For example, if I won the stock market game in my state, would I say “1st place in the statewide stock market game” or “Ranked 1/300 teams in the stock market game” to show how many teams there were. Also do I include the organization that runs it?
If you are applying through commonapp then there is a space to write the organization.
Most of the time you have a very limited word/character count, so try to pack as much valuable information as you can into it. This can mean dropping unecessary words and avoiding full sentences.
In your example you should definitely include the level of the competition, your placement, any specific role you held in the team, what you did to support your team, and the overall number of participating students.
I would say something like "<organization> statewide stock market game; 1st place of 300 teams; organized team spreadsheets and tracked results"
general rule of thumb is to think of what someone who never heard of the competition would find impressive. I would reccomend having a friend or family member who wasn't involved in your extracurricular (but knows what you did) look over how you put it and let you know if you missed anything that struck them as impressive.
I think statewide is more important and you should definitely include the org
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