modular arithmetic - Modulus of Fraction

I just want to confirm I am doing this problem correctly.

The problem asks to compute without a calculator:
$$3 * \frac{2}{5} \pmod 7$$
The way I am solving the problem:
$$3 * \frac{2}{5} \bmod 7 \\ 3 * 2 * \frac{1}{5} \bmod 7\\ \gcd(5,7) = 1 (\text{inverse exists})$$
$$(((3 \bmod 7) * (2 \bmod 7) * (1 \bmod 7)) \bmod 7) \\ (3 * 2 * 1) \bmod 7$$
$$6 \bmod 7 = 6$$

Am I doing this correctly? Just started learning this in class. This is an even number practice problem out the book so I cannot check the answer lol.

Answer

We want to find $(3)(2)(x)\pmod{7}$, where $x$ is the inverse of $5$ modulo $7$, that is, where $5x\pmod 7=1$.

There are general procedures for finding inverses modulo $m$, but $7$ is a very small number, so we can do it efficiently by trial and error. Note that $(5)(3)$ has remainder $1$ on division by $7$. So $x\pmod 7=3$.

Thus we want to compute $(3)(2)(3)\pmod 7$. This is $4$.