A more mathematical answer shows that working with parts of 60 is actually a very efficient mathematical choice, especially in Babylonean times. To this day we still work with '60', such as in our system of time, the 12 star signs, etc.
To understand why parts of 60 work, you need to understand what a mathematical base is. Each digit raises the base to power based on the position (going right to left). The first digit is the base to the power 0, the second digit is the base to the power 1, etc.
We are used to a base 10 system, so when we add things we count up to nine, before starting anew in a cycle. 11 means '10 and 1'. Most are also familiar with the base 2 system, aka the binary system. You count from zero to one, before 'restarting the clock'. 11 means '2 and 1'.
It is known from ancient texts that in the Babylonian region a base 60 system was used.
This system has some incredible properties in regards to counting without tools, such as an abacus. In a base 10 system, we count on our fingers to ten, and then things start to get difficult. How do you count to 11 on your fingers? In a base 60 system that is much easier, and the Babylonian writings reflect their 'counting on fingers system'.
Take your left hand, and don't count on your thumb. Start counting the sections of your fingers. You have four fingers (if you exclude the thumb), with three parts each, so twelve parts in total. The trick is to count up to twelve by pointing on your 12 finger-parts with your index finger. As soon as you have counted 12, you start anew by counting with your index and middle finger. This allows you to count up to 24. After adding your ring finger and ultimately your pinky, you will have reliably counted up to 60.
The base 60 system also has other big advantages for a world without computers. 60 is easily divided into parts, which is why it is still popular in our time-system, which is also a base-60 system (60 minutes in one hour). If you need to divide it in half, you get 30, which can be divided in quarters to get 15 etc. But most importantly: 60 has a wonderful property woefully lacking in the base-10 system, and that is that it is divisible by 3. The fraction 10/3 is much more difficult to work with than 60/3, because 10/3 is not an integer, whereas 20 is a whole number.
Base 10 can be divided into whole numbers by two factors only, namely, '5' and '2'. Other systems have advantages over base-10, especially in the ancient world that hadn't invented the decimal points yet. Base 30, the smallest base that is divisible by 2, 3, and 5. Base 60 is at the tipping point where it can be counted 'naturally' such as explained above, and is divisible by 2, 3, 4, 5, 6. Higher than sixty doesn't really help improve upon base-60. To get a clean fraction like 1/7 using an analogous representation, you’d have to go up all the way to the basically unusable base 210 system.
A bit more technical: base 10 only has the prime factors '2 x 5', whereas the prime factors of 60 are '2 × 2 × 3 × 5'. More prime factors are a good thing if you need to divide things in parts. Any system that allows you to divide things in clean equal parts is superior to a system to forces you to 'wing it' when you need to divide it in anything other than 2 or 5. As mentioned, you need to go up to 210 to get a qualitative improvement upon 60 with the prime factors '2 × 3 × 5 × 7'. But counting in cycles of 210 is a cognitive strain of biblical proportions.
So, in a world where you count, and most importantly, divide a lot of 'real' things (as opposed to the more abstract things we use mathematics for nowadays), and you don't have the luxury of computers, the base-60 system has a lot of advantages over other systems of counting, mostly because it has a lot of prime factors, i.e. is easy to divide into whole numbers. Base-10 has other advantages, but in the old world of the Babylonian Talmud the base-60 system is superior.