To solve this problem, we can use the principles of set theory and Venn diagrams. Let’s denote:
- \( A \) = students who speak Hausa
- \( B \) = students who speak Igbo
- \( C \) = students who speak Yoruba
From the problem, we have the following information:
- \( |A| = 25 \)
- \( |B| = 16 \)
- \( |C| = 21 \)
- \( |A \cap B| = 8 \) (students who speak both Hausa and Igbo)
- \( |A \cap C| = 11 \) (students who speak both Hausa and Yoruba)
- \( |B \cap C| = 8 \) (students who speak both Igbo and Yoruba)
- \( |A \cap B \cap C| = x \) (students who speak all three languages)
Step A: Venn Diagram Representation
In a Venn diagram with three circles (Hausa, Igbo, and Yoruba), the area where all three circles overlap will contain \( x \).
We can set up the equations based on the information provided:
- Students who speak only Hausa: \( |A| - |A \cap B| - |A \cap C| + |A \cap B \cap C| = 25 - 8 - 11 + x = 6 + x \)
- Students who speak only Igbo: \( |B| - |A \cap B| - |B \cap C| + |A \cap B \cap C| = 16 - 8 - 8 + x = 0 + x = x \)
- Students who speak only Yoruba: \( |C| - |A \cap C| - |B \cap C| + |A \cap B \cap C| = 21 - 11 - 8 + x = 2 + x \)
Next, we can write expressions for the number of students who speak at least one language (\( |A \cup B \cup C| \)):
\[ |A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| + |A \cap B \cap C| \]
Substituting the numbers and our expressions:
\[ 40 = 25 + 16 + 21 - 8 - 11 - 8 + x \] \[ 40 = 25 + 16 + 21 - 27 + x \] \[ 40 = 35 + x \]
Solving for \( x \): \[ x = 40 - 35 = 5 \]
So:
- Those who speak all three languages (Hausa, Igbo, and Yoruba), \( x = 5 \).
Step B: Number of students who speak Igbo
Now, using \( x \), we can find the number of students who speak only Igbo and those who speak Igbo with others:
\[ |B| = (|B| - |A \cap B| - |B \cap C| + x) + |A \cap B| + |B \cap C| + x \] \[ |B| = 16 = (16 - 8 - 8 + 5) + 8 + 8 - 5 \]
- Hence, students who speak only Igbo = \( x = 5 \)
Total students who speak Igbo: \[ = 0 + 8 + 5 + 8 = 16 \]
Step C: Number of students who speak Yoruba
Using \( x \): \[ |C| = (|C| - |A \cap C| - |B \cap C| + x) + |A \cap C| + |B \cap C| + x \]
- Total students who speak Yoruba: \[ = (21 - 11 - 8 + 5) + 11 + 8 + 5 \]
- Which results in : \[ = 2 + 11 + 8 + 5 = 21 \]
Step D: Number of students who speak Hausa
Using the same method for Hausa: \[ |A| = (|A| - |A \cap B| - |A \cap C| + x) + |A \cap B| + |A \cap C| + x \] \[ = (25 - 8 - 11 + 5) + 8 + 11 + 5 \]
- Which results in: \[ = 6 + 8 + 11 + 5 = 25 \]
Final Answers
The results for each part are as follows:
B. Number of students who speak Igbo: 16
C. Number of students who speak Yoruba: 21
D. Number of students who speak Hausa: 25