Salget af den gamle cykel hjælper hende med at betale huslejen.

Because -et is the Danish definite article attached to the noun.

• salg = sale
• salget = the sale

Danish often adds the to the end of the noun instead of using a separate word like English does.

So:

• Salget af den gamle cykel = the sale of the old bicycle

Af here means of.

So:

• salget af cyklen = the sale of the bicycle

This is a very common pattern in Danish:

• taget af huset = the roof of the house
• farven af bilen = the color of the car

Danish can also sometimes use a possessive/genitive structure, but with something like the sale of the old bicycle, af is the normal and natural choice.

This is the normal way to say the old bicycle in Danish when an adjective comes before a noun.

You need:

1. den = the
2. gamle = old
3. cykel = bicycle

So:

• cyklen = the bicycle
• but the old bicycle = den gamle cykel

This is a very important Danish pattern:

• bilen = the car
• den røde bil = the red car

When there is an adjective, Danish often uses double definiteness in this kind of phrase: a separate definite word (den/det/de) plus the adjective.

Because adjectives change form in Danish.

The basic form is:

• en gammel cykel = an old bicycle

But after den/det/de in a definite phrase, the adjective usually takes -e:

• den gamle cykel = the old bicycle

Compare:

• en gammel cykel = an old bicycle
• den gamle cykel = the old bicycle

So gamle here is the correct definite/common-gender adjective form.

Because hende is the object form of hun.

• hun = she
• hende = her

Since she is the person being helped, Danish uses the object pronoun:

• Hun hjælper ham = She helps him
• Det hjælper hende = It helps her

In your sentence:

• Salget ... hjælper hende = The sale ... helps her

So hende is exactly what you would expect grammatically.

hjælpe nogen med at ... is a very common Danish structure meaning help someone to ... or help someone with ...-ing.

So:

• hjælper hende med at betale = helps her pay / helps her to pay

Breakdown:

• med = with
• at betale = to pay

Very literally, it is something like helps her with paying.

Other examples:

• Han hjælper mig med at lave mad = He helps me cook
• De hjalp os med at flytte = They helped us move

So this structure is normal and natural.

Usually, hjælpe nogen med at ... is the more natural pattern here.

So:

• hjælper hende med at betale huslejen = natural

Danish does sometimes allow hjælpe nogen at ... in some contexts, but for learners, med at is the safest and most idiomatic choice in a sentence like this.

Because huslejen means the rent.

• husleje = rent
• huslejen = the rent

In English, we often just say pay rent, but Danish frequently uses the definite form in cases where English does not.

So:

• betale husleje can exist in some contexts
• betale huslejen often means pay the rent, referring to the expected rent payment

In this sentence, huslejen sounds very natural.

hjælper is the present tense of hjælpe.

• at hjælpe = to help
• hjælper = help / helps
• hjalp = helped
• har hjulpet = have/has helped

In the sentence:

• Salget ... hjælper hende ... = The sale ... helps her ...

Danish present tense verbs usually end in -r.

This is standard Danish main-clause word order:

1. subject
2. verb
3. other elements

So:

• Salget af den gamle cykel = subject
• hjælper = verb
• hende = object
• med at betale huslejen = extra information

A Danish learner should notice that this sentence is quite straightforward because it starts with the subject. If something else came first, Danish would usually force the verb into second position.

Yes, in theory that means the old bicycle’s sale, but it sounds less natural here.

Danish often prefers af phrases instead of the English-style possessive idea in this kind of abstract expression.

So:

• salget af den gamle cykel = the natural phrasing

That is the form learners should remember.

Cykel is an en word:

• en cykel = a bicycle
• cyklen = the bicycle

Yes, it matters, because that is why the definite phrase uses den:

• den gamle cykel

If it were an et word, you would use det instead:

• det gamle hus = the old house

So the gender of the noun affects the form of the article and sometimes the adjective.

A fairly literal version would be:

The sale of the old bicycle helps her with to pay the rent.

That is not natural English, but it helps show the Danish structure:

• Salget = the sale
• af den gamle cykel = of the old bicycle
• hjælper hende = helps her
• med at betale = with to pay / with paying
• huslejen = the rent

This can be useful if you are trying to understand how the grammar fits together.