Squaring up 25 !

After squaring up numbers ending with 5, me and my daughter have moved onto squaring numbers ending with 25. We proceed exactly the same way we did with 5, but with a larger number. That is all.

We will begin with same equation of (a + b)2= a2 + 2ab + b2.

Here b = 25.

Now the equation gets simplified like this:  

a2 + (2 x a x 25) + 625

That is further simplified into: 

 a2 + (50a) + 625. 

Now we will leave out 625 to be added at the end.

a2 + (50a) is further simplified as:

a (a + 50)

Here 'a' is any number that ends with 00, like 100, 200, 300 and so on. We have 625 at the end of the number. 

a (a + 50) is a round figure that will end with at least '000' because we are multiplying 100 x 150 or 200 x 250 and so on. Leaving out the 'three zeros' we will further simplify the equation a (a + 50).

We will take the two zeros [00] from 'a' and arrive at a simpler number 'n'. We will remove the third zero [0] from the [a + 50]. We remove these three zeros [000] because we have 625 at the end of the answer and hence we are simplifying the multiplication leaving out the '0's.

That means: 150 becomes 15, 250 becomes 25, 350 becomes 35 and so on.

So what we have is following simplified equation:

n (n5) - *n5* here is NOT a product here, but we just write n and place 5 next to it to derive the number. So if 'n' is 7, n5 is 75.

Now the n (n5) product is placed before 625 and we have the whole square.

Like: n25 whole squared = n(n5) followed by 625 

Some examples: 

For Whole square of 125: 1 x 15 = 15. Whole square: 15625

For Whole square of 225: 2 x 25 = 50. Whole square: 50625

For Whole square of 425: 4 x 45 = 180. Whole square: 180625

For Whole square of 725: 7 x 75 = 525. Whole square: 525625

For Whole square of 1025: 10 x 105 = 1050. Whole square: 1050625

For Whole square of 1325: 13 x 135 = 1755. Whole square: 1755625

This is quicker, accurate and time saving way of doing Arithmetic for school kids. This again can be worked out without a paper and pencil, if we know our simple multiplication well.

 

Dr. Punned-it