What points in $[0,1)$ will have two binary expansions? What points in $[0,1)$ will have two binary expansions?
I know that $\frac{1}{2}$ has the two expansions $0.1\bar{0}$ and $0.0\bar{1}$ 
$0.1\bar{0}$ is found by starting with $\frac{1}{2}$ and finding the binary expansion
$0.0\bar{1}$ is found by working backwards from the expansion to find the fraction. 
But how do I go about finding other fractions that have two decimal expansions?

Partial answer: Any nonzero rational in $[0,1)$ with a terminating representation has two binary expansions. 
Suppose $x\in (0,1)$ is rational, $x\ne 0$, and
$$
x = 0.d_1 \dotsm d_n
$$
for binary digits $d_i$. As $x\ne 0$, we have $d_n = 1$:
$$
x = 0.d_1 \dotsm d_{n-1} 1
$$
But now it's easy to see and show that 
$$
x = 0.d_1 \dotsm d_{n-1} 0 \overline{1}
$$
is another representation of $x$.
