What is the fastest way to perform mental division?

Question

I am looking for a way to divide relatively small numbers with high accuracy, though I can't seem to find a way around long division. For instance, say we are trying to convert $7/11$ or $23/17$ to a decimal value. How would we compute a value accurate to the second decimal place $\pm 0.01$ or so?

Answer 1

I'm afraid that long division really is the only way; fortunately it's not too long when all you want is an accuracy of $\pm 0.01$. Take $23/17$ for example. $17$ goes into $23$ once with a remainder of $6$; thus the answer so far is one point something. Multiply $6$ by $10$ and divide by $17$; $17$ goes into $60$ three times with a remainder of $9$. Repeat the previous step; $17$ goes into $90$ five times. This is the value of the hundredths place, which is all we want so we can stop here. The answer is $1.35$.

For expressions in which the denominator is larger than the numerator it is much the same, only you would first multiply the numerator by the smallest power of $10$ such that it is larger than the denominator. You would then proceed as usual, remembering to divide the answer you get by the power of $10$ you used in the first step.