1. 31 X 12

Multiply vertically on the left: 3 x 1 = 3 (first figure)

Multiply cross-wise and add: (3 x 2) + (1 x 1) = 7 (middle figure)

Multiply vertically on the right 2 x 1 = 2 (last figure)

So the answer is 372

2. 275 x 513

Multiply vertically 5 x 3 =15 write 5 and carry 1 ==> 5

Multiply and add crosswise last two digits  (3x7)+(1x5) = 26+ carry 1 =27

write 7 and carry 2 ==> 75

Multiply and add  vertically and cross wise all digits

(2x3) + (5x5) + (7x1)=38+ carry 2 = 40 write 0 and carry 4 ==>075

Multiply and add crosswise first two digits (1x2)+(5x7)=37+ carry 4=41

write 1 and carry 4 ==> 1075

Multiply vertically 2x5=10 + carry 4 ==> 14

write 141075

So 275x513 = 141075

 

3. Divide (12X² –8X-32) by (X-2) using Urdhva-Tiryak

The quotient can be written as (12X+k)

We also know that –8x = kx - 24x Hence k = 16 cross-check –2k = -32 third term

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Some rules of thumb

1. Multiplying a number by 11

To multiply any two digit number by 11 we just put the total of the two figures between the 2 digits.

36 x 11 = 3 (3+6) 6 = 396
74 x 11 = 7 (7+4) 4 = 7+carry 1 (1) 4 = 814
234 x 11 = 2 (2+3) (3+4) 4 = 2574

2. Diving by 9.

23 / 9 = 2 remainder 5
The first digit of 23 i.e, 2 is the answer and  remainder is sum of 2 and 3!

134 / 9 = 14 remainder 8
Answer is (1st digit of 134)(1+3) remainder (1+3+4)