URDHVA-TIRYAK SUTRA

This sutra says -"Vertically and Crosswise". That's all to multiply two numbers!

Till now, you were multiplying like this:
Question: Multiply 432 by 617.
Answer:
       432
    x 617
     3024
     432
 2592   
 266544

More the number of digits in the numbers, more lines and time you consume. No more! Using the Sutra "Vertically and Crosswise", you have
Step 1 (mentally, don't write on notebook) : vertically (last digits) :
          2x7=14; write 4 carry 1
Step 2 (mentally) : crosswise (last two digits) :
          3x7 +2x1 = 23 +carry 1 = 24; write 4 carry 2
Step 3 : vertically and crosswise (three digits) :
           4x7 + 3x1 +2x6 = 43 +carry 2 = 45; write 5 carry 4
Step 4 : (move left; first two digits) :
             4x1 +3x6 = 22 +carry 4 = 26; write 6 carry 2
Step 5 : (move left; first digit of each number) :
             4x6 = 24 +carry 2 = 26. End.
Write answer : 266544
This is how it appears on notebook :
        432
     x 617
  266544

No matter how big the numbers are, you will need to write only the final answer. All other steps are easily carried out mentally. If the two numbers have different number of digits, write smaller number below the other and pad it on left side with zeros. Thetheory behind above example is :

     ax² +bx +c
     dx² +ex +f                                                         
adx4 +(ae+bd)x³ +(af+be+cd)x² +(bf+ce)x +cf

Observe that coefficient of x0 (units digit) is cf, which is obtained by multiplying last two coefficients (vertically). The coefficient of x1 (tens digit) is bf+ce, which is obtained by crosswise multiplication of last two coefficients. The coefficient of x² (hundreds digit) isaf+be+cd, which is obtained by crosswise and vertical multiplication of last three coefficients. Now as all coefficients are used up, we leave last coefficients and use the remaining, and so on.

Here are a few more examples:

Multiply 92 by 67
    92
 x 67
(Mentally) 2x7 is 14; write 4 carry 1;
9x7 +2x6 = 75 +carry 1 = 76; write 6, carry 7
9x6 is 54, add carry 7 to get 61 -so answer is 6164

Multiply 2376 by 4060
     2376
 x 4060
6x0 = 0; write 0;
7x0 +6x6 = 36; write 6 carry 3;
3x0 +7x6 +6x0 = 42 +carry 3 = 45; write 5 carry 4
2x0 +3x6 +7x0 +6x4 = 42 +carry 4 = 46; write 6 carry 4
2x6 +3x0 +7x4 = 40 +carry 4 = 44; write 4 carry 4
2x0 +3x4 = 12 +carry 4 = 16; write 6 carry 1
2x4 = 8 +carry 1 = 9; write 9. End. Answer is 9646560

Note that all the calculations can be easily done in mind; you just go on writing a digit of answer one at a time (from right to left). So on notebook you will just write:
     2376
 x 4060
9646560

Please do the following multiplications by Sutra "Vertically and Crosswise" :
32 x 54?
50 x 98?
123 x 987?
654 x 84?
749 x 302?
3112 x 8735?
3022 x 7004?