Squaring a 3 Digit Number mentally

Can you work out the square of a 3 digit no mentally?

Yes if you can add two 3 digit numbers mentally and multiply the result by a number from 1 to 5 and add the product mentally to a 5 or 6 digit number.

The following steps will show how?

First, let us find the square of a 2 digit number.

Consider the no  73

732  =  702   +  3 (70 + 73)

= 4900  + 429

=  5329

The same can also be worked by the formula

732    =  802  – 7 (73 +80)

=  6400  – 171

=  6329

But we would prefer to use the first form as the multiplicand is 3 for the second terms while it is 7 in the second calculation.

In both  formulae, for a given no N we choose a convenient no which is a nearest multiple of 10, say m.

Then         N2    =  m2 + (N –m ) (N + m)

By taking m close to N so that N-m is from 1 to 5, the square can be worked out mentally and easily.

Note:

When m is smaller than N, (N-m) is positive.

When m is greater than N, (N–m) is negative.

We select m so that the numerical value of N-m is one of the numbers from 1 to 5.

It is easy to find the square of m as its unit digit is 0 and 10s digit will be from 1 to 9.

Examples:

492   =  502  – 1X (50 +49) = 2401

It is quite easy to do this mentally because 52  is 25. By adding two 0’s and subtracting 99 mentally we get the answer

862    =  902 – 4 X (90 +86) = 8100 –  704 = 7396

Let us try a 3 digit no    between 100 and 200, say 1632

Applying the above formula we get

1632  =  1602  +  3 (160 + 163)  = 25600  +  969  =  26569

We can easily remember squares of numbers  upto 30. By practice we can mentally add two 3 digit numbers and multiply the sum by numbers upto 5 and then add the product mentally to a 5 digit or 6 digit number.

For numbers above 200 we can use the following approach

4392   =  4402   –  1 ×  (440 + 439) = (442) ×  100  +  879

= (402 + 4 × 84) ×  100 + 879

= 193600  –  879

= 192721

9932     =  9902  + 3 × 1983

              =  980100 + 5949

=  986049

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