MSC #24: Vertically and Crosswise:
“When using Vertically and crosswise to calculate 367 × 482 , what is the result of the third step before any carry digits are included?
- 76; B) 80; C) 82; D) 87; E) 92”
A question of this type is given in every level of the 2nd International Vedic Mathematics Olympiad which was held last November 26, 2022. Perhaps, this is due to the fact that the Sutra Vertically and Crosswise is the general method of multiplication in Vedic Math and this question, as crafted, can not be solved using the conventional method.
The 24th of our 30 MATH-Inic Specials for Christmas series deals with the application of Vertically and Crosswise in operations involving fractions and mixed numbers which young students generally dislike.
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.
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.
In solving 10 1/4 x 12 3/5 using the conventional method, we need to convert the multiplicands into 41/4 and 63/5, Then we have to multiply two large numbers 41 and 63 and divide their product by (5 x 4 = 20).
If we rewrite this problem into (10 + 1/4)(12 + 3/5), we can see that we can use the vertically and crosswise method which is just like the “FOIL” method we use in multiplying binomials in algebra to get the answer.
We can follow these easy mental steps:
- 10 x 12 = 120
- 10 x 3/5 = 6; 6 + 120 = 126
- 1/4 x 12 = 3: 3 + 126 = 129
- 1/4 x 3/5 = 3/20; 129 + 3/20 = 129 3/20
Now, let us consider a “difficult” example, 7 4/7 x 9 5/9, which can be rewritten as (7 + 4/7)(9 + 5/9).
Using the conventional method, this would be equivalent to 53/7 x 86/9. Multiplying 53 by 86 and later dividing the product by 63 will certainly require the use of a written solution or a calculator.
Let us try to solve it by using these mental steps:
- 7 x 9 = 63
- 7 x 5/9 = 35/9 or 3 8/9; 3 + 63 = 66, remember 8/9.
- 4/7 x 9 = 36/7 or 5 1/7; 66 + 5 = 71
- (1/7 + 8/9) + (4/7 x 5/9) = (65 + 20)/63 = 85/63 or 1 22/63
- 71 + 1 22/63 = 72 22/63
The third step in IVMO question, 367 × 482, is simply, (3 x 2) + (6 x 8) + (7 x 4) = 82
