In the following sentence, the part in brackets may be incorrect. Choose the option which best improves the phrase “had finished” in the sentence: “After they (had finished) eating they went for a walk.” If no improvement is needed, select “no improvement”.

Introduction / Context:
This question examines correct tense sequencing in complex sentences. English often uses past perfect tense to show that one action was completed before another past action. Here, the sentence describes finishing a meal first and then going for a walk. Candidates must decide whether “had finished” is the appropriate tense or whether a different form such as “have finished”, “finish”, or “will finish” should replace it.

Given Data / Assumptions:
Sentence: “After they had finished eating they went for a walk.”
Bracketed part: “had finished”. Options include “no improvement”, “have finished”, “finish”, and “will finish”. We assume that the entire sentence is set in the past and that “went for a walk” definitely occurs after the earlier action of finishing eating.

Concept / Approach:
When two past actions are described and we want to show clearly that one happened before the other, English often uses the past perfect (had + past participle) for the earlier action and the simple past for the later action. For example, “After they had finished their work, they went home.” In this question, finishing eating clearly happens before going for a walk. Therefore “had finished” is grammatically correct and clearly expresses the sequence, so we should choose “no improvement”.

Step-by-Step Solution:
Step 1: Identify the two actions: finishing eating, and going for a walk. Step 2: Decide which action happens first. They obviously finish eating before they go for a walk. Step 3: Note that the later action “went for a walk” uses simple past tense. Step 4: For the earlier action, the standard pattern is past perfect: “had finished”. Step 5: Check the alternative tenses: “have finished” (present perfect), “finish” (simple present), and “will finish” (simple future) all conflict with the past time reference “went”. Step 6: Conclude that the original form “had finished” is already correct; hence the best answer is “no improvement”.

Verification / Alternative check:
We can rewrite the sentence slightly: “They went for a walk after they had finished eating.” This still sounds natural and preserves the clear time order. If we change it to “After they have finished eating they went for a walk”, the mixture of present perfect and simple past feels wrong because it mixes time frames. “After they finish eating they went for a walk” mixes present and past in the same sentence. “After they will finish eating they went for a walk” uses future with past, which is also incorrect. These checks confirm that the original past perfect is the best choice.

Why Other Options Are Wrong:
“Have finished” (present perfect) would imply a connection to the present time, which clashes with “went” in the past.
“Finish” (simple present) is inconsistent with the past action “went” and would suit habitual situations, not one specific past event.
“Will finish” (simple future) is clearly wrong because the walk has already taken place, and we are not predicting a future event.
Thus, none of the changes improve the sentence.

Common Pitfalls:
Students sometimes avoid using past perfect because they feel any double past structure is wrong, and they may incorrectly change “had finished” to simple past. However, when two events are clearly ordered in the past, the past perfect is an important tool to avoid ambiguity. Another pitfall is selecting an option simply because it looks simpler; simplicity should not override correct tense usage.

Final Answer:
The correct choice is no improvement; the sentence “After they had finished eating they went for a walk” is grammatically correct.