Is it permissible to exchange half a dirham for half a dirham and exchange the other half of the original dirham for *fulūs* (base metal coins) or something else?

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This is permissible because the exchange is equivalent to buying half for half, which are equal, making the transaction valid. This is analogous to giving two dirhams and requesting that one be exchanged for *fulūs* and the other half be exchanged for two halves.