I know that college essays generally have a word limit (e.g. no more than 500 words). However, what should the minimum length of a college essay be? (relative to its maximum length)
This is an interesting question. If the word limit is 500, I think the ideal length is around 450. 400-500 is good. I always try to have at least 80% of the maximum length (so, if the essay is 100 words, you should write 80 or more).
Given that we only get ~1,000 words for our essays to most schools, I think it’s in your best interest to use most of that space. Why leave words on the table (obviously, it’s better to be efficient than rambling on though.
Although I do agree with the above post, a good essay doesn’t have to take up a certain number of words. A student got into Yale a few years ago with a 78-word essay. 
That being said, if you don’t think you can creatively express yourself in under 100 words, I’d recommend a length of 400-500. As long as you fit in what you want to tell the admissions officers, length shouldn’t be too important.
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