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Are formal writing and academic writing the same? I am writing a paragraph and I do not want to repeat the same word over and over. so at times, I substitute academic with formal. Is it alright?
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Generally academic writing is a type of formal writing. Academic writing is for school / university. Formal writing is a more general term for writing that has stricter grammatical rules, and includes such things as business writing or legal writing. It would depend entirely on the context you are writing as to whether you can use the two words interchangeably. However, it is probably not a good idea.

See the Wiktionary entry on "Academic" to check the meaning: http://en.wiktionary.org/wiki/academic
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How about scholarly as a different word?
Formal is OK.
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Academic writing may or may not be formal. For PhD thesis and dissertations, it is better that the writing may be formal or at least semi formal. But essays are also considered a form of academic writing and they may not be so formal. For some examples of formal academic writing, you can visit this site: http://www.researchomatic.com/thesis.html

You should read this blog where you can find lots of cool articles on the thematic of education and writing.