Anna, Britta and Christina work at a fruit stand at the market. A customer asks them about the prices of apples, pears and peaches.
Anna says, “Seven apples and five pears cost the same as six peaches.”
Britta says, “Four apples and nine peaches cost the same as five pears.”
Christina says, “Six peaches and three pears cost the same as four apples.”
One of the three salespeople is lying. Which one?
If we denote the prices of an apple, a pear and a peach by a, b and p, respectively, we can express the salespeople’s claims using three equations.
Anna: 7a + 5b = 6p Britta: 4a + 9p = 5b Christina: 6p + 3b = 4a
If we substitute Britta’s equation into Anna’s, we get 11a = –3p, which is meaningless. So one of those two salespeople is lying. If we now substitute Anna’s equation into Christina’s, we get 3a = –8b, which is also meaningless. So Anna is lying. If we substitute Christina’s equation into Britta’s, however, we get the valid equation 2b = 15p.
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This puzzle originally appeared in Spektrum der Wissenschaft and was reproduced with permission.