In this sentence improvement question on word choice, you must decide whether the phrase "by a mutual friend" in the sentence "We were introduced to each other by a mutual friend" should be replaced, and if so, choose the most appropriate alternative.

Introduction / Context:
This question examines fine shades of meaning and traditional prescriptive usage in English phrases like "mutual friend" and "common friend". The sentence "We were introduced to each other by a mutual friend" is common in everyday English. However, older examination and grammar traditions often prefer "common friend" to avoid a debated usage of "mutual".

Given Data / Assumptions:

• Sentence given: We were introduced to each other by a mutual friend.
• The underlined part is "by a mutual friend".
• The intention is to say that one person was friends with both speakers and introduced them.
• The exam pattern tends to follow conservative grammar references that treat some uses of "mutual" as questionable.

Concept / Approach:
Strictly traditional usage defines "mutual" as "felt or experienced by each of two or more people". For example, "mutual respect" or "mutual understanding". On that view, some older style guides state that "mutual friend" is not ideal, because the friendship itself is not mutual in the same sense. Instead, they recommend the phrase "common friend" to mean "a friend shared by two people". Modern English widely accepts "mutual friend", but examinations based on older norms often expect "common friend" as the correct improvement.

Step-by-Step Solution:
Identify that the friend in the sentence is shared by both people being introduced.Recall that traditional grammar books use the phrase "a common friend" for this idea.Note that the preposition "by" is appropriate with "introduced".Form the improved phrase: "by a common friend".Check that the complete sentence reads: "We were introduced to each other by a common friend."

Verification / Alternative check:
The improved sentence now matches the wording used in many classic grammar questions: "We were introduced to each other by a common friend." It clearly expresses that there is one friend who knows both people and serves as the link between them.

Why Other Options Are Wrong:
Option A: "through a mutual friend" changes the preposition and keeps "mutual friend", which some exam setters consider less correct, although it is common in modern speech. Option C: "by mutual friends" changes the number to plural and suggests several friends instead of one, which does not match the original meaning. Option D: "No improvement" is rejected under the traditional exam key because it leaves "mutual friend" unchanged.

Common Pitfalls:
Many learners now use "mutual friend" because it is widespread in social media and everyday conversation, and this usage is increasingly accepted. However, in conservative examination contexts, it is safer to choose "common friend" when you see this kind of question. Always pay attention to how grammar is treated in your syllabus and in past papers for the specific exam you are preparing for.

Final Answer:
The most appropriate improvement according to standard examination practice is by a common friend.