fraction, faction

Fraction or fraction means a part of something that has been cut up or divided; or a part in relation to the whole--
The fraction familiar to most Americans is the quarter, which says "one quarter dollar" on it.
When you multiply a number by a fraction, you get a smaller number.

Fraction is used to mean a small part, in a general sense--
After paying the hospital bill, only a small fraction of their savings remained.
Only a small fraction of the group agreed with him.

Faction or faction means a group of people having ideas that differ from  the ideas of a larger group; a group within a group--
Each faction is trying to drown out the other; the noise is deafening.
He belongs to a faction that believes we ought to spend more money on space research.

Faction may mean party politics in general, or political intrigue and deal-making--
"Liberty is to faction, what air is to fire..."
Each member wants to win a gain for its own faction.

Now that you know that, you can say--
"Each faction wants the whole sum, and refuses to settle for a fraction."

Factious or factious tending to faction; or to dividing people from the main group, based on a single issue or set of issues--
That issue is always factious, so let's not bring it up for discussion.
His factious speech divided the house into two opposing camps.

Fractious or fractious means testy or irritable--
The old man is fractious when he doesn't take a nap in the afternoon.
The baby is fractious when his diaper is wet.

Note-- "factitious" means artificial or man-made--
The factitious news report caused a sensation, until it was discovered that the reporter had invented the story.
He wrote a factitious account of the event that many people mistook for the truth.
Note # 2 -- "fraction" is a noun ( thing or idea ), with a singular ( one ) and a plural ( more than one ) form--
The dealer offered him a fraction of what it was worth.
We are studying fractions on school.
Note # 2 -- "faction" is also a noun ( thing or idea ), with a singular ( one )  and a plural ( more than one ) form--
He belongs to a different faction, and we have an argumnent every time we meet.
There are so many factions, no one can keep track of them all.