Proportion Combinations

[KWF]

There are eight different combinations for expressing the proportion 2/4 =3/6.

2/4 = 3/6
4/2 = 6/3
2/3 = 4/6
3/2 = 6/4
3/6 = 2/4
6/3 = 4/2
4/6 = 2/3
6/4 = 3/2

Is there a logical arrangement of them? If so, what is it and why?

 

[tkhunny]

First, why do you think there are only eight (8)? Are you not telling us something?

 

[soroban]

Hello, KWF!

There are eight different combinations for expressing the proportion 2/4 =3/6.

2/4 = 3/6
4/2 = 6/3
2/3 = 4/6
3/2 = 6/4
3/6 = 2/4
6/3 = 4/2
4/6 = 2/3
6/4 = 3/2

Is there a logical arrangement of them?
If so, what is it and why?

I'd say you did a great job!


I would probably use this pattern . . .

We begin with: .\(\displaystyle \dfrac{a}{b} \:=\:\dfrac{c}{d}\)

Switch the numbers on this \(\displaystyle ^{\searrow}_{\nwarrow}\) diagonal: .\(\displaystyle \dfrac{d}{b} \:=\:\dfrac{c}{a}\)

Switch the numbers on this \(\displaystyle ^{\nearrow}_{\swarrow}\) diagonal: .\(\displaystyle \dfrac{a}{c} \:=\:\dfrac{b}{d}\)

Switch the numbers on both diagonals: .\(\displaystyle \dfrac{d}{c} \:=\:\dfrac{b}{a}\)


Now reverse the equaltiies: .\(\displaystyle \displaystyle \frac{c}{d}=\frac{a}{b},\;\;\frac{c}{a}=\frac{d}{b},\;\;\frac{b}{d}=\frac{a}{c},\;\;\frac{b}{a}=\frac{d}{c}\)