How to show the following is true?

Question

I need a proof of the following. If $$\frac{a_1}{b_1}=\frac{a_2}{b_2}=\cdots =\frac{a_K}{b_K}=T$$ then $$\frac{a_i}{b_i}=\frac{\sum_{i=1}^Ka_i}{\sum_{i=1}^Kb_k}$$ where all the $a_i$'s and $b_i$'s are positive values.

Answer 1

Hint: $a_j=Tb_j \forall j$...

Answer 2

You have $\sum a_i = T\sum b_i\implies \dfrac{a_i}{b_i}= T = \dfrac{\sum a_i}{\sum b_i}$

Answer 3

$$\frac{\sum\limits_{i=1}^Ka_i}{\sum\limits_{i=1}^Kb_i}=\frac{\sum\limits_{i=1}^K(Tb_i)}{\sum\limits_{i=1}^Kb_i}=\frac{T\sum\limits_{i=1}^Kb_i}{\sum\limits_{i=1}^Kb_i}=T=\frac{a_i}{b_i}.$$