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The phrase "almost one sixth of" is correct and usable in written English.
It can be used when referring to a quantity that is nearly one sixth of a whole or total amount.
Example: "The survey revealed that almost one sixth of the participants preferred option A over option B."
Alternatives: "nearly one sixth of" or "approximately one sixth of".
Exact(5)
WJN is based in Milwaukee, the state's largest city, which holds almost one sixth of the state's population.
But Shell's U.S. offshore Arctic expedition has cost $4.5 billion since 2005, almost one sixth of Shell's annual capital spending budget, and has faced a series of setbacks.
Carson the candidate was extremely popular with the senior set: Almost one sixth of the $63 million he raised for his campaign came from retired individuals.
In those 60 69 years of age, almost one sixth of the population had immunity to SINV.
However, the data show that even in countries with better environmental conditions almost one sixth of the disease burden could be prevented.
Similar(55)
Almost one third of the Alliance membership lives in Ontario.
In partial CCA analyses, the geomorphology accounted for almost one third of explained variance.
Almost one third of nTBI patients in acute care died.
Almost one third of MenY case-patients had pneumonia.
They revealed that almost one third of 327 patients developed shoulder pain during the first year.
One or several concomitant prophylactic agents were used by almost one fifth of the residents.
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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