Find aspect ratio of rectangle made of multiple and spaced by percentage

Question

I have two rectangles with known aspect ratios:

• First one is 1:1
• Second one is 3:4

They're vertically stacked under each other. Finding aspect ratio of combined rectangle is easy.

BUT there is a a catch - space between vertical blocks equals to 10% of the final height (including the percentage increase itself).

For example, if the first block height is 100 units and second block height is 400 units - the total space between will NOT be (100+400) * 10% = 50, since the total height will be 550, and 50 is not 10% of 550.

Can someone assist on how to calculate this percentage? as I'm losing my mind...

Thank you.

Answer 1

If the gap is $10\%$ of the total height, then the actual rectangles constitute $90\%$ of the height. That means that whatever the height $h$ is, $90\%$ of it is $500$: $$ h\cdot 90\% = 500\\ h = \frac{500}{90\%} = 555.55\ldots $$ which makes the gap $555.55\ldots - 500 = 55.55\ldots$, which you can easily see is a tenth of the total height.

Answer 2

Let the heights of the rectangles by $h_1$ and $h_2$. You need the space between the rectangles, $s$, to equal $(h_1+h_2+s) \cdot 0.1$. That suggests that:

$s = (h_1+h_2+s) \cdot 0.1$

The above simplifies to:

$s = \frac{h_1+h_2}{9}$