A student who missed the last class was confronted by this problem on a quiz.
He had either a vague recollection of a solution method or a strong understanding of the properties of proportions and decided (in my words) to use common numerators. Thus,
He noodled around and came up with x = 13 and then clearly checked it.
Full credit. Clear understanding of the problem and algebraic correct steps toward the solution. The written explanation could have been clearer but this was a quiz after all.
I’m not going to start teaching the “least common numerator” method for solving rational expressions (It doesn’t generalize.) and there are easier ways of solving proportions which I am sure this student will “get” quickly. So a quick compliment and on with the class.
Students’ ways of thinking are so individual and unexpected. Teachers get the pleasure of such small discoveries everyday.