What are the methods of dividing numbers to get weird values like $16\over 17$ without a calculator? I tried estimating it to somewhere near $16\over 20$, but it's a far stretch from getting the actual $16\over 17$. How can one do so? Conventionally, I think for numbers such as $50\over 17$, or for any large numbers, we have methods to do division and we can get $2+{16\over 17}$, but we're still left with $16\over 17$, which I have no idea how to find (or at least estimate till a good accuracy).
More interestingly, how would computers or calculators even find these values?

I notice that $17 \cdot 2 = 34 \approx 33$, and since $3 \cdot 33 = 99 \approx 100$, it makes sense that $17 \cdot 6 = 102$ is going to be close to $100$. This is nice, because once we know that $17 \cdot 6 \approx 100$, we have that
$$17 \approx \frac{100}{6}$$ and in turn by flipping everything over,
$$\frac{1}{17} \approx \frac{6}{100} = 0.06.$$
Thus I would estimate $$\dfrac{16}{17} = \dfrac{17 - 1}{17} = 1 - \dfrac{1}{17} \approx 1 - 0.06 = 0.94.$$
I'm sure we could reckon whether I've over- or under-estimated, and by how much, but this was just a quick calculation (that could conceivably done completely in one's head) that hinged on the fact that there's a multiple of $17$ that's near $100$.
