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

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? $$\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$$

• you can use the same visualization? after all $\frac a b = ab^{-1}$ and $\frac c d = cd^{-1}$ Jan 31, 2021 at 23:39
• Yep, divide through by $bd$ in your visualisation. Even better, re-draw or re-label each relevant quantity in your visualisation (in effect, dividing by $bd$), so that your second equation is visualised Jan 31, 2021 at 23:40
• If I'm understanding correctly, these would basically just be using the rectangles to visualize distributivity? That does work, but IMO it feels a lot less elegant than the other proof. Maybe my question is unreasonable though, and there just isn't a better way to look at it. Jan 31, 2021 at 23:50

Maybe something like this: • I really like this, thank you! It feels like I'm going back to elementary school :'). Feb 1, 2021 at 0:21
• Yes, this is the way we teach elementary school teachers to teach (but most of them don't). But isn't this exactly what @BenjaminWang suggested above? Feb 1, 2021 at 3:48
• @Frank: you are welcome!
– user9464
Feb 1, 2021 at 13:06
• TedShifrin: well, I think the picture is a "visualization" literally... :)
– user9464
Feb 1, 2021 at 13:07 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 $$d$$. $$\frac{ad+bc}{bd}$$

Here is my version which depends on similar triangles. 