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Graph showing gain and loss

If a sum invested gains $10\%$ each year how long will it be before it has doubled its value?

If an object depreciates in value by $10\%$ each year how long will it take until only half of the original value remains?

Why aren't these two answers the same?

Is there a rate, used for both gain and depreciation, for which those two answers would actually be the same?

If you send in a solution please use mathematics that a Stage 4 reader can follow.

If this problem caught your interest and you know some Stage 5 mathematics this Plus article on Carbon Dating could be a good next step for you.