Please forgive my ignorance, I am unsure of the proper terminology here.
A 10 base number system can easily give us 5ths (2) and halfs (5).
A 12 base number system can easily give us 6ths (2), quarters (3), thirds (4) and halfs (6).
What I am after is a number system that can do 5ths, thirds and quarters (and if possible, 6ths)
On
What you want is a number system with a base that is easily divided by all those numbers, $5,3,4$ and, if possible, $6$.
It is possible so let's build it!
You already have two systems that are quite close to what you want. Is there a good way of combining them? There is! If you combine them correctly, the base $10$ system and the base $12$ together will give you what you need. So, how to combine them? You could try a few things but you want to be able to make some divisions, so let us try multiplying them. Indeed $120$ is good enough. It will let you calculate the good things you want, and some more, like the eighths ($15$).
You wonder then "is $120$ the smallest base that fulfills my requirements"? And the answer is no. For the smallest base you need to find the least common multiple of all the numbers $3,4,5,6$. Can you do the maths to find out what is the smallest base you re after?
Spoiler:
$60$
On
Note that base $10$ can do quarters: ${1\over4}={25\over10^2}=0.25$, as well as ${1\over25}={4\over10^2}=0.04$. It follows that you don't need base $60$ to realize ${1\over3}$, ${1\over4}$, and ${1\over5}$. Base $30$ will do: $${1\over3}={10\over30}\>,\qquad {1\over4}={225\over 30^2}\>,\qquad{1\over5}={6\over30}\ .$$
Take LCM of 3, 4, 5 and 6 to find base.
You got base = 60.
Called sexagesimal system.