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The phrase "approximately one third" is correct and usable in written English.
It can be used when you want to express an estimate of a quantity that is about one third of a whole.
Example: "Approximately one third of the participants completed the survey by the deadline."
Alternatives: "roughly a third" or "about one third".
Exact(22)
Approximately one third of the world's food is wasted before it is consumed (pdf).
The book, Mekas says, features "approximately one third" of his Voice pieces.
Approximately one third reported depression and dyspnea.
Intramuscular 17-alpha hydroxyprogesterone caproate injections reduce the risk of recurrent prematurity by approximately one third.
In Australia, approximately one third of fatal crashes occur in speed zones of 100 km/h.
Barn owls occupied approximately one third of the nest boxes we monitored each year.
Similar(38)
Approximately one-third of all junior and senior high schools have no on-site health services.
Approximately one-third of them are over age 50.
That is approximately one-third of U.S. GDP.
Approximately, one-third to half of all food produced globally is wasted.
However, approximately one-third of bariatric patients will experience significant post surgical weight gain.
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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