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Proportionately

Most grade school learners can easily get the correct answer to this question which was given in the Lower Primary group of the recently held 3rd International Vedic Mathematics Olympiad(IVMO 2023) using pen and paper. 16400 ÷ 25 = ? The aim, however, of Vedic Math competitions, which usually require participants to solve 40 or more problems in 1 hour,  is to encourage the use of mental calculations as much as possible. Doubling and halving, which we have discussed extensively in our book, 25 Math Short Cuts, are among the easiest to master but have extensive applications. Dividing by 25 cannot be easily done mentally if we will use the conventional algorithm taught in schools.   But if we realize that in division, the dividend is directly proportional to the divisor with the quotient as the proportionality constant, this problem can be solved just by doubling. (16400 ÷ 25) x 2  = 32800 ÷ 50 (32800 ÷ 50) x 2  = 65600 ÷ 100 (65600 ÷ 100) ÷ 100 = 656 ÷ 1                                  = 656 This solution ca be further simplified by factoring the dividend:  

16400 ÷ 25 = (164 x 100) ÷ 25

                     = 164 x (100 ÷ 25)

                     = 164 x 4

                     = 328 x 2

                         = 656
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