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The phrase "nearly one third" is grammatically correct and can be used in written English.
It is typically used to indicate a value that is close to, but not exactly, one third. Example: "Nearly one third of the students in the class received A's on the final exam." This means that slightly less than 33% of the students received A's on the exam. Here, "nearly one third" is used to give a general estimate without specifying an exact number.
Exact(15)
Nearly one third of tickets involve drugs or alcohol.
Nearly one third of all injuries (29.3%) were considered clinically insignificant.
Residential buildings consume nearly one third of Total National Energy Consumption.
Motor vehicle crashes account for nearly one third of U.S. military fatalities annually.
Nearly one third of the respondents (28.9%) reported some bodily discomfort associated with electronic game use.
This accounts for nearly one third of Scottish households.
Similar(45)
Nearly one-third of the population lives in poverty.
Mangrove swamps cover nearly one-third of the land area.
Forests (mainly coniferous) cover nearly one-third of the province.
Nearly one-third of them are overweight or obese.
Such products accounted for nearly one-third of total exports.
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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