# 30MSC 2023 #26: Using Vertically and Crosswise on Fractions and Mixed Numbers

The conventional way of multiplying mixed numbers is to first convert them into improper fractions then follow the usual way of multiplying fractions (numerator x numerator/denominator x denominator) and then converting back the product into a mixed number.

This works well for mixed numbers composed of small figures. But when the whole numbers and the denominators are large the resulting procedure would produce large products and consequently lead to lengthy and error-prone calculations. We can see this in our featured example which was one of the questions in the Seniors group of the 4th Philippine National Vedic Mathematics Olympiad held last Oct 2023: 8 1/6   x   12 3/4 =?

This can be solved easy by those proficient in mental calculations but would be difficult for the majority of students.  Converting the multiplicand into improper fractions is relatively easy. 8 1/6 = (8 x 6 + 1)/6 = 49/6 while 12 3/ 4 = (12 x 4 + 3)/ 4 = 51/4. Multiplying the resulting two improper fractions 49/6 and 51/4 is also easy for those who usually use the product of the sum and difference formula. 49 x 51 = (50 -1) x (50 + 1) = 2500 – 1 = 2499 and 6 x 4 = 24.

So, we have 49/ 6 x 51/ 4 = 2499/ 24

= (2400 + 96 + 3)/ 24

= 100 + 4 + 3/24

= 104 1/8

Now if we use the vertically and crosswise multiplication which is just like the FOIL method we use in multiplying binomials we will have:

8 1/6 x 12 3/ 4   = (8 + 1/6)( 12 + 3/ 4)

= 8(12) + 8(3/ 4) + 12(1/6)  + (1/6)(3/4)

= 96 + 6 + 2 + 1/8

= 104 + 1/8

Other examples of handling fractions using vertically and crosswise are presented in chapter 6 of our forthcoming book, 30 Master Strategies in Computing, Vol III.

Follow us on our Facebook page, MATH-Inic Philippines at https://www.facebook.com/MATHInicPhils to see tomorrow’s 30MSC 2023 #27: Vedic Methods of Solving Quadratic Equations.