finding squares and close multiplications quickly...

Today's concept....finding squares and close multiplications quickly...

In a hurry guys....so few small concepts that'll help u save sum time...n avoid cramming....which i've always maintained is the best ill-preparation for cat.

#1 finding squares.

step 1. think of a base which is a multiple of 10 or 100 (whichever nearer to the no. whose square is to be determined.)

suppose 34^2.

a gud base wud be 30.

express 34 as 30 + 4 (i.e. base + 4)...i'll call this base + deviation

step 2.

ldigit(s) before 0 in base x (no. + deviation) | deviation^2 where | is an imaginary line separating 2 parts of calculation.

i.e. 3x(34 + 4) | 4^2

i.e 114 | 16.

now the important thing....no. of digits on right side of the imaginary line shud be exactly same as the no. of zeros in our base..dont forget this!!!!

since our base 30 has only one 0 at the end, v can have only one digit on right side of the imaginary line....the less powerful digit...thats 6.

and thus, one has to be carried to the left side....so 114 will become 115.

so the expression becomes...115 | 6

hence, the answer is 1156.


look at few more examples for practice...

28^2

base 20

2x(28 + 8 ) | 8x8

= 72 | 64

= 784


106^2

106 + 6 | 6x6

base 100

=112 | 36

=11236 . since 100 has 2 zeroes, there shud be 2 digits on right side of the line....


103^2

base 100

103 + 3 | 3x3

=106 | 9

= 10609. ensure the no. of digits are exactly same as the no. of zeroes in base....so expressed 9 as 09....a 2 digit no.

312^2

base 300

3x(312 + 12) | 12^2

3x324 | 144

972 | 144

= 97344....1 of 144 carried.



smart ways of using this method....



98^2. now if v take base 100, v can avoid multiplication by 9.

so base 100.

98 - 2 | -2^2

= 96 | 04

= 9604....funny ha?



197^2

base 200

2x(197 - 3) | -3^2

=388 | 09

= 38809.


#2 : finding product of nos when they lie nearby.


all digits before 0 in base x (first no. + deviation of second no.) | deviation of first no. x deviation of second no.

rest, everything is same...

examples...

27 x 22 =

base 20

2x(27 + 2) | 7x2

= 58 | 14

= 594


103 x 108

base 100

= 103 + 8 | 3x8

= 111 | 24

=11124


97 x 102

base 200

97 + 2 | -3 x 2

=99 | -6

= 9900 - 6

= 9894



197 x 199

base 200

= 2x(197 - 1) | -3 x -1

= 392 | 03

= 39203


37 x 31

base 30

37 + 1) x 3 | 7x1

= 114 | 7

= 1147.


32 x 24

base 30

32 - 8 | 2 x -8

= 24 | -16

= 240 - 16

= 226.