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The phrase "almost one fifth of" is correct and usable in written English.
It can be used when referring to a quantity that is close to 20% of a whole.
Example: "Almost one fifth of the participants reported feeling satisfied with the service."
Alternatives: "nearly 20 percent of" or "approximately one fifth of".
Exact(21)
In our sample, records on vital signs were missing in almost one fifth of the patients.
The Changjiang drainage basin is situated within 24°30'–35°45'N and 90°33'–122°25'E, draining almost one fifth of the total area of China.
Duell et al. (2009b) showed that almost one fifth of the working-age population in Norway in 2007 was receiving long-term sickness or disability benefits.
Almost one fifth of Nicaragua's landmass is protected in national parks or nature reserves.
Combined the stores bring in almost one fifth of the profit for the entire DABC.
These illnesses, notwithstanding, account for almost one fifth of the health burden in women [ 1].
Similar(39)
At one point, Bangkok's SET index had lost almost one-fifth of its value.
Almost one-fifth of American workers have bad jobs.
Roman Catholics constitute almost one-fifth of the population.
Almost one-fifth of current domestic flights provide matching service.
The manufacturing and mining sector employs almost one-fifth of the total labour force.
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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