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One of the retailers says he needs to be able to buy product at a 50% margin from his vendors. Does that mean he needs to sell something for twice what he pays for it, or does he need to mark up 50% of what he pays for it. In the first case, he would buy a $20 shirt for $10. In the second, he would buy a $15 shirt for $10.

Margin is the amount of profit that is made by a business, so this guy is saying that he wants to make 50% profit. That means that he will buy a shirt from you for $10, and sell it for $15. The math works like this:

Margin = (selling price-cost)/cost

So for your example, margin =(15-10)/10 or 50%.

I wouldn't even worry about mark-up. You should, however, be worrying about your own profit margin and making sure that it is appropriate for your market, and that you really are making a profit.
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