# The final digits add up to 10

7 2/13 x 7 11/13 =

a) 49 22/169

b) 51 2/13

c) 51 11/13

d) 51 22/169

e) 56 22/169

Only 30% of the 349 participants in the Intermediate age group in the 1st Math2Shine International Vedic Mathematics Competition got the correct answer in this question.

The Vedic solution to this problem is so simple that it would take less than 3 seconds to pick the right choice. It is just a matter of recognizing that the Sub-Sutra The final digits add up to 10 can be applied to the fractional parts of the multiplicands. This is the basis for using By One more than the One Before in squaring numbers ending in 5 and in multiplying complementary numbers.

For example, since 2 + 8 = 10; and in 72 x 78, the number that precedes 2 and 8 are both 7, then the product is just 7 x 8 | 2 x 8 = 56 | 16 = 5616.

And in 352 , where 5 + 5 = 10, we have 352  = 3 x 4 | 5 x 5 = 12 | 25 or 1225.

It is easy to see that 3.52 = 12.25 so that this sutra can also be applied in cases like this when .5 + .5 = 1.

The conventional solution to this type of problem is to convert the mixed numbers into improper fractions before performing the multiplication. If we know that 13 x 7 = 91, then converting the factors into improper fractions can be done mentally.

7 2/13 x 7 11/13  =  93/13 x 102/13 =

The next step is easy for us who practice base multiplication: 93 x 102 = 9486 and 13 x 13 = 169

93/13 x 102/13 = 9486/ 169

But dividing 9486 by 169 is difficult without a calculator or pen and paper.

An easier way to solve this would be to treat the mixed numbers as binomials and then apply the FOIL method:

7 2/13 x 7 11/13 = (7 + 2/13) x ( 7 + 11/13) = 7(7) + 7 (11/3) + 7 (2/13) + (2/13)(11/13)

= 7(7) + 7( 11/13 + 2/13) + (2/13)(11/13) = 7(7) + 7(1) + 22/169 = 49 + 7 + 22/169 = 56 22/169

The Vedic solution is, of course, the easiest. After seeing that the fractional parts 2/13 and 11/13 add up to 1, we can extend  the application of  the sub-Sutra, The final digits add up to 10  to a power of 10 which is 1. Then we can use the Sutra By one more than the one before to immediately get the answer, 7 x 8| (2/13)(11/13) = 56 | 22/169 = 56 22/169.

The logic behind the By one more than the one before Sutra can be seen in the Foil solution above.

7(7) + 7(1) = 7 x (7 +1) = 7 x 8.