das Verhältnis
relationship
The type of relationship or connection between people, groups, or things.
Sie hat ein sehr gutes Verhältnis zu ihren Kollegen.
She has a very good relationship with her colleagues.

relationship / ratio
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relationship
The type of relationship or connection between people, groups, or things.
Sie hat ein sehr gutes Verhältnis zu ihren Kollegen.
She has a very good relationship with her colleagues.

'Verhältnis' (relationship / ratio) is the noun related to the verb 'sich verhalten' (to behave / to relate). It describes how people or things relate to one another or what relationship they have to one another.
Das Verhältnis zwischen den beiden Abteilungen hat sich verbessert.
The relationship between the two departments has improved.
Careful: the plural 'die Verhältnisse' (circumstances) does not simply mean 'several relationships'; it describes general living conditions (e.g. social or economic). This meaning only exists in the plural.
Sie kommt aus ärmlichen Verhältnissen.
She comes from poor circumstances.
To describe a personal relationship, 'Verhältnis' (relationship) is often used with the preposition 'zu' (to) + dative. Typical verbs are 'haben' (to have), 'verbessern' (to improve), or 'belasten' (to strain).
Ich habe ein angespanntes Verhältnis zu meinem Nachbarn.
I have a tense relationship with my neighbor.
In this meaning, 'Verhältnis' (affair) often has a negative or secretive undertone. The typical construction is 'ein Verhältnis mit jemandem haben' (to have an affair with someone).
In der Firma wird gemunkelt, dass er ein Verhältnis mit einer Kollegin hat.
People at the company are whispering that he is having an affair with a colleague.
When referring to a proportion, the structure 'im Verhältnis von ... zu ...' (in a ratio of ... to ...) is used. This describes the quantitative ratio between two quantities.
Mischen Sie Mehl und Zucker im Verhältnis von zwei zu eins.
Mix flour and sugar in a ratio of two to one.
The expression 'im Verhältnis zu' (in relation to / compared with) is a fixed unit and means 'verglichen mit' (compared with). The preposition 'zu' is always followed by the dative.
Im Verhältnis zu den Kosten ist der Nutzen sehr gering.
In relation to the costs, the benefit is very small.
percent
'Prozent' and 'Verhältnis' can both connect numbers with each other. 'Prozent' gives a share out of one hundred, for example 25 percent. 'Verhältnis' describes more generally how two quantities relate to each other, for example 1 to 3 or 'im Verhältnis zu den Kosten' (in relation to the costs). Learners may confuse the words because a ratio can often be expressed as a percentage. But 'Verhältnis' is broader: it does not always have to be a percentage.

Das Verhältnis von Wasser zu Saft sollte zwei zu eins sein.
The ratio of water to juice should be two to one.
Nur zehn Prozent der Befragten wollten den Termin verschieben.
Only ten percent of the respondents wanted to postpone the appointment.