Visual proof for $\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$

Question

I've found visualizations of simple mathematical concepts to be a really useful tool in building intuition for more complex mathematical concepts. For example, this visualization of $(a + b)(c + d) = ac + ad + bc + bd$ can be used to visualize the calculus product rule.

That being said, does anyone know of a visual way to show the following equation? $$ \frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd} $$

Answer 1

Maybe something like this:

See also this one: https://www.geogebra.org/m/DV6Ehjnx#material/aEsvBzN2

Answer 2

This image is intended to show that $\frac{a}{b}+\frac{c}{d} = \frac{ad+bc}{bd}$. The width of this cuboid can be computed in two ways, one way by simple addition. $$\frac{a}{b}+\frac{c}{d}$$ Another way is to compute the volume of the cuboid first. The volume of the green cuboid is the area $a$ times the depth $d$, the volume of the blue cuboid is the area $c$ times the height $b$. Adding these volumes gives $ad+bc$. Therefore the width of this cuboid is the volume of the cuboid divided by the height $b$ and depth $d$. $$\frac{ad+bc}{bd}$$

Answer 3

Here is my version which depends on similar triangles.