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The phrase "almost one third of" is correct and usable in written English.
It can be used when referring to a quantity that is close to one third of a total amount or group.
Example: "Almost one third of the participants in the survey reported feeling satisfied with the service."
Alternatives: "nearly a third of" or "approximately one third of".
Exact(20)
Almost one third of the Alliance membership lives in Ontario.
In partial CCA analyses, the geomorphology accounted for almost one third of explained variance.
This represents almost one third of all State School Lands in Oklahoma OklahomaAdAd Valorem 2006).
Nonetheless, almost one third of participants reported headache effectively lasting throughout the whole day.
Almost one third of nTBI patients in acute care died.
Almost one third of MenY case-patients had pneumonia.
Similar(40)
Almost one-third of California's 15m Latinos live in poverty.
Together, they account for almost one-third of Israel's population.
As a result, almost one-third of the samples had a perfect appearance as designed.
Almost one-third of Lake Erie's United States coastline is vulnerable to groundwater sources of nutrients.
Almost one-third of Americans now identify with a racial/ethnic minority group.
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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