Approximation of rational numbers The question is stated thus:
Find fractions of the form $\frac{a}{b}$ where $|b| \leq 10$ which are as close as possible to the following fractions:
(a) $\frac{660}{563}$
(b) $\frac{2344}{733}$
(c) $0.150493827$
No other information was provided with the question besides the fact that from context it may be inferred that $\frac{a}{b}$ must be a rational number.
My first impulse is to grab a calculator and find the $a,b \in \mathbb{N}\ , \ 0 < b \leq 10\ $ whose quotient $\frac{a}{b}$ has the closest decimal value to that of the fraction in the question by guessing and checking. That method might provide a solution which is technically correct but there isn't really any mathematical thought behind it and "I tried like twenty different combinations and I promise this is the best one" is obviously not a valid explanation for said answer.
My next thought was to divide the numbers in the numerator and denominator by the appropriate power of $10$ , round the resulting real numbers up or down to integers and hoping for the best. While slightly more theoretically methodical than checking at random, it still involves a degree of guesswork which I don't believe this problem necessitates. Moreover, this method worked for the fraction (a) but not for (b), so it can't be applicable for any fraction.
I've also tried solving inequalities of the form $\frac{a-1}{b-1} < \frac{660}{563} < \frac{a+1}{b+1}$ and several variations thereof and they all seem to reach a methodical conclusion similar to the second approach outlined above.
Basically, how can I solve this problem with 100% fidelity for any given fraction using a method that isn't based on intuitive guesswork?

Every best rational approximation $r$ of a real number $x$ (in the sense that $|r - x| < |s-x|$ for every other rational number $s$ with the same or smaller denominator) is either a convergent or an intermediate fraction of the continued fraction representation of $x$. 
