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The phrase "a very tiny fraction of the" is correct and usable in written English.
It can be used when discussing a small part or percentage of a larger whole, often in contexts involving statistics or comparisons.
Example: "A very tiny fraction of the population is affected by this rare condition."
Alternatives: "a minuscule portion of the" or "a small percentage of the".
Exact(3)
(Though a very tiny fraction of the state's massive gross domestic product).
Only a very tiny fraction of the contigs were mapped to 5′UTR/3′UTR and regions 1 kb upstream/1 kb downstream of them.
First, the advance of knowledge is incremental, with many false steps, and with breakthroughs few and far between, so that only a very tiny fraction of the reports in the medical literature signal new knowledge that is both adequately tested and important enough for practitioners to depend upon and apply.
Similar(57)
It's a very tiny fraction of what's available.
Often, only a very tiny fraction of all these possible assignments participates in a satisfactory solution.
Yet we are paying teachers who are in charge of our human capital, arguably more important than our financial capital, a very tiny fraction of what Wall Streeters are paid.
Of all the billions of pounds of BPA in common use, Senator Pavley's bill only phases out only a very tiny fraction: that used in food and drink containers designed for children ages three and younger.
FDPIR services about 90,000 households a month, a tiny fraction of the number that SNAP supports.
Genomic resources are currently very limited to a tiny fraction of the animal species available for study.
This modeling provides a very efficient way for predicting unsteady flowfields with varying parameters because it needs only a tiny fraction of the cost of a full-order modeling for each new condition the more cases studied, the more savings rendered.
So far it is just a tiny, tiny fraction of the ice sheets that has melted.
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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