An absolutely lovely fraction result

In my new job at Twinkl I get the time to dig into a lot more of what people are saying in the world of maths education than I have in the past. An absolutely lovely result came up today in my google alert for maths education that, despite 20 years teaching mathematics, I had never come across – or forgotten if I have. 

It might be well known to others, I don't know, but I was flabbergasted that there was something so simple about fractions that I wasn't already aware of, and I felt compelled to share it.

Take two equivalent fractions, say  and . If you subtract the numerators and denominators, the resulting fraction is also equivalent to these two fractions. In this case .

The result is equally valid when neither fraction is in its simplest form, for example  and  : .

The proof of this is actually relatively straightforward. Given two equivalent fractions , then we have

However, if . Substituting  gives .

I think this is a lovely proof to offer GCSE or A-Level pupils, although you would probably have to provide a hint for the  step.

It also suggested to me a nice (if inefficient) route to determining whether two fractions are equivalent - particularly if one of the fractions has a numerator and denominator that is relatively large. Simply subtract the smaller numerator/denominator from the larger (or multiples thereof), and if the result is equivalent to the smaller fraction, then so is the original fraction.

For example, given the fractions  and , one might do the following:

At this point we can see that both  and  both simplify to , and so as these are equivalent, then  is also equivalent.

Now, of course, in the age of calculators that can simplify these fractions immediately, this is not something we would potentially encourage pupils to carry out as a process but, I think, with the right pupil at the right time, showing something like this could be a nice way to opening them up to a wider view of mathematics than that which is covered by efficient calculation alone.

Credit to ‘Math with Bruna’ whose YouTube video was included in my google alert which brought this lovely result to my attention.