
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.

circumstances
The general living conditions, the social or economic situation of a person or society.
Er wuchs in einfachen Verhältnissen auf.
He grew up in modest circumstances.

affair
A romantic, often sexual, and usually secret relationship; an affair.
Es kam heraus, dass der Chef ein Verhältnis mit seiner Sekretärin hatte.
It came out that the boss was having an affair with his secretary.

ratio
A mathematical or quantitative relationship between two quantities; a proportion or share.
Das Verhältnis von Wasser zu Saft in der Mischung ist drei zu eins.
The ratio of water to juice in the mixture is three to one.

in relation to
In comparison with something else; relative to.
Im Verhältnis zu seiner geringen Größe ist der Motor erstaunlich stark.
In relation to its small size, the engine is surprisingly powerful.
'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.
connection / context
Zusammenhang and Verhältnis can both describe how two circumstances are connected. Zusammenhang tends to emphasize that a substantive or possibly causal connection exists, while Verhältnis refers to the nature of their relationship to each other. Use Zusammenhang for the interplay and Verhältnis for the more precise relation between the two.

Das Verhältnis zwischen Arbeitszeit und Erholung ist bei ihm unausgeglichen.
The balance between working hours and time for relaxation is out of balance for him.
Der Zusammenhang zwischen Bewegung und Gesundheit ist gut zu erkennen.
The connection between exercise and health is easy to see.