I need to build a formula for the following. I have a sheet of cardboard $m$ cm long and $n$ cm wide and inside there are stickers or stamps $p$ cm long and $q$ wide separated by $a$ mm. I need to know the full number of stickers per sheet.
Let's see: The area of the sheet is $m \cdot n$. The area of the stickers is $p \cdot q$.
The separation $\frac{a}{10}$ in cm.
I can't find how to put the separation in the formula and 2 cases are presented
a) leave $a$ mm of separation at the edges per sheet
b) do not leave any separation on the edges
can you help me, I hope you understand
edit the idea is like this but with rectangular stickers
The general formula will be (stickers that fit on the top border) $\times$ (stickers that fit on the left border)
Depending on the case the algebraic form will be different.
Let´s start with case a), the number of stickers that fit on the top border is the integer part of (paper´s width) $\div$ (sticker´s width $+$ gap) however the gap won´t be just a, instead, we need $a + \frac{a}{2}$ because, otherwise, our formula won´t take into account the separation needed between both edges. The number of stickers that fit on the left border is just the integer part of (paper´s height) $\div$ (sticker´s height $+$ gap). So the general formula for case a) is $\lfloor \frac{m}{p+a+\frac{a}{2}}\rfloor \times \lfloor \frac{n}{q + a + \frac{a}{2}}\rfloor = \lfloor \frac{m}{p+\frac{3a}{2}}\rfloor \times \lfloor \frac{n}{q + \frac{3a}{2}}\rfloor = \lfloor \frac{2m}{2p+3a}\rfloor \times \lfloor \frac{2n}{2q + 3a}\rfloor$
Case b) is very similar, however, now the gap we will consider is only $\frac{a}{2}$, so the general formula in this case is $\lfloor \frac{m}{p+\frac{a}{2}}\rfloor \times \lfloor \frac{n}{q+\frac{a}{2}}\rfloor = \lfloor \frac{2m}{2p+a}\rfloor \times \lfloor \frac{2n}{2q+a}\rfloor$