Complete the sentence "I have been _____ this problem since noon" by choosing the correct phrasal verb.

Introduction / Context:
This question tests knowledge of phrasal verbs and preposition choice with the verb work. The speaker wants to say that they have been spending time dealing with or trying to solve a particular problem since noon. The correct preposition must show this close, focused effort on a task.

Given Data / Assumptions:

The sentence pattern is I have been _____ this problem since noon.

The verb is work in the form working.

The object is this problem, which indicates a task or issue to be solved.

The preposition following working must form a standard phrasal verb.

Options include working into, working on, working for, working after and working around.

Concept / Approach:
In English, the normal expression for spending time dealing with a problem or project is work on something. The phrasal verb work on means attempt to solve, improve or develop. Other combinations like work for, work into or work around have different meanings and are not used in this particular sentence pattern. Therefore, working on is the correct choice.

Step-by-Step Solution:
Step 1: Substitute working on into the sentence: I have been working on this problem since noon. This clearly states that the speaker has been trying to solve or handle the problem for a period of time. Step 2: Test working into. I have been working into this problem sounds unnatural and does not match standard usage. Step 3: Test working for. I have been working for this problem is wrong because work for is used with organisations or people, not with problems. Step 4: Test working after. This phrase does not form a recognised phrasal verb in this context. Step 5: Test working around. Work around a problem means to avoid it or find a way that does not directly solve it, so it changes the intended meaning and would require a different sentence structure.

Verification / Alternative check:
Consider similar sentences: She is working on her project, they are working on the new design, and he has been working on his presentation all day. In all cases, work on is used for tasks. Replace this problem with another task, such as this report, and working on still fits naturally. Check that the present perfect continuous form have been working on correctly expresses an action that began in the past and continues till now.

Why Other Options Are Wrong:
Working into does not combine naturally with this problem and has no standard meaning here. Working for normally takes a person or organisation as object, as in working for a company, not a problem. Working after is not a recognised phrasal verb in this type of sentence and sounds incorrect. Working around a problem has a different meaning, implying avoidance rather than directly dealing with the issue.

Common Pitfalls:
Students sometimes overgeneralise prepositions, assuming that any preposition can follow a verb. In reality, phrasal verbs are often fixed expressions, and work on is one such pattern that must be memorised. It helps to learn common verb plus preposition combinations in groups, such as depend on, insist on, and work on.

Final Answer:
The correct phrasal verb is working on, so the sentence should read I have been working on this problem since noon.