Find a three-term ratio from three two-term ratios

Question

I want to know if solutions to this problem exist. Let's say I have known ratios in percentages between each pair of A, B, and C. For example:

A:B = 30:70, A:C = 40:60, B:C = 20:80,

How might I then find a three-term ratio A:B:C such that the components of the ratio add to 100%, using the information for the ratio of each pair?

Answer 1

$$A=\frac{3B}7$$

$$A=\frac{2C}3$$

$$\implies\frac{\frac{3B}7}{ \frac{2C}3}=1$$

$$\implies \frac BC=\frac {14}9\ne\frac14$$

So, no solution exists

Answer 2

we have $$\frac{A}{B}=\frac{3}{7}$$ and $$\frac{A}{C}=\frac{2}{3}$$ and $$\frac{B}{C}=\frac{1}{4}$$ from here can wew deririve $$A=\frac{3}{7}B$$ and $$A=\frac{2}{3}C$$ and $$B=\frac{1}{4}C$$ combining equation (1) and (3) we have $$A=\frac{3}{28}C$$ and we have also from (2)$$A=\frac{2}{3}C$$ this can not be