#Algebra Help
frail horizon
icy vector
Well, let's just do the general case.
In any case, the total amount of wine is constant. Let V be the volume of each solution and φ be the volume fraction of wine in the solution. Then, supposing that the volume of mixture is the sum of volumes, we get a system of equations:
V1 + V2 = V
V1φ1 + V2φ2 = Vφ
We are looking for the values of V1 and V2.
First, let's express V2 from the first equation.
V2 = V - V1
Then, let's substitute it into the second equation.
V1φ1 + (V - V1)φ2 = Vφ
Now we can solve for V1.
V1(φ1 - φ2) = V(φ - φ2)
V1 = V(φ - φ2)/(φ1 - φ2)
And V2 then is:
V2 = V - V1 = V - V(φ - φ2)/(φ1 - φ2) = V(φ1 - φ2 - φ + φ2)/(φ1 - φ2) = V(φ1 - φ)/(φ1 - φ2)
So, we got:
V1 = V(φ - φ2)/(φ1 - φ2)
V2 = V(φ1 - φ)/(φ1 - φ2)
Now, by the statement of the problem, we have φ1 = m/(m + n), φ2 = p/(p + q), φ = 1/2, V = 1 l. So:
V1 = 1 l*(1/2 - p/(p + q))/(m/(m + n) - p/(p + q)) = (m + n)(p - q)/(2(mq - np)) l
V2 = 1 l*(m/(m + n) - 1/2)/(m/(m + n) - p/(p + q)) = (m - n)(p + q)/(2(mq - np)) l
Using the general result, you can now solve a and b, too.
icy vector
js rq whats this mean "φ"
As I said, volume fraction: φ = V(wine)/V(solution) = V(wine)/(V(wine) + V(water)).
I'm used to using φ for it, as such is standard in chemistry.
frail horizon
ohh i see isee thanks a lot