Is it permissible for me to take one hundred dirhams in exchange for only eighty dirhams, after my relative waived the remaining twenty dirhams, knowing that I did not intend to exploit her or take an excess?

This transaction involves both an exchange (sarf) and a gift (hiba). The exchange of eighty for eighty is permissible as its conditions are met. The gift of the remaining twenty, completing the hundred, is permissible and legitimate. Islamic law has encouraged and promoted gifting.

Al-Nawawi, in Al-Majmu', quoted Al-Shafi'i's view that it is permissible for a man to take from someone he lent money to less than what he lent, as a gift of beneficence, or for a judge to voluntarily give more than the weight of his gold. This is permissible because it does not fall under the meaning of sales.