How would one solve this equation which uses floor?

Question

I am stuck solving the following:
$$ S =\lfloor\lfloor(2*B+I+E) * L / 100 + 5\rfloor * N \rfloor $$ I am trying to isolate $I$ in the equation (like $I = $ {everything else}), but I do not know how to deal with the floors in the equation. Any help and explanation would be much appreciated. Thank you!

Answer 1

The floor function doesn't have an inverse, since it's not bijective, as Dando18 pointed out, so you won't be able to invert it directly. At best, you'll be able to get an inequality for $I$, since you can use the fact that $\lfloor x \rfloor \le x$ - so $S \le\lfloor (2*B+I+E)*L/100 + 5\rfloor * N$. If any of your variables can be negative, then be careful with the direction of your inequality.

I get $I \ge (\frac{S}{N} - 5)*\frac{100}{L} - (2B+E)$, provided $L>0$, and you already state $N \in \{0.9,1,1.1\}$. If $L<0$, reverse the inequality. If $L=0$, you can't make $I$ the subject since the whole expression reduces to $S=\lfloor 5N \rfloor$.