# 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 1^{st} 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 35^{2 }, where 5 + 5 = 10, we have 35^{2 } = 3 x 4 | 5 x 5 = 12 | 25 or 1225.

It is easy to see that 3.5^{2} = 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**.