Here is an alternative method for subtracting fractions that have a different denominator.
That is a cool idea!
In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.
As a teacher I have mixed view on this method.
I only use it as a last resort for these reasons:
a) Although your pupils will be able to get the correct answer they will not understand what they have done.
b) As you don't always use the lowest common multiple it is not always the most efficient method.
c) When subtracting the pupils have to remember which numerator to subtract from the other, they always get this wrong if the answer happens to be negative!
Having said this, when you have tried and tried to teach with understanding and their exam is approaching fast, I must admit to have used this as a quick win!