Answer
∴ 1012 × 98 = 99224 = 99176 [ As 992200 – 24 = 99176]
Checking:1012 = 1 + 1 + 2 = 4, 98 = 8
LHS = 4 × 8 = 32 = 3 + 2 = 5
RHS = 99176 = 1 + 7 + 6 = 14 = 1 + 4 = 5
As RHS = LHS so, answer is correct
Try These:
(i) 1015 × 89 (ii) 103 × 97 (iii) 1005 × 96 (iv) 1234 × 92 (v) 1223 × 92 (vi) 1051 × 9 (vii) 9899 × 87
(viii) 9998 × 103 (ix) 998 × 96 (x) 1005 × 107
Sub – base method:
Till now we have all the numbers which are either less than or more than base numbers. (i.e.10, 100, 1000,
10000 etc. , now we will consider the numbers which are nearer to the multiple of 10, 100, 10000 etc.
i.e. 50, 600, 7000 etc. these are called sub-base.
Example: 213 × 202
Step1: Here the sub base is 200 obtained by multiplying base 100 by 2
Step 2: R. H. S. and L.H.S. of answer is obtained using base- method.
213 + 13
202 + 02
215 13 × 02 = 26
Step 3: Multiply L.H.S. of answer by 2 to get 215 × 2 = 430
∴ 213 × 202 = 43026
∴
Example 2: 497 × 493
Step1: The Sub-base here is 500 obtained by multiplying base 100 by 5.
Step2: The right hand and left hand sides of the answer are obtained by using base method.
Step3: Multiplying the left hand side of the answer by 5.
497 –03
493 –07
Same 497–07 = 490 21
493 – 03 = 490
490 × 5
= 2450
∴ 497 × 493 = 245021
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Example 3: 206 × 197
Sub-base here is 200 so, multiply L.H.S. by 2
206 + 06
197 – 03
206 – 3 = 203 –18
197 + 06 = 203 × 2 = 18
= 406
∴ 206 × 197 = 40618 = 40582
Example 4: 212 × 188
Sub – base here is 200
212 + 12
188 -12
200 – 12 = 200 (1)44
188 + 12 = 200
×2
400 –1 = 399
∴ 212 × 188 = 399 44 = 39856
Checking:(11 – check method)
+–+
212=2+2–1=3
+–+
188=1–8+8=1
L.H.S. = 3 × 1 = 3
+–+–+
R.H.S. = 3 9 8 5 6 = 3
As L.H.S = R.H.S. So, answer is correct.
Try these
(1) 42 × 43 (2) 61 × 63 (3) 8004 × 8012 (4) 397 × 398 (5) 583 × 593
(6) 7005 × 6998 (7) 499 × 502 (8) 3012 × 3001 (9) 3122 × 2997 (10) 2999 × 2998
Doubling and Making halves
Sometimes while doing calculations we observe that we can calculate easily by multiplying the number
by 2 than the larger number (which is again a multiple of 2). This procedure in called doubling:
44
35 × 4 = 35 × 2 + 2 × 35 = 70 + 70 = 140
26 × 8 = 26 × 2 + 26 × 2 + 26 × 2 + 26 × 2 = 52 + 52 + 52 + 52
= 52 × 2 + 52 × 2 = 104 × 2 = 208
53 × 4 = 53 × 2 + 53 × 2 = 106 × 2 = 212
Sometimes situation is reverse and we observe that it is easier to find half of the number than calculating
5 times or multiples of 5. This process is called
Making halves:
4. (1) 87 × 5 = 87 × 5 × 2/2 = 870/2 = 435
(2) 27 × 50 = 27 × 50 × 2/2 = 2700/2 = 1350
(3) 82 × 25 = 82 × 25 × 4/4 = 8200/4 = 2050
Try These:
(1) 18 × 4
(2) 14 × 18
(3) 16 × 7
(4) 16 × 12
(5) 52 × 8
(6) 68 × 5
(7) 36 × 5
(8) 46 × 50
(9) 85 × 25
(10) 223 × 50
(11) 1235 × 20
(12) 256 × 125
(13) 85 × 4
(14) 102 × 8
(15) 521 × 25
Multiplication of Complimentary numbers :
Sutra: By one more than the previous one.
This special type of multiplication is for multiplying numbers whose first digits(figure) are same and
whose last digits(figures)add up to 10,100 etc.
Example 1: 45 × 45
Step I: 5 × 5 = 25 which form R.H.S. part of answer
Step II: 4 × (next consecutive number)
45
