This answer might depend upon which organization's rules the kids are dancing under. I'll give the rules for the two major bodies in the US, since that's what I know. Canada, England, Australia, etc. might have slightly different rules.
Under USA Dance rules, the age you are for the entire year is the age you will be turning that year. So, if it is March and you are 12 but will be turning 13 in November, you are counted as being 13 for the whole entire year from January until December.
Under NDCA rules, the dancers' ages are the age they are on the day of the competition. Also, the rules on who may dance "up" are a little different from USA Dance's rules.
In USA Dance, if someone is 12 but will be turning 13, then they are Junior I. If someone is 14 but will be turning 15, then they are Junior II The couple will have to dance in Junior II. If there are only Junior events, with no I or II specified, then they may dance in those. They may also dance "up" into Youth.
If someone is 13 but will be turning 14, then they are Junior II under USA Dance rules. If someone is 15 but will be turning 16, then they are Youth. The couple will have to dance Youth. I believe they may also dance up into Adult but I'm not 100% sure on that.
Under NDCA rules, if one dancer is 13 and the other is 15, then they are both Juniors and they may dance "up" into Youth.
