Of all fractions with a denominator of 17 and a whole number numerator, how many are between 1/3 and 2/3?
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If $\frac13 <{n\over17}<\frac23$ what can you say about $n$? (Hint: Clear denominators). – saulspatz Oct 14 '19 at 23:33
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$$\frac13<\frac n{17}<\frac23\iff \frac{17}3<n<\frac{34}3\iff \biggl\lceil\frac{17}3\biggr\rceil\le n\le \biggl\lfloor\frac{34}3\biggr\rfloor\;\text{ i.e. }\enspace6 \le n\le 11. $$
Bernard
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$\frac13<\frac{n}{17}$ 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 $(6,7,8,9,10,11)$.