Our 24th Math Special for Christmas is about the use of vertically and crosswise in a topic which many elementary students would want to avoid.
Vertically and crosswise is one of the Sutras with the widest range of applications. I have a book “Vertically and Crosswise” by A.P. Nicholas, K.R. Williams and J. Pickles which has 16 chapters and only 3 chapters deal with arithmetic and elementary Algebra. The rest cover topics such as determinants and inversion of matrices, trigonometric, inverse trigonometric, logarithmic and exponential functions and transcendental, differential, integral and partial differential equations.
In MSC #5 we discussed how vertically and crosswise is applied in multiplication of whole numbers. This time we will use it to simplify multiplication of mixed numbers.
We were trained to convert mixed numbers into improper fractions first before performing multiplication and later change back the product into mixed numbers. Oftentimes direct application of the distributive law, which is a form of vertically and crosswise multiplication, offers a quicker solution.
We purposely chose “easy-to-compute” figures in our featured example to emphasize the method not the actual computation. But this technique works well for most cases.
Let us consider a “difficult” example 7 4/7 x 9 5/9. If we rewrite this problem into (7 + 4/7)(9 + 5/9), we can see that we can use vertically and crosswise method which is just like the “FOIL” method we use in multiplying binomials in algebra to get the answer.
7 4/7 x 9 5/9 = 7(9) + [7(5/9) + 9(4/7)] + (4/7)(5/9)
= 63 + [35/9 + 36/7] + 20/63
= 63 + [3 8/9 + 5 1/7 ]+ 20/63
= (63 + 3 + 5) + (8/9 + 1/7) + 20/63)
= 71 + (56 + 9)/63 +20/63
= 71 + 65/63 +20/63
= 71 + 85/63 = 72 22/63
This solution may look long and complicated but it can be actually performed mentally.