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The saying and the explanation as written here makes no sense to me.
The saying that I know is 'It would be cheap at twice the price.', meaning that it is so cheap you could double the price and it still would be cheap.
I completely agree! Cheap at half the price doesn't mean anything. I can understand people getting it muddled but the idiom dictionary says half the price as well....how ridiculous.
No, idioms are not always logical, though I tend not to use the illogical ones.
For example, both "I couldn't care less" and "I could care less" mean the same thing to different people. (I believe the second version in AmE).
This would mean that saying 'It would be cheap at twice the price' means that an item is already cheap and even if you doubled the price it would still be cheap.
Cheap at half the price is used as a phrase in the UK, though it makes no sense.