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Area in feet squared and inches squared

9. What is the area of a rectangular floor measuring 15 ft 3 in. by 12 ft 9 in.? (1 foot = 12 inches)

A) 162 ft^2 93 in^2

B) 178 ft^2 23 in^2

C) 194 ft^2 63 in^2

D) 180 ft^2 27 in^2

E) 192 ft^2 36 in^2

This question is from the Open Division of the 3rd International Vedic Mathematics Olympiad held last Nov 25, 2023. The conventional solution would be to convert the measurements to inches, perform the multiplication and convert the answer to ft^2 and in^2.

Converting the measurements to inches can be done easily mentally: 15 ft 3 inches = 15(12) + 3 = 183 inches while 12 feet 9 inches = 12(12) + 9 = 149 inches. But the following computations will not be easy without using pen and paper: (183 x 149)/144.

Using the Vertically and Crosswise general multiplication method of Vedic Math however, this problem can be solved mentally:

15 ft                +                3 in

12 ft                +                 9 in

180 ft^2 + 171 ft-in + 27 in^2

= 180 ft^2 + 168 ft- in + 3 ft-in + 27 in^2

= 180 ft^2 + 168/12 ft^2 + 3(12) in^2 + 27 in^2

= 194 ft^2  + 63 in^2

Now if we treat 15 ft and 3 inches as 15 1/4  feet and 12 ft and 9 inches as 12 3/4  feet and using the V&C method like the FOIL method of multiplying binomials, an easier solution is available.

(15  1/4)  ft x (12 3/4  ft) = [15(12) + 15(3/4) + 1/4 (12) + (1/4)(3/4)] ft^2

                                             = [180 + 45/4 + 3 + 3/16] ft^2

                                             = 180 + 11 + 1/4  + 3 + 3/16] ft^2

                                             = [194 + 7/16] ft^2

                                             = 194 ft^2 63 in^2      

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