Question
In a class of 40 students, 25 speak Hausa, 16 speak Igbo, 21 speak Yoruba and each of the students speak at least one of this three languages. If 8 speak Hausa and Igbo, 11 speak Hausa and Yoruba and 6 speak Igbo and Yoruba
A draw a venn diagram to illustrate this information using x to represent the number of students who speak all three languages
B find
1 x
2 the number speak only one language
A draw a venn diagram to illustrate this information using x to represent the number of students who speak all three languages
B find
1 x
2 the number speak only one language
Answers
GPT-4o mini
To solve this problem, let's first summarize the information given about language speakers in the class:
- Let \( H \) be the set of students speaking Hausa.
- Let \( I \) be the set of students speaking Igbo.
- Let \( Y \) be the set of students speaking Yoruba.
From the information provided:
- \( |H| = 25 \)
- \( |I| = 16 \)
- \( |Y| = 21 \)
- \( |H \cap I| = 8 \) (speak both Hausa and Igbo)
- \( |H \cap Y| = 11 \) (speak both Hausa and Yoruba)
- \( |I \cap Y| = 6 \) (speak both Igbo and Yoruba)
- Let \( x \) be the number of students who speak all three languages.
Using the principle of inclusion-exclusion for three sets, we can calculate the total number of students (which is 40) who speak at least one of the languages:
\[
|H \cup I \cup Y| = |H| + |I| + |Y| - |H \cap I| - |H \cap Y| - |I \cap Y| + |H \cap I \cap Y|
\]
Substituting the known values:
\[
40 = 25 + 16 + 21 - 8 - 11 - 6 + x
\]
Simplifying the equation:
\[
40 = 62 - 25 + x
\]
\[
40 = 37 + x
\]
\[
x = 40 - 37 = 3
\]
### A: Venn Diagram Representation
To draw the Venn diagram:
- In the center (intersection of all three circles), fill in 3 for students who speak all three languages.
- Next, calculate the remaining numbers for each intersection:
1. Students who speak Hausa and Igbo but not Yoruba:
\[
|H \cap I| - |H \cap I \cap Y| = 8 - 3 = 5
\]
2. Students who speak Hausa and Yoruba but not Igbo:
\[
|H \cap Y| - |H \cap I \cap Y| = 11 - 3 = 8
\]
3. Students who speak Igbo and Yoruba but not Hausa:
\[
|I \cap Y| - |H \cap I \cap Y| = 6 - 3 = 3
\]
Now we can determine the students who speak only one language:
- Let \( a \) be the number of students who speak only Hausa:
\[
a = |H| - (|H \cap I| + |H \cap Y| - |H \cap I \cap Y|) = 25 - (8 + 11 - 3) = 25 - 16 = 9
\]
- Let \( b \) be the number of students who speak only Igbo:
\[
b = |I| - (|H \cap I| + |I \cap Y| - |H \cap I \cap Y|) = 16 - (8 + 6 - 3) = 16 - 11 = 5
\]
- Let \( c \) be the number of students who speak only Yoruba:
\[
c = |Y| - (|H \cap Y| + |I \cap Y| - |H \cap I \cap Y|) = 21 - (11 + 6 - 3) = 21 - 14 = 7
\]
### B: Finding Numbers
1. **Value of \( x \):**
\[
x = 3
\]
2. **Number of students who speak only one language:**
- Only Hausa: \( 9 \)
- Only Igbo: \( 5 \)
- Only Yoruba: \( 7 \)
Total number who speak only one language:
\[
9 + 5 + 7 = 21
\]
### Final Answers
- \( x = 3 \)
- The number of students who speak only one language = \( 21 \)
- Let \( H \) be the set of students speaking Hausa.
- Let \( I \) be the set of students speaking Igbo.
- Let \( Y \) be the set of students speaking Yoruba.
From the information provided:
- \( |H| = 25 \)
- \( |I| = 16 \)
- \( |Y| = 21 \)
- \( |H \cap I| = 8 \) (speak both Hausa and Igbo)
- \( |H \cap Y| = 11 \) (speak both Hausa and Yoruba)
- \( |I \cap Y| = 6 \) (speak both Igbo and Yoruba)
- Let \( x \) be the number of students who speak all three languages.
Using the principle of inclusion-exclusion for three sets, we can calculate the total number of students (which is 40) who speak at least one of the languages:
\[
|H \cup I \cup Y| = |H| + |I| + |Y| - |H \cap I| - |H \cap Y| - |I \cap Y| + |H \cap I \cap Y|
\]
Substituting the known values:
\[
40 = 25 + 16 + 21 - 8 - 11 - 6 + x
\]
Simplifying the equation:
\[
40 = 62 - 25 + x
\]
\[
40 = 37 + x
\]
\[
x = 40 - 37 = 3
\]
### A: Venn Diagram Representation
To draw the Venn diagram:
- In the center (intersection of all three circles), fill in 3 for students who speak all three languages.
- Next, calculate the remaining numbers for each intersection:
1. Students who speak Hausa and Igbo but not Yoruba:
\[
|H \cap I| - |H \cap I \cap Y| = 8 - 3 = 5
\]
2. Students who speak Hausa and Yoruba but not Igbo:
\[
|H \cap Y| - |H \cap I \cap Y| = 11 - 3 = 8
\]
3. Students who speak Igbo and Yoruba but not Hausa:
\[
|I \cap Y| - |H \cap I \cap Y| = 6 - 3 = 3
\]
Now we can determine the students who speak only one language:
- Let \( a \) be the number of students who speak only Hausa:
\[
a = |H| - (|H \cap I| + |H \cap Y| - |H \cap I \cap Y|) = 25 - (8 + 11 - 3) = 25 - 16 = 9
\]
- Let \( b \) be the number of students who speak only Igbo:
\[
b = |I| - (|H \cap I| + |I \cap Y| - |H \cap I \cap Y|) = 16 - (8 + 6 - 3) = 16 - 11 = 5
\]
- Let \( c \) be the number of students who speak only Yoruba:
\[
c = |Y| - (|H \cap Y| + |I \cap Y| - |H \cap I \cap Y|) = 21 - (11 + 6 - 3) = 21 - 14 = 7
\]
### B: Finding Numbers
1. **Value of \( x \):**
\[
x = 3
\]
2. **Number of students who speak only one language:**
- Only Hausa: \( 9 \)
- Only Igbo: \( 5 \)
- Only Yoruba: \( 7 \)
Total number who speak only one language:
\[
9 + 5 + 7 = 21
\]
### Final Answers
- \( x = 3 \)
- The number of students who speak only one language = \( 21 \)