Squaring numbers ending in 5

The Sutra “By one more than the one before” is used to square numbers ending in 5.

Example 1: What is 35²?

The number before 5 is 3 so we will multiply it by one more than 3 or 4  to get the first part of the answer. 3 x 4 is 12. And the last part of the answer is always 5 x 5 or 25. 

35² = 3 x 4 | 5 x 5 = 12|25 = 1225

See proof of this in p. 53 of 25 Math Short Cuts byVirgilio Prudente.

This technique can be used with larger numbers.

Example 2: What is the square of 1005?

The number before 5 here is 100 so one more than this is 101.

1005 x 1005 = 100 x 101 | 5 x 5 = 10100| 25 = 1,010,025

Example 3: (3335)²=

= 333 x 334 |25 = 111222 | 25 = 11,122,225

And also with decimals.

Example 4: What is (3.5)²?

This is like Example 3 but 3.5 is exactly a tenth of 35. Whereas before we have 5 + 5 = 10, now we have 0.5 + 0.5 = 1, also a unity

(3.5)² = 3 x 4| 0.5 x 0.5 = 12| 0.25 = 12.25

And fractions.

Example 5: 4 1/2 x 4 1/ 2 =?

 Here we have 1/ 2 + 1/ 2 = 1

4 1/2 x 4 1/ 2 = 4 x 5 | 1/ 2 x 1/ 2 = 20 | 1/ 4 = 20 1/ 4

Even in area computations:

Example 6: What is floor area of a square room which is 10’ 6” on a side?

Since 6” = 1/ 2 foot, the sutra can also be used here.

10’ 6” x 10’ 6” = 10 x 11 | 6 x 6 = 110 square feet and 36 square inches  

(Taken from pp 3-5 of 30 Master Strategies in Computing, Vol I e-book)