When one action happens before another action we use having (done)for the first &

When one action happens before another action we use having (done) for the first action? Why?

“Having found a hotel, we looked for somewhere to have dinner “

But can we say?

After finding hotel, we looked for somewhere to have dinner

“Having finished her work, she went home “

But can we say?

After finishing her work , she went to home

You need some slight changes to the sentences to make them correct:

After finding a/the hotel, we looked for somewhere to have dinner.
After finishing her work, she went home.

Even though I am a learner I am going to tell you my opinion.

We use the perfected past participle for actions or events that were completed in the past.
In combination with the simple past tense used for one more action or event that happened in the simple past we usually describe two events which lasted one after another without time difference between them.

We use the “ing” forms that are not perfected for actions that are not completed.

Thanks

Madam Beeesneees ,Your explaining is brilliant thank you ever so much :slight_smile: -
also Thank you @E2e4, Good Luck :slight_smile: