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The phrase "a fraction of a" is correct and usable in written English.
It can be used to describe a small part or portion of something, often in a quantitative context.
Example: "Only a fraction of a second is needed to make a decision in a fast-paced environment."
Alternatives: "a small part of" or "a tiny portion of".
Exact(60)
Each patch is a fraction of a full moon wide.
(A fraction of a per cent actually do believe that).
All that in a fraction of a second.
"Everything changes within a fraction of a second".
It turns out that winemakers are most interested in a fraction of a fraction of a fraction of something that is quantified in parts per million.
In a fraction of a millisecond, electrons jump from one atomic nucleus to the other.
"In a fraction of a second I lost all sense of time and space.
The valves switch on or off in a fraction of a second.
"Probably just a water moccasin," he said — poisonous, but only a fraction of a python's size.
Afflalo stopped with him, but a fraction of a second late.
The Ends Justify the Meanness" represent just a fraction of a contemporary, best-selling literary genre.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com