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

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  • $\begingroup$ you can use the same visualization? after all $\frac a b = ab^{-1}$ and $\frac c d = cd^{-1}$ $\endgroup$
    – lmaosome
    Commented Jan 31, 2021 at 23:39
  • $\begingroup$ 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 $\endgroup$
    – Benjamin Wang
    Commented Jan 31, 2021 at 23:40
  • $\begingroup$ 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. $\endgroup$
    – Frank
    Commented Jan 31, 2021 at 23:50

3 Answers 3

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Maybe something like this:

enter image description here

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

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  • $\begingroup$ I really like this, thank you! It feels like I'm going back to elementary school :'). $\endgroup$
    – Frank
    Commented Feb 1, 2021 at 0:21
  • $\begingroup$ 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? $\endgroup$
    – Ted Shifrin
    Commented Feb 1, 2021 at 3:48
  • $\begingroup$ @Frank: you are welcome! $\endgroup$
    – user9464
    Commented Feb 1, 2021 at 13:06
  • $\begingroup$ TedShifrin: well, I think the picture is a "visualization" literally... :) $\endgroup$
    – user9464
    Commented Feb 1, 2021 at 13:07
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Visualization of sum of two quotients

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 $d$. $$\frac{ad+bc}{bd}$$

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Here is my version which depends on similar triangles.

enter image description here

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