1:1 Solution from Solutions of Two Different Ratios

Jan 2015
I searched to see if I could find a similar problem posted to no avail, so apologies if this is a repeat. It's been a few years since my chemistry degree, and anyway I don't think I had ever seen this problem before now.

I have two solutions with two compounds of interest, each, call the solutions 1 and 2; the compounds within them A and B. Solution 1 will be used in toto, it's volume V1, and concentrations of both solutions are known. The question is, what volume of solution 2, V2, is required to create a final solution in which [A]= (1:1 solution, not to be confused with i:f notation of dilutions). This seems like a straightforward question, but my mind creates several ways to attack it and I'd like to bounce this off some peers as it actually matters whether or not I do it correctly.

Option I:

[A final]/[B final] = 1
= V1[A1]/[B1] + V2[A2]/[B2]
V2 = [B2]/[A2] - V1[A1]/[B1]
This doesn't look right and I don't trust it. I'm guessing that's because 1 = is not a valid argument once I incorporate volume into the equation. I think it should though, because volume*concentration should give you content, or mass, and the ratio of the compounds' masses must be 1.

Actually I get why this doesn't work -- it is stating that if volume increases, the ration of A to B will change, which is untrue. In reality it should be V1[A1]/V1[B1], which is just A/B, which means that this will not help me figure out the volume I need of the second solution.

Option II:

Really just a reworking of Option I...
[A final] = [B final]
V1[A1] + V2[A2] = V1[B1] + V2[B2]
V2 = V1 ([A1]-[B1])/([B2]-[A2])

This is what I ended up going with, but today I am rethinking myself.

Option III:

Graph the damn thing. Y1 is the total mass of compound A per volume (X) of added solution 2. Y2 is the same for compound B. Where they intersect, they will have the same total mass (or # particles, whichever value m or M is handy) of each compound at the given volume of added solution 2. Concentrations of solution 2 will serve as slope coefficients; concentrations of solution 1 will serve as y-intercepts, respectively.

Y1 = [A1] + [A2]X
Y2 = [B1] + [B2]X

Y1 = Y2
[A1] + [A2]X = [B1] + [B2]X
X = ([A1]-[B1])/([A2]-[B2])
Hmm that looks kinda familiar...

Okay, so which (if any) are right? What am I missing?