【Remainder】 , 【residue】 ,  【residuum】 ,  【remains】 ,  【leavings】 ,  【rest】 ,  【balance】 ,  【remnant】  can all mean what is left after the subtraction or removal of a part.

【Remainder】 is the technical term for the result in the arithmetical process of subtraction. It is otherwise a comprehensive term for things that remain after the others of a collection, assemblage, or stock have been taken away, used up, or accounted for, or for any persons that remain after the others of the group have departed.

【Residue】  and  【residuum】  are often interchanged with  【remainder】 , but they usually imply whatever may be left of a former whole, often a previously intact whole, after it has been subjected to some process which depletes or diminishes it but does not annihilate it.

Both terms, but especially  【residue】 , have acquired specific meanings; thus, a testator, after making certain bequests and providing for the payment of all his debts and charges, usually leaves the  【residue】  of his estate to a legatee, or to legatees, of his choice; water after evaporation often leaves a  【residue】  of mineral material; the  【residue】  of something destroyed by burning is called ash or ashes.

【Residuum】  is frequently used in place of  【residue】 , especially when evaporation or combustion is implied, and it may be preferred to  【residue】 when what is left after a process, whether physical or chemical or mental, is such that it cannot be ignored or left out of account or may have value as a product or significance as a result.

【Remains】 is chiefly used of what is left after death, decay, decline, disintegration, or consumption; the term is specifically applied to a corpse, to the unpublished works of a dead author, and to the ruins of an ancient civilization.

【Leavings】 usually implies that the valuable or useful parts or things have been culled out and used up or taken away or that what is left has been rejected or discarded.

【Rest】  is seldom distinguishable from  【remainder】  (except in the latter's technical arithmetical sense), and the two are commonly used interchangeably without loss. However it may be preferred to  【remainder】 when it means simply the persons or things not previously referred to or mentioned (as in an enumeration or list) and carries no implication of subtraction, deduction, or depletion.

【Balance】  is sometimes used in the simple sense of  【remainder】  or  【rest】 . But  【balance】  is more often found in technical and especially commercial use; thus, in reference to a banking account,  【balance】  usually is applied only to the amount left after withdrawals and other charges have been deducted from the deposits and accumulated interest; in a mercantile charge account,  【balance】 is usually applied to the amount owed after credits have been deducted from the debits.

【Remnant】  and its plural remnants are applied to a 【remainder】 that is small in size or numbers or that represents only an insignificant part or piece left from a former whole.