elementary number theory - Difficulty in finding modulus of fraction

It is quite easy to evaluate $\frac{a}{b}\bmod m$ when $a$, $b$ and $m$ are integers and $\gcd(b,m)=1$ by replacing $\frac{1}{b}$ with an inverse of $b$ modulo $m$.

But, is it possible to evaluate that when $\gcd(b,m)\neq1$? For example I tried to find $\frac{16}{7}\bmod 7$ following way:

We are actually to find some $x$ such that $\frac{16}{7}\equiv x \pmod 7$
As modular arithmetic permits multiplication$16\equiv 7x \pmod 7$
But, $7x\equiv0 \pmod 7$
Hence $16\equiv0 \pmod 7$ which is absolutely wrong.

What wrong am I doing or should I conclude no such $x$ exists?