Vedic Mathematics: How to find out square of a two-digit number ending with 5? | Examite

Let us start with an example. Consider the two-digit number is 25 and we are going to find out its square i.e; 25² = ?.

25 X  25

?

We are going to follow the following simple steps to find the answer.

1. We multiply 5 by 5 and put (25) as right hand side of the answer.
2. Add 1 to top left digit 2 to make it 3.
3. Now multiply (3) with bottom left 2 which gives 6. Place (6) as the left hand side of the answer.

Voila!!! We got the answer.  It’s 625.

25 X  25

6 / 25

 So, 25² = 2 X (2+1) /  5 X 5 = 625

In the same way,

35² = 3 X (3+1) / 25 = 3 X 4 / 25 = 1225;

Algebraic proof:

Consider (ax + b)² = a². x²+ 2abx + b².

 Assume, x = 10 and b = 5 becomes

(10a + 5) ² = a² . 10² + 2. 10a . 5 + 5²

= a² . 10²+ a.10²+ 5²

= (a²+ a ) . 10² +5²

= a (a + 1) . 10² + 25.

Here, 10a + 5 represents two-digit numbers 15, 25, 35, ….,95 for the values a = 1, 2, 3, ….,9 respectively. In such a case the number (10a + 5)²  is of the form whose L.H.S is
a (a + 1) and R.H.S is 25,

that is, a (a + 1) / 25.

Exercises:

1.

45 X     45

2.

75  X    75

3.

95 X    95

4.

55 X    55