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The phrase "a tiny fraction of their" is correct and usable in written English.
It can be used when discussing a small part or percentage of something belonging to a group or entity.
Example: "A tiny fraction of their profits is reinvested into community projects."
Alternatives: "a small portion of their" or "a minuscule part of their".
Exact(58)
The auction sites say that fraud occurs in only a tiny fraction of their sales.
Corporate media gives alternative parties just a tiny fraction of their election coverage.
Never mind that BMW only recycles only a tiny fraction of their own cars.
At the same time, other states had just a tiny fraction of their inmates in solitary confinement.
Still, the numbers for three species have shrunk to only a few thousand, a tiny fraction of their former levels.
NEW YORK — For many legacy retailers, online sales still make up only a tiny fraction of their overall sales haul.
Astronomers discovered that most types of stars emit X-rays but usually as a tiny fraction of their energy output.
However, most local health authorities have traditionally devoted only a tiny fraction of their budgets to drug treatment programmes.
There are 23m North Koreans to the South's 48m, with only a tiny fraction of their average income.
Similar(2)
I guarantee you that any money a shareholder would see years from now as a result of a successful shareholder lawsuit (where individuals typically get pennies on the dollar after all the lawyer's fees) will be a tiny fraction of what their shares would be worth if they had just held onto them.
Medical devices that connect to smartphones already cost a tiny fraction of what their hospital counterparts do; 3D printers will become as affordable as laser printers are; energy prices will fall exponentially in price through access to sunlight.
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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