der Rest
rest / remainder
The part that remains after something else has been taken away or used up; the remainder.
Den Rest des Kuchens hebe ich für morgen auf.
I'll save the rest of the cake for tomorrow.

rest / remainder
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rest / remainder
The part that remains after something else has been taken away or used up; the remainder.
Den Rest des Kuchens hebe ich für morgen auf.
I'll save the rest of the cake for tomorrow.

'Der Rest' means 'the remainder' or 'the rest of it'. It should not be confused with the English verb 'to rest' ('sich ausruhen') or the noun 'rest' ('Ruhe', 'Erholung').
Den Rest des Tages möchte ich einfach nur entspannen.
For the rest of the day, I just want to relax.
'Der Rest' can refer both to remaining things and to the remaining people in a group. The context makes clear what is meant.
Ich esse die Suppe, du kannst den Rest haben.
I'll eat the soup; you can have the rest.
To say what something is the rest of, you use the genitive or 'von' + dative. The construction with 'von' is more common in spoken language.
Der Rest des Films war leider nicht so spannend.
The rest of the film was unfortunately not that exciting.
The idiom 'jemandem den Rest geben' means that one final negative thing is too much and brings someone to their emotional or physical limit. The person affected is in the dative.
Die unerwartete Rechnung hat mir für diesen Monat wirklich den Rest gegeben.
The unexpected bill really finished me off for this month.
backlog / deficit / residue
'der Rest' and 'der Rückstand' can both refer to something that remains. 'der Rest' is the general word: it works for food, money, time, or things that are still there after something has been used up. 'der Rückstand' is more specific. It can mean unfinished work or an outstanding payment. In a technical or chemical context, it can also mean a small amount of a substance that remains after a process. Learners may confuse the words because both can express 'something left over'. In everyday language, 'Rest' is usually more natural; 'Rückstand' fits better for debts, work that is overdue, or substances left after cleaning or a reaction.

Den Rest der Suppe hebt Maria für morgen auf.
Maria saves the rest of the soup for tomorrow.
Nach dem Urlaub hat sie einen großen Rückstand bei den E-Mails.
After her vacation, she has a large backlog of emails.