Ours Not to Reason Why, Just Flip and Multiply

I still encounter math teachers, especially in the middle grades, who are not aware that there is a fractional division approach analogous to the multiplication method commonly used for fractions. In the same way that multiplication can be thought of as "multiply top times top, bottom times bottom", division can be done by using, "divide top by top, bottom by bottom". Of course this often leads to the result of a fraction divided by a fraction again, and is not of practical use without one additional step which may have once been more common than the current "reciprocals" approach. That additional step was simply to do what was done with addition and subtraction of fractions, find a common denominator. It is the preferred or "best" method in the "Text-book of arithmetic, for the use of teachers" by John Hunter, 1847; as shown in the clip from page 50.



This is not an isolated usage as David E. Smith pointed out in his "History of Mathematics: Vol. I, General Survey of the History of Elementary Mathematics; Vol. II, Special Topics of Elementary Mathematics (1923)

In "The elements of arithmetic in theory and practice", 1903 By John William Hopkins, and Patrick Healy Underwood we see the reciprocal method used, but a historical note points out that the method is very old, probably not what would happen if it was a common part of textbooks for a long period.



An even earlier reference is given by Smith, who credits it to "both the Hindus and the Arabs" of the early Middle Ages.

A translation of "Līlāvatī of Bhāskarācārya" which was written in 1150 contains the following:

Smith adds a Historical footnote to his description of the second common method, using the cross.