Sometimes I see people add preposition before "which", sometimes not. How should I understand the rule?

Question

Three sentences with preposition before "which":

1. I was surprised by the speed at which he completed the project.
2. The rate at which trees absorb CO2 varies by species.
3. The degree to which they agreed was astonishing.

Two sentences without:

1. I read the book, which was very interesting.
2. They’re selling the house which they bought just last year.

Can I say these?

1. I was surprised by the speed which he completed the project.
2. I was surprised by the speed that he completed the project.
3. They’re selling the house for which they bought just last year.

Thanks in advance for explanation of the rules.

Answer 1

Your versions of the sentences are not correct -- the prepositions are required.

Your three sentences can be reworded to look like this:

1. He completed the project at a speed that surprised me.
2. Trees absorb CO2 at a rate that varies by species.
3. They agreed to an astonishing degree.

In those sentences, if you omit the prepositions, you end up with bad grammar. For instance, *"He completed the project a speed that surprised me" is nonsense because the noun phrase, "a speed that surprised me" isn't connected to the rest of the sentence. This is what your version of sentence 1 sounds like without the preposition.

When you reword the sentences to include a relative clause modifying a noun, you replace the noun with "which" and move it to the front of the clause. English allows moving the preposition before the relative pronoun. This is what has happened here. It's not optional.