Past tense of scrape: to remove something from a surface by rubbing it with something sharp or rough, or to barely manage to achieve something.
From Old Norse 'skrapa,' the word has been in English since around the 13th century, originally meaning to scratch or grate. The figurative meaning (barely achieving) developed later.
Your knees know this word well—but 'scraped by' (meaning you barely passed) comes from the image of barely removing the outer layer of something, which is why we use it for just barely making it!
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