Derivation of proportion of population given rates

Question

There are only blue and red cars driving on a road. Given the average speed of all of the cars $v_0$, the average speed of the blue cars $v_b$ and the average speed of the red cars $v_r$ derive a formula for the proportion $p_b$ of blue cars on the road.

I know that the formula is given by:

$$p_b = \frac{v_0 - v_r}{v_b - v_r}$$

However I do not understand how to derive it. Can you show me how?

Answer 1

Let $n$ be the total number of cars, with $n_b$ blue, and $n_r$ red, so that $n_b+n_r=n$. Then:

$v_0=\frac{v_r\cdot n_r+v_b\cdot n_b}{n}$.

Define $p_b=n_b/n=1-n_r/n$, and solve the above.

Answer 2

Let the number of red cars be $R$ and the number of blue cars be $B$. To get $v_r$ we need to add up the speeds of all the red cars and divide by $R$. Similarly for the blu cars and for all cars.$$ v_r=\frac{S_r}{R},\\ v_b=\frac{S_b}{B},\\ v_0=\frac{S_r+S_b}{R+B}. $$

Now the proportion of blue cars is $$ p_b=\frac{B}{R+B}$$

Take it from here.