MENTAL SQUARING

Hi folks, today we are going to learn, squaring a number and that too mentally.

Square of a number is multiplication of a number by itself. So square of 16 will be 16X16.

Now lets do the trick. Suppose you have to find square of 8 (i.e. 8X8).

Now notice 8, its nearest neighbour to a multiple of 10 is indeed 10, and its 2 greater than 10. So if you want to go to 10 (8+2)all you will do is to subtract 2 from the number (8-2=6) and then multiply the numbers and add the square of difference.

That is,

|--------(8+2)---- 10

8-----| 10X6 = 60 + (2²)=60+4=64

|--------(8-2)----- 6

So, all in all you have done is 8X8=(8+2)(8-2)+ 2² (Try to do it mentally like 8 is close to 10 and is 2 less than it, so you will subtract 2 from 8 and multiply it by 10 and add 2² to it)

Now lets square two digit number, let it be 14

so 14 is close to 10 (10+4=14) so what we would do is add 4 to 14 (14+4=18) and subtract 4 from 14(14-4=10), then multiply them and add the square of 4 to it

So 14²= 10X18+4²

=180+16

=196

Now you can square any two digit number like 82 as

82²=80X84+2² (Try to do it mentally like 82 is close to 80 and is 2 greater than it, so you will add 2 to 82 and multiply it by 80 and add 2² to it)

=6720+4

=6724

For three digits, the rule is same

For example 127²=(130X124)+3² (Try to do it mentally like 127 is close to 130 and is 3 less than it, so you will subtract 3 from 127 and multiply it by 130 and add 3² to it)

=16120+9

=16129

Corollary 1:

Squaring a number ending in 5

25²=20X30+5²

=600+25

=625

Now this can be seen as 25²= 2X(2+1)X100 +25

Or in other words just write 25, and perpend it with numbers multiplied with its next number

like 85²=85 [ (8X9)25 ] Or 115²=115 [ (11X12)25 ]

X 85 X 115

(8X9) (11X12)

(5X5) (5X5)

72 25 132 25

Easy Isn't it?

Now Solve these (Select between <<>> to see the answer):

77²=<<5929>>

36²=<<1296>>

95²=<<9025>>

87²=<<7569>>

229²=<<52441>>

156²=<<24336>>

28²=<<784>>

How can it help in CAT,GMAT,GRE,SAT etc.

Some of the common questions are

Q. Calculate the value of X if X=990²

This one is easy, but do you think that you will have that much paper/time to waste for such a simple problem?

Do it mentally like 1000X980+10²

980100

Q. A machine produces 24 bottle in an hour, how many would it produce in a day? A. <<24²=576>>

Q. There are 448 schools in a city, and each school has 448 students. Calculate total number of students in city? A. <<200704>>

Apart from this there are a lot of problems where you will find application of this method.

If you want to know why this method works then mail me.

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friggere.Sumit@gmail.com