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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)

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    Use base $60$, as was done by the ancient Sumerians https://en.wikipedia.org/wiki/Sexagesimal and is still in (anachronical) use when we count time or angles. Because $5\cdot3\cdot4=60$. $6^{ths}$ and others come for free. Folklore says that these guys had thirty-fingered hands. – 2017-01-27
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    A sexagesimal system. Look at an (analog) clock. – 2017-01-27

3 Answers 3

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Take LCM of 3, 4, 5 and 6 to find base.

You got base = 60.

Called sexagesimal system.

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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:

$60$

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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: $${1\over3}={10\over30}\>,\qquad {1\over4}={225\over 30^2}\>,\qquad{1\over5}={6\over30}\ .$$