Time scaling and shifting of delta function
Note that the time-scaled and shifted Dirac delta impulse $\delta(2t-1)$ is non-zero where $2t-1=0$ is satisfied, i.e., at $t=\frac12$. It's area is indeed $\frac12$ because we have

$$\delta(at)=\frac{1}{|a|}\delta(t)\tag{1}$$

So for your example you get

$$\delta(2t-1)=\frac12\delta\left(t-\frac12\right)\tag{2}$$