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To be more specific, fixed exchange rate, here, fixed which means fastened. But how am I suppose to explain the difference between fixed or fastened and stabilized?

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Stabilization refers to measures taken to prevent wide fluctuations in a value.

Fixing refers to making something unable to move physically or, by figurative extension, unable to change value.

Fastening refers to attaching something to something else. It can move physically only if the other thing moves, or by figurative extension, it can change value only if the other thing changes value.

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