Not sure what you are trying to say. Inevitably you are going to lose some of the duration when you force the clip to play faster than it is recorded and uneven framerates will do so even more due to quantization. The only thing that is odd is actually the large amount of time going missing, but without more info about the source files this is difficult to pin down. It should be more like 2 minutes instead of 5.
I did the quickmatch and I discovered that going from 51,965fps to 52fps should give me a 2 second loss. What happens with my footage at the moment is exactly like if it cut the last 5 minutes before applying the Frame Rate change. If you take a look at my first 2 screenshots, you can see that the exact same fps still results in a 5 minute shorter footage. Hope this clarifies what I am trying to explain!
I don't think you have any understanding of frame rates and time.
A frame is a frame. Time is time. If you change the frame rate of a clip using File>Interpret Footage then you change the length of time it will take to playback the clip. The frame rate of a composition has absolutely no bearing on how long a clip will take to playback. Let me give you some examples.
A 10-second clip that was shot at 59.94 fps placed in a comp that is 15 fps will take 10 seconds to playback but you will not see every frame. A 10-second clip that was shot at 59.94 fps but is interpreted as 29.97 fps will take 20 seconds to playback when it is placed in any comp of any frame rate because timelines are based on time and not frame rate.
Media players always playback at a set frame rate. Always. The playback frame rate is assigned through metadata in the file. You cannot start a media player at 24 fps and then switch it to 60 fps in the middle of the clip. It can't be done. Unless you have a specific reason and specific knowledge that you can use a non-standard frame rate in a specific application you have no business setting a custom frame rate for a composition. Video and media players have standards and if you mess with them it is highly unlikely that you will achieve success. Even if you are successful in creating a comp and rendering a video with a 52 fps frame rate, a 10-second clip originally shot at 59.94 or 51.965 will still take exactly 10 seconds to playback, it will just have some missing or blended frames.
That said if you have footage that is reporting a non-standard frame rate when imported it probably came from something like a phone. The best practice is to open up File>Interpret Footage and set the frame rate to the nearest standard frame rate. The pitch of the audio may change a little bit, but everything will remain in sync. If you want to change the playback time then interpret the footage at a whole number multiple of the comp's frame rate so it will either playback faster or slower than real time. The most common use of this kind of manipulation is creating slow-motion footage. You shoot at 59.94 and interpret at 29.97 so one second of real time takes 2 seconds to playback. Then to preserve every frame you put that clip in a 29.97 fps comp. If further time changes are required you use tools like Time Remapping (my favorite for simple changes) to change the time. If you need to manufacture additional frames, especially a lot of additional frames then a 3rd party effect like Twixtor is called for. This is only really necessary if you could only shoot at say 60p but you want to make one second of real time take 10 seconds to playback. Time changes larger than that can only be successfully handled with specific types of shots and 3rd party software because the calculations become so complicated as the software tries to predict the motion of the pixels, correct for motion blur and keep everything lined up so you don't end up with a mess.
I'm not exactly sure what you are trying to do but from the paragraph, I could tell that you did not understand how frame rate and time work. I hope this helps. The error comes from something other than just the frame rate.
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