Among a group of 50 students, 18 like both tea (T) and coffee (C), 12 like both
coffee (C) and juice (J), 10 like both tea (T) and juice (J), and 5 students like
all three types of beverages; but no one drinks juice only.
(i) Represent this information in a Venn Diagram.
(ii) If the same number of students drink tea as those who drink
coffee, then how many students like coffee?

### Given Information:

- Total number of students: 50
- Students who like both tea (T) and coffee (C): 18
- Students who like both coffee (C) and juice (J): 12
- Students who like both tea (T) and juice (J): 10
- Students who like all three beverages (T, C, and J): 5
- No students drink juice only.
- The number of students who drink tea is equal to the number of students who drink coffee.

### (i) Venn Diagram Representation:

To construct a Venn diagram:

1. **Label the sets:**
   - Let \( T \) represent the set of students who like tea.
   - Let \( C \) represent the set of students who like coffee.
   - Let \( J \) represent the set of students who like juice.

2. **Set intersections:**
   - \( T \cap C \) (students who like both tea and coffee) = 18
   - \( C \cap J \) (students who like both coffee and juice) = 12
   - \( T \cap J \) (students who like both tea and juice) = 10
   - \( T \cap C \cap J \) (students who like all three beverages) = 5

3. **Breakdown the intersections:**
   - \( T \cap C \cap J \) = 5
   - \( T \cap C \) but not \( J \) = 18 - 5 = 13
   - \( C \cap J \) but not \( T \) = 12 - 5 = 7
   - \( T \cap J \) but not \( C \) = 10 - 5 = 5

4. **No students drink juice only, so the set \( J \) has no elements outside of its intersections with \( T \) or \( C \).**

Let’s now calculate the number of students in each set:

- Total students in \( T \) = \( |T| = x \) 
- Total students in \( C \) = \( |C| = y \) 
- Total students in \( J \) = \( 5 + 5 + 7 = 17 \) (because no one drinks juice only)

### (ii) Number of Students Who Like Coffee:

Since the number of students who like tea is equal to the number of students who like coffee (\( x = y \)), we calculate as follows:

1. Total students who like tea = \( x = 13 + 5 + 5 + a \) where \( a \) represents those who like only tea.
2. Total students who like coffee = \( y = 13 + 5 + 7 + b \) where \( b \) represents those who like only coffee.
3. Students who like juice = \( 5 + 5 + 7 = 17 \).

Now, the sum of all the regions in the Venn diagram must equal the total number of students:

\[
x + b + c + 17 - 5 = 50 
\]

However, let's break it down into a more straightforward calculation.

Since the total number of students who like tea equals those who like coffee:

\[
x = y \implies x + b + c + 17 = 50 
\]
\[
x = 25
\]

Thus, since there are a total of \( 25 \) students who drink coffee.

Would you like further details or have any questions?

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### Relative Questions:
1. How would the scenario change if students could drink juice only?
2. How do you calculate the number of students who like only tea?
3. What is the role of set theory in solving this problem?
4. How could you express the given conditions using set notation?
5. What if we introduced a new beverage? How would it impact the Venn diagram?

---

**Tip:** In problems involving sets and their intersections, always clearly break down the overlapping regions to avoid confusion and ensure accurate calculations.

Among a group of 50 students, 18 like both tea (T) and coffee (C), 12 like both coffee (C) and juice (J), 10 like both tea (T) and juice (J), and 5 students like all three types of beverages; but no one drinks juice only.
(i) Represent this information in a Venn Diagram.
(ii) If the same number of students drink tea as those who drink coffee, then how many students like coffee?

Solve a problem involving a group of 50 students and their beverage preferences using Venn Diagrams and Set Theory. This problem explores the inclusion-exclusion principle to determine how many students like coffee when the number of tea and coffee drinkers is the same. Ideal for students in Grades 9-11.

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