Anurupyena: proportionality

Anurupyena is the corollary to Ekadhikina Purvena, the first sutra of Vedic mathematics.

What follows is a brief and incomplete summary of the math shortcuts this corollary contains:

Squaring a number that does not end in 5. (to square a number ending in 5, see Ekadhikina Purvena)

This method requires rounding a number up and down based on the nearest base of 10 or 100, multiplying the two numbers, then adding the square of the number added and subtracted. I'll explain with two examples:

Rounding to base-100: To find the square of 96, you would round up to 100. Since you added 4, you now subtract 4 from 96 to yield 92. Multiply 92 and 100. This can be easily done in one's head: 9200. Since you added and subtracted 4, square the 4 to yield 16. Now add 16 to 9200. Thus, 96 squared is 9216.

Rounding to base-10: To find the square of 57, you would round up to 60. Since you added 3, you now subtract 3 from 57 to yield 54. Multiply 60 by 54 (some people can do this in their head). The answer is 3240. Since you added and subtracted 3, square it. That makes 9. Add 9 to 3240. The square of 57 is 3249.

Multiplying two different numbers by rounding to base 10, base 100, base 1000, etc.

This method can also be used to multiply two different numbers, but it requires several more steps and is sometimes no faster then multiplying by hand. If you want to learn the method anyway, check out the resources listed below:

Vedic Mathematics by Sri Bharati Krisna Tirthaji
Mathemagics by Arthur Benjamin and Michael B. Shermer