Let’s start by organizing the information given.

1. Let $n$ be the number of students who prefer both tea and coffee.
2. The number of students who prefer coffee is $2n$.
3. The number of students who prefer tea is $3n$.
4. The number of students who prefer neither tea nor coffee is $4n$.
5. The total number of students in the class is 200.

We need to use the principle of inclusion-exclusion to find the value of $n$.

Step 1: Setting up the inclusion-exclusion formula

The total number of students is given by: $\text{Total} = (\text{Students who prefer coffee}) + (\text{Students who prefer tea}) - (\text{Students who prefer both tea and coffee}) + (\text{Students who prefer neither tea nor coffee})$

Substitute the given values: $200 = (2n) + (3n) - (n) + (4n)$

Step 2: Simplify the equation

Combine like terms: $200 = 2n + 3n - n + 4n$ $200 = 8n$

Step 3: Solve for $n$

Now, divide both sides by 8: $n = \frac{200}{8} = 25$

Conclusion

The value of $n$ is $25$.

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Would you like further details on this solution or have any additional questions?

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Related Questions:

1. If 25 students prefer both tea and coffee, how many students prefer only tea?
2. How many students prefer only coffee?
3. How many students prefer either tea or coffee, but not both?
4. What percentage of the students prefer neither tea nor coffee?
5. If 10 more students joined the class, and they all prefer only coffee, how would this change the percentage of coffee drinkers?

Tip: Using a Venn diagram can help visualize inclusion-exclusion problems like this one.