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left after death, decay, decline, disintegration, or consumption; the term is specifically applied to a corpse, to the unpublished works of a dead author, and to the ruins of an ancient civilization.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