Rationals of the form $\frac{p}{q}$ where $p,q$ are primes in $[a,b]$
Solution 1:
Roughly speaking this asks whether the quotients of two primes are dense in the positive reals. The answer is yes.
Let $0 < a < b$ and let $q$ be a prime. Then there will a a prime $p$ with $a < p/q\le b$ if and only if $\pi(bq) > \pi(aq)$ where $\pi$ is the prime-counting function. But by the prime number theorem, as $q\to\infty$, $$\frac{\pi(bq)}{\pi(aq)}\sim\frac{b\log(aq)}{a\log(bq)} =\frac{b(\log q+\log a)}{a(\log q+\log b)}\sim\frac ba>1.$$ For all large enough $q$, $\pi(bq)/\pi(aq) > 1$ as required.