# Why do two relations when divided become an equation?

I have problems with understanding how to operate with proportionality the first problem is with a question above.

Say we have two relations A proportional to B and C is proportional to D then if we divide the two relations why do we get A upon C is equals to B upon D and not A/C = KB/QD k and q being some numbers

Another question is why can a term be substituted in one relation from another for example:-

   a is proportional to p
a + (n-1)x is proportional to q
Then why is
p + (n-1)x proportional to q


Also, Why can we substitute relations in equations

I don't seem to understand how proportionality works if there are any other rules for applications that I've missed can you also please tell me those?

• No wonder you don't understand this. The claim is false in general. – Hagen von Eitzen Oct 1 '18 at 13:45

The relation "$a$ is proportional to $b$" actually represents an equation already, namely: $a=k\cdot b$ for some number $k$. So all of the rules of equal signs can be used, including transitivity and substitution.

• Then why in the relations given in question, a is substituted by p and not kp. – user25614 Oct 1 '18 at 13:52
• Say we have two relations A proportional to B and C is proportional to D then if we divide the two relations why do we get A upon C is equals to B upon D and not A/C = KB/QD k and q being some numbers – user25614 Oct 1 '18 at 13:54

Saying that one variable is proportional to another is simply another way os saying that there is a linear relationship between them. In other words the following statements:

1. $$a$$ is proportional to $$b$$

2. $$a=kb$$ for some constant $$k$$

3. $$\frac{a}{b}=k$$ for some constant $$k$$

all say the same thing.

If $$a$$ is proportional to $$b$$ and $$a$$ is also proportional to $$c$$, then we have $$a=k_1b$$ and $$a=k_2c$$. So $$b=\frac{k_2}{k_1}c$$. But since $$k_1$$ and $$k_2$$ are constants then so is $$\frac{k_2}{k_1}$$. So we can conclude that $$b$$ is proportional to $$c$$.