Is it permissible to give a dirham and request half of it in its original form and the other half in *fulūs* or something else?

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This is also permissible because it conveys the same meaning. It is valid because this manner of distribution does not lead to unequal exchange upon valuation, as the value of the half retained in the dirham is certainly equal to the value of the half given as *fulūs*, and the value of the *fulūs* is exactly equal to the value of the other half.