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 (ax

^{2}+ bx + c) and (dx

^{2}+ ex + f).

Note that x=10

Now the product is

= (ax

^{2}+ bx + c) (dx

^{2}+ ex + f)

= ad.x

^{4 }+ bd.x

^{3 }+ cd.x

^{2 }+ ae. x

^{3 }+ be.x

^{2 }+ ce.x + af.x

^{2 }+ bf.x + cf

= ad.x

^{4}+ (bd + ae). x

^{3}+ (cd + be + af).x

^{2}+ (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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