Question

if in a class of 40 students 25 speak Hausa, 16 speak Igbo, 21 Speak yoruba and eaxh of the students speaks at least one of these three languages.nif 8 speak Hausa and Igbo, 11 speak Hausa and Yoruba and 8 speak Igno and yoruba
1. Use a venn diagram to illustrate the information using x to represent the number of those who speak all the three languages.
B. How many students speak igbo
C. How many students speak Yoruba
D. How many students speak Hausa

Answer 1

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**