To rearrange the surface area equation for the quantity of interest \( B \), we start with the original equation:
\[ S = \frac{1}{2} l p + B \]
Here, \( S \) is the surface area, \( l \) is the length of the base, \( p \) is the slant height, and \( B \) is the area of the base. To isolate \( B \), we need to move \(\frac{1}{2} l p\) to the other side of the equation. This can be done by subtracting \(\frac{1}{2} l p\) from both sides:
\[ B = S - \frac{1}{2} l p \]
Thus, the rearranged equation for \( B \) is:
\[ B = S - \frac{1}{2} l p \]