In a class of 40 students, 25 speak Hausa, 16 speak Igbo, 21 speak Yoruba and each of the students speaks at least one of these three languages. Of 8 speaks Hausa and Igbo, 11 speaks Hausa and Yoruba and 6 speaks Igbo and Yoruba.

Question

a. Draw a Venn diagram to illustrate this information, using X to represent the number of students who speaks all three.
b. Calculate the value of x

Answer 1

To establish the values for a Venn diagram based on the given information, we can start by defining some variables and using the principle of inclusion-exclusion:

1. Let \( |A| \) be the number of students that speak Hausa = 25
2. Let \( |B| \) be the number of students that speak Igbo = 16
3. Let \( |C| \) be the number of students that speak Yoruba = 21
4. Let \( |A \cap B| \) be the number of students that speak both Hausa and Igbo = 8
5. Let \( |A \cap C| \) be the number of students that speak both Hausa and Yoruba = 11
6. Let \( |B \cap C| \) be the number of students that speak both Igbo and Yoruba = 6
7. Let \( x \) be the number of students who speak all three languages (Hausa, Igbo, and Yoruba).

Using the formula for the union of three sets, we have:

\[ |A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| + |A \cap B \cap C| \]

Since all 40 students speak at least one language:

\[ 40 = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| + x \]

Substituting in the known values:

\[ 40 = 25 + 16 + 21 - 8 - 11 - 6 + x \]

Calculating the right-hand side:

\[ 40 = 25 + 16 + 21 = 62 \ 62 - 8 - 11 - 6 = 62 - 25 = 37 \]

Now substituting back into our equation, we have:

\[ 40 = 37 + x \]

To find \( x \):

\[ x = 40 - 37 \ x = 3 \]

Summary of the Calculation:

1. The number of students who speak all three languages \( x = 3 \).

Venn Diagram Representation:

• The section representing students who speak all three languages (Hausa, Igbo, and Yoruba) will have the value 3.
• Then, we need to find out the remaining numbers in the individual sections of the Venn diagram (Hausa only, Igbo only, Yoruba only, etc.) by considering the overlaps and the total counts of each language, but that is the calculation for \( x \).

Venn Diagram Sections:

• Hausa only: \( |A| - |A \cap B| - |A \cap C| + |A \cap B \cap C| = 25 - 8 - 11 + 3 = 9 \)
• Igbo only: \( |B| - |A \cap B| - |B \cap C| + |A \cap B \cap C| = 16 - 8 - 6 + 3 = 5 \)
• Yoruba only: \( |C| - |A \cap C| - |B \cap C| + |A \cap B \cap C| = 21 - 11 - 6 + 3 = 7 \)
• Hausa and Igbo only: \( |A \cap B| - |A \cap B \cap C| = 8 - 3 = 5 \)
• Hausa and Yoruba only: \( |A \cap C| - |A \cap B \cap C| = 11 - 3 = 8 \)
• Igbo and Yoruba only: \( |B \cap C| - |A \cap B \cap C| = 6 - 3 = 3 \)

So, the complete layout in the Venn diagram would provide a complete account of the 40 students that match the given conditions.