Not Continued Fractions Age 14 to 18 Challenge Level:
The key here is that $x$ has to be the integer part of $N$ because the 'continued fraction' part of the expression gives a value less than one.
As $y$ and $z$ are positive integers (whole numbers), $y + 1/z > 1$ and $1/(y+1/z) < 1$ so we know that this must equal $3/7$ and $x = 1$.
Hence $y + 1/z = 7/3$. Again $y$ has to be the integer part of $7/3$ so $y = 2$ and $z = 3$.
As in the first part, if $N = 8/5$, then we must have $x = 1$ and $y + 1/z = 5/3$.
To make $y$ and $z$ positive integers we must have $1/z < 1$ and $y = 1$.
It then follows that $1/z = 2/3$ so it is impossible to find positive integer values for $x$, $y$ and $z$.