In the previous article, we have discussed few cases of Multiplication with "vertically and crosswise" sutra. In this article, we shall learn the remaining cases.
E. Multiplying three digit by three digit numbers
For example: abc x pqr
Multiply:
1) vertically (a x p)
2) crosswise in both directions and add (a x q) + (b x p)
3) crosswise in both directions and add (a x r) + (b x q) + (c x p)
4) crosswise in both directions and add (b x r) + (c x q)
5) vertically (c x r)
Step2 and Step4 are same as we did in the previous sections i.e. multiplying crosswise with two columns at a time. But in Step3, we multiply crosswise using the outer columns, then multiply vertically in the middle column and add these numbers.
Example: 456 x 258
4 5 6
2 5 8
--------------------------------------
8 | 20+10 | 32+25+12 | 40+30 | 48
= 8 | 30 | 69 | 70 | 48
= 8 | 30 | 69 | (70+4) | 8
= 8 | 30 | 69 | 74 | 8
= 8 | 30 | (69+7) | 4 | 8
= 8 | 30 | 76 | 4 | 8
= 8 | (30+7) | 6 | 4 | 8
= 8 | 37 | 6 | 4 | 8
= (8+3) | 7 | 6 | 4 | 8
= 11 | 7 | 6 | 4 | 8
Hence 456 x 258 = 117648
The Algebraic Expression is:
Let the two numbers be (ax2 + bx + c) and (dx2 + ex + f).
Note that x=10
Now the product is
= (ax2 + bx + c) (dx2 + ex + f)
= ad.x4 + bd.x3 + cd.x2 + ae. x3 + be.x2 + ce.x + af.x2 + bf.x + cf
= ad.x4 + (bd + ae). x3 + (cd + be + af).x2 + (ce + bf)x + cf
The Vertically and Crosswise formula can also be Extended into 2 by 2 method. See the following example:
123 × 132
We can split the numbers up into 12 | 3 and 13 | 2 , treating 12 and 13 as if they are single figures:
12 3
13 2
-------------
156 | 63 | 6
= (156+6) | 3 | 6
= 162 | 3 | 6
= 16236
F. Multiplying four digit by three digit numbers
For abcd x pqr
Multiply:
1) vertically (a x p)
2) crosswise in both directions and add (a x q) + (b x p)
3) crosswise in both directions and add (a x r) + (b x q) + (c x p)
4) crosswise in both directions and add (b x r) + (c x q) + (d x p)
5) crosswise in both directions and add (c x r) + (d x q)
6) vertically (d x r)
Example: 4562 x 258
4 5 6 2
2 5 8
----------------------------------------------
8 | 20+10 | 32+25+12 | 40+30+4 | 48+10 | 16
= 8 | 30 | 69 | 74 | 58 | 16
= 8 | 30 | 69 | 74 | (58+1) | 6
= 8 | 30 | 69 | 74 | 59 | 6
= 8 | 30 | 69 | (74+5) | 9 | 6
= 8 | 30 | 69 | 79 | 9 | 6
= 8 | 30 | (69+7) | 9 | 9 | 6
= 8 | 30 | 76 | 9 | 9 | 6
= 8 | (30+7) | 6 | 9 | 9 | 6
= 8 | 37 | 6 | 9 | 9 | 6
= (8+3) | 7 | 6 | 9 | 9 | 6
= 11 | 7 | 6 | 9 | 9 | 6
4562 x 258 = 1176996
I hope this would help you in quick multiplication. If you have any doubt, you can welcome to post your queries.
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