10 Vikings vs 10 Voids, is a hands-down victory for the Void Rays.
5 Void Rays Vs 10 Void Rays is also a hands down victory to the 10 Void Rays. A Viking is half the cost of a Void Ray so you will always see twice as many.
I found that the only time that Vikings won with or without micro was when they had a 2:1 ratio on Void Rays, which is ridiculous, consider Vikings hard counter Void Rays... Or at least are supposed to.
That is because a Void Ray costs twice as much as a Viking. Yes 2 Vikings cost 50 more Minerals than 1 Void Ray but those Minerals can be ignored since you will be Gas bound anyway and all this means is the Protoss player could make a few more Zealots or Photon Cannons (which you have no problem handling anyway as terran).
2:1 ratio however is somewhat cost effective, it still is more expensive to counter Voids with Vikings than it is to counter Vikings with Voids.
Not by much. As you are Gas bound those extra Minerals mean nothing.
Conclusion, Void Rays are more cost effective, macro effective, and micro effective versus Vikings than vice versa.
Yet you state otherwise in your argument when you agree that a 2:1 ratio wins hands down (and costs the same, which you fail to see for some reason).
Cost Effective is anything that kills more worth of units than it loses. As long as only 1 Viking survives in a 2:1 ratio, Vikings are a counter unit.
The entire problem with Vikings against Void Rays is they both counter each other mechanically (extra damage to Armored). As I mentioned earlier, Void Rays only out DPS per cost Vikings when both
upgraded and charged, otherwise Vikings are more cost effective against them. If they are not upgraded, they are always weaker than Vikings. If they are not charged, they are always weaker than Vikings.
In small numbers and upgraded the Void Rays will win as they can charge off Vikings before they kill them (the unit has to live 3 seconds to charge, changing targets resets this timer, you need to gain 2 charges to become charged) and so enter a state where they out damage them. In large numbers it is highly unlikely that Void Rays will be able to charge off Vikings forcing them to remain uncharged and outputting considerably less damage than the Vikings.
One fact I mentioned before is Vikings are faster than Void Rays. This may not be too effective to strafe Void Rays with due to the 1 range difference, but it is effective to stop Void Rays retreating as you can keep hitting them indefinitely. If your opponent goes full Void Rays and you counter with full Vikings, there is a high chance that you may have more than a 2:1 ratio by the time he attacks. As this will easily kill his Void Rays, he loses the game as he is forced to fight since retreating will cause him to lose his Void Rays anyway. If he keeps Void Ray production up, keep Vikings up and start pushing into him. If he changes army composition, consider landing Vikings in the case of reduced air as they will still add extra damage which your opponent lost.