# What was the gain?

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A merchant sold a bicycle for \$ 50, then bought it again for \$ 40, clearly earning \$ 10, since he had the same bike and also \$ 10. After having bought it for \$ 40, he resold it for \$ 45, earning \$ 5 more, or \$ 15 in total.

“But,” says an accountant, “the man starts with a bike worth \$ 50, and at the end of the second sale he only has \$ 55! How, then, how could he have won more than \$ 5? The sale of the bicycle at \$ 50 is a mere exchange that does not yield profit or losses, but when you buy it for \$ 40 and sell it for \$ 45, you earn \$ 5, and that's it. ”

“I affirm,” says a bookkeeper, “that when he sells it for \$ 50 and buys it again for \$ 40, he has clearly won \$ 10, because he has the same bike and also \$ 10, but when he sells it for \$ 45 it is when he makes that mere exchange of which we speak, which does not yield profit or losses. This fact does not affect your first win, so it is clear that you have won exactly \$ 10. ”

It is a simple transaction that any first grader could solve mentally and yet we face three very different answers! In your opinion, Which one is the right one?

#### Solution

The problem has an ambiguous solution, unless one knows how much the merchant originally paid for that bicycle. As this data is not supplied, the problem cannot be solved in any way that is satisfactory.