Which of these two ways of simplifying $y=\dfrac{a\times\frac{b}{2}\times c}{d}$ is correct, and why?
1st way:
$$\begin{align} y &=\frac{a \times \frac{b}{2} \times c}{d} \\[4pt] \implies\quad y &=\frac{\frac{abc}{2}}{d} \tag1 \\[4pt] \implies\quad y &=\frac{abc}{2} \div d \tag2 \\[4pt] \implies\quad y &=\frac{abc}{2d} \tag3 \end{align}$$
2nd way:
$$\begin{align} y&=\frac{a \times \frac{b}{2} \times c}{d} \\[4pt] \implies\quad y&=\frac{\frac{abc}{2}}{d} \tag4 \\[4pt] \implies\quad y&=abc \div \frac{2}{d} \tag5 \\[4pt] \implies\quad y&=\frac{abcd}{2} \tag6 \end{align}$$
The correct method is the first way. It all comes down to order of operations. Recall that mathematical expressions in parentheses come before multiplication and division. So when we write $$\frac{\frac{abc}{2}}{d}$$ we're saying: $$(abc/2)/d$$ because the fraction bar between the $2$ and $d$ is largest.
This fractional expression is equivalent to $$\frac{\frac{abc}{2}}{\frac{d}{1}} = \frac{abc}{2}\cdot\frac{1}{d} = \frac{abc}{2d}$$
The second way is evaluating $$abc/(2/d),$$ which changes the location of the parentheses.