In a class containing 200 students, n students prefer both tea and coffee; 2n students prefer coffee, 3n students prefer tea; 4n students prefer neither tea nor coffee. What is the value of n?

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 \( 25 \).

---

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

---

**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.

Learn how to solve a problem involving student preferences using the inclusion-exclusion principle. In a class of 200 students, determine the value of n given that some prefer both tea and coffee, while others prefer only one or neither. This problem is suitable for students in Grades 8-10 and covers algebraic concepts and linear equations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation