Eccoci.

Consider the fraction, $\dfrac n d$, where $n$ and $d$ are positive integers. If $n \lt d$ and $\operatorname{HCF}(n, d)=1$, it is called a reduced proper fraction.

If we list the set of reduced proper fractions for $d \le 8$ in ascending order of size, we get: $$\frac{1}{8},\frac{1}{7},\frac{1}{6},\frac{1}{5},\frac{1}{4},\frac{2}{7},\frac{1}{3},\frac{\mathbf{3}}{\mathbf{8}}\mathbf{,}\frac{\mathbf{2}}{\mathbf{5}}\mathbf{,}\frac{\mathbf{3}}{\mathbf{7}},\frac{1}{2},\frac{4}{7},\frac{3}{5},\frac{5}{8},\frac{2}{3},\frac{5}{7},\frac{3}{4},\frac{4}{5},\frac{5}{6},\frac{6}{7},\frac{7}{8}$$

It can be seen that there are $3$ fractions between $\dfrac 1 3$ and $\dfrac 1 2$.

How many fractions lie between $\dfrac 1 3$ and $\dfrac 1 2$ in the sorted set of reduced proper fractions for $d \le 12\,000$?