I am completing my second year of Master in mathematics in France and applying for a PhD in the UK, I wonder how to present my grades. I had average grades of 17.1/20 and 17.4/20. Shall I just write this (I'm afraid the 0-20 scale might be confusing), add that it is equivalent to First Class Honours (I was for sure in the first 10% of the class, maybe first, but there is no record of the ranking), or do you have any other suggestions? I appreciate your answers
I would be careful when trying to make a direct comparison between the grades. Having "honours" after your degree in the UK generally means that you didn't fail any of your courses, so a 1st class and a 1st class hons. are not equivalent. Furthermore, research-based Masters degrees in the UK are generally marked Fail/ Pass/ Merit/ Distinction rather than 1st/ 2:1/ 2:2 etc. The exception to this are so-called "integrated Masters" which are common in the sciences. These are four year courses where the candidate finishes with (for example) an MSci rather than a BSc, and are graded with the usual classifications.
So, I think there is no direct equivalent, so you should simply write your grade as you have it. Understanding a mark out of twenty is not too challenging and chances are there will be academics familiar with the French system wherever you apply, or at least they will have dealt with French applicants before.
One additional detail that would be helpful is to write next to your grade "top 10% in my year" or "one of the top 5 students", if you have that information. You could also ask one of your referees to write that in their reference letter too.