The problem said that there are red socks and black socks (plural). So perhaps one black sock does not qualify. With that as a (very thin) excuse, we look at the general problem.
Let $n$ be the total number of socks, and let $r$ be the number of red socks. With the changed notation, your equation becomes
$$\frac{r}{n}\frac{(r-1)}{(n-1)}=\frac{1}{2},$$
which can be rewritten as $n(n-1)=2r(r-1)$. Multiply both sides by $4$, and complete the squares. We arrive at the equation
$$(2n-1)^2-2(2r-1)^2=-1,$$
which is a special case of a (negative) Pell Equation. So we look for solutions of the equation
$$x^2-2y^2=-1$$
in odd integers $x$ and $y$. (Odd because we want to set $2n-1=x$ and $2r-1=y$.) It turns out that all integer solutions of $x^2-2y^2=-1$ are odd, so we don't need to worry about this detail.
There is an obvious solution $(x_1,y_1)=(1,1)$. It is part of the standard theory of Pell equations that if $(x_n,y_n)$ is a positive solution of the equation, then $(x_{n+1},y_{n+1})$ is also a solution, where
$$x_{n+1}=3x_n+4y_n \quad\text{and}\quad y_{n+1}=2x_n+3y_n.\tag{$1$}$$
Moreover, all positive solutions can be obtained in this way. This comes from the fact that in general $x_k+y_k\sqrt{2}=(1+\sqrt{2})^{2k-1}$. Or else, if we are more number-theoretically minded, we can talk about fundamental units of $\mathbb{Z}[\sqrt{2}]$. From this, we can obtain general formulas for $x_k$ and $y_k$ analogous to the familiar "Binet" formula for the Fibonacci numbers.
So starting at $(x_1,y_1)=(1,1)$, we get the solution $(x_2,y_2)=(7,5)$. This gives $2n-1=7$, $2r-1=5$, so $3$ red socks and a total of $4$ socks, so $1$ lone black sock.
The next solution is $(x_3,y_3)=(41,29)$, that is, $15$ red socks and a total of $21$. Here we have more than one sock of each kind, so perhaps $21$ is the intended answer.
But let's go on. Next comes $(x_4,y_4)=(239,169)$, so $85$ red socks and a total of $120$. And for people with more drawer space, there are arbitrarily large solutions.
There is a little more information about this particular Pell equation in an answer to a recent question. The subject is quite beautiful, and connects with many branches of mathematics.
