. Take 7985 is 7985 as 100%
2, 10% of 7985 is 798.5 and 1% of 7985 is 79.85
3. Just by seeing we can say the answer is b/w 1% & 2%
4. Divide 132 as 80 + 52.1% is 80 (approximately) & 0.5% is 40.
5. Answer will be nearly 1.6 %
In case of traditional method, first we cancel 7985 and 100 with 5. Then we
keep on doing it until the time gets over. By this method, just by seeing, we can
say it comes between 1% and 2%. If you see a little bit deeper, you can guess it
is more than 1.5% and less than 2%.
In exam, if there is any option in the specific range, just mark it.
