A Faulty Balance!

A faulty beam balance has pans of different masses, and its beam remains horizontal when the pans are empty. On such a beam balance, a mass of \(m\) is put on one pan and it is balanced by a weight of 9 grams. When the same mass is put on the other pan, it can be balanced by a weight of 16 grams.

Find the mass \(m\) in grams.

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