How do you know when a key name needs an accidental?
Take the following example from musictheory.net. If we go by the "cheat" rule where you can take the last sharp, then go one half note up, it will tell us that the major key is A.
Next, if we use the "cheat" rule of counting down a minor 3rd to find the relative minor key, we get F. That's okay, but how do I know when the key name needs an accidental? I would have guessed F here, not F#.
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You know because you have to count down a minor third.
A minor third interval consists of one whole tone and one half tone.
Between A and G there is one whole tone.
Between G and F there is another whole tone. To make the interval between G and F a half tone you have to raise the F by a half tone by putting a sharp in front of it.
You also have an important hint by seeing that the F# is already in the key signature. Which probably is a faster way of getting it right.
I don't think the other answers caught on to your mistake. The minor 3rd is from the Major key name (A), not the last accidental (G#) which is what you did.
Because it tells you right there that the key has 3 sharps and which notes are sharp. F, G, AND C are sharp.
When you do your "cheat" you land on F which you can see needs to be sharp by the key signature. But also if you knew your intervals you'd know a minor third below A is F# and a major third below A is F
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