How many fractions of the form $\frac m{17}$ are between $\frac13$ and $\frac23$?

Question

Of all fractions with a denominator of 17 and a whole number numerator, how many are between 1/3 and 2/3?

Answer 1

$\frac13<\frac{n}{17}$ if and only if $3n>17$, or $n>5$. $\frac{n}{17}<\frac23$ if and only if $3n<2\cdot17=34$, or $n<12$. Therefore, there are $6$ solutions $(6,7,8,9,10,11)$.

Answer 2

$$\frac13<\frac n{17}<\frac23\iff \frac{17}3<n<\frac{34}3\iff \biggl\lceil\frac{17}3\biggr\rceil\le n\le \biggl\lfloor\frac{34}3\biggr\rfloor\;\text{ i.e. }\enspace6 \le n\le 11. $$