Proof without actual numbers. Just knowns and a piece of string.
My aquarium is scratched and I will replace it. That is the mathematic problem at hand.
Found a nice one on Craigslist for a good price. I can just go there and buy it. But I want to keep my base for it. So the new tank's dimensions must match. I told the guy the numeric dimensions, because that's how those things are communicated, and he texted back saying he'll measure tomorrow. Today. And I haven't heard back yet. Maybe he doesn't have a tape measure or a ruler.
Pffft. Who cares about exact scientific measurement by numbers in inches?
We're talking about a 50 gallon aquarium so its dimensions are largely intuitive. It's going to be big. We're checking to see the tank its long, tall and narrow and not squat and wide. And we're checking to make sure it will fit my base exactly. Any differences between tanks will be a matter of only a few inches in either direction. So we want the proper ratio of sides, without caring about precise numerical measurements. We can act as if inches don't even exist. But ratios do. So no need for a tape measure or ruler.
I texted back suggesting he could use a piece of string to confirm. The numerical dimensions in inches are 48"x12"x18".
Beautiful numbers, don't you think?
That's 4 to 1 to 1.5
Knowing that solves everything. A short string can be pulled to shortest measurement, the width, presumably, hopefully, 12 inches for his tank.
Then that string length can be used to pinch off 4 sections of the length, presumably 48 inches.
Proof, right there.
If the top and bottom are 4:1 then the height must be 18 inches.
To prove that too, the seller can use the short string for a new ratio. Width is presumably 12" and height is presumably 18" or 2x6 : 3x6.
So the short string can be folded in half and used to pinch off three sections of height.
QED, right there. Ta-daaaaaah. Graciously bows.
Conversely, a long piece of string can be pulled off to the full length, then folded in half twice, for 4 folded lengths, and that used to compare with the width.
And that same 4 folded string halved again (for 6 inches presumably) and used to pinch off 3 sections up for the tank's height for 18 inches presumably.
If that works then his tank will definitely fit my base. And if that doesn't work then the tank will not fit my base.
Now this right here is the sort of thing they teach you in school when you pay attention and in books when you read them. But what they don't teach you is that you don't have to go digging around in your toolbox for a tape measure all the time thinking that is your only option and comparing numbers all over the place holding them in mind and confusing you when a piece of dental floss is sitting right there.
I'm pretty sure this is how cavemen and cavewomen confirmed their aquarium dimensions.