For example, 6/8 is mathematically the same as 3/4, but to say they are the same is wrong.
In 3/4 the beat is typically on the 1st note (ONE two three ONE two three etc) while in 6/8
you usually have the beats on the 1st and the 3rd note, so it's more like it consists of
two [U]dotted[/U] quarters than 3 regular ones.
You also group the notes together in two groups in 6/8 and 3 groups in 3/4.
What I'm trying to say is that you don't want to simplify time signatures too
much, because there might be some vital information about accents, beats etc
that you miss if you do that. There's often a purpose to why it says 15/16
instead of something else.