Some universities I'll apply to ask for a writing supplement. Am I supposed to send it with the whole application, or I'm allowed to send it later? And does it influence the application decision?
If a college wants a writing supplement, you have to submit it before the college application deadline. The only add'l docs that are permitted in general after the applications are due is if they didn't receive a transcript because of a technical error from your HS school etc. Then they will send you an email saying, we didn't get your dual enrollment transcript from your HS, please contact your HS counselor and make sure they send it along. They usually send a follow-up to the school as well.
For example, Princeton asks for a graded paper in addition to a writing supplement so here is the link for that.
If colleges permitted supplements to be submitted on a rolling basis after the application deadline, the admissions process would be a complete mess. These days some colleges get 100,000 applications and doling them out to readers is a precision process like a factory. If applications were only 75% complete and readers had to wait for supplements then no one would get their admissions letters of acceptance until the end of summer.
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