Learn Polish: Musimy dokładnie analizować te proporcje. - We must analyze these proportions carefully.

Using the imperfective analizować implies that this is an ongoing, habitual, or general process. It means "We must analyze these proportions" as a general rule or continuous practice. If you were talking about a specific, one-time task that needs to be completed, you would use the perfective verb przeanalizować.

What case is te proporcje and how is it formed?

It is in the accusative case because it is the direct object of the verb analizować. The singular noun proporcja is feminine, so its plural form is proporcje. Because it is plural and does not refer to a group of men, it takes the non-virile (non-masculine-personal) demonstrative pronoun te (these).

Why is the adverb dokładnie placed before the verb it modifies?

In Polish, adverbs commonly go right before the verb they describe, making dokładnie analizować the most natural, neutral word order. Putting dokładnie at the very end of the sentence is grammatically fine, but it would put a heavy, specific emphasis on the word "carefully".

Since this lesson is about fractions, how do words like proporcje relate to reading fractions like "two fifths"?