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 the these three languages. If 8 spe...
Question 1 Report
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 the these 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 the information, using x to represent the number of students that speak all three languages.
(b) calculate the value of x.
Answer Details
(a) Let \(H\), \(I\) and \(Y\) represent the sets of students who speak Hausa, Igbo and Yoruba respectively. Since the given pairwise intersections include those who speak all three languages, the Venn diagram is:
(b) Since every student speaks at least one of the languages,
\[n(H\cup I\cup Y)=40.\]
Using the inclusion-exclusion principle,
\[40=25+16+21-8-11-6+x\]
\[40=37+x\]
\[x=3.\]
Therefore, 3 students speak all three languages.
Hence the numerical entries in the Venn diagram are:
\[H\cap I\cap Y=3,\quad H\cap I\text{ only}=5,\quad H\cap Y\text{ only}=8,\quad I\cap Y\text{ only}=3,\]
\[H\text{ only}=9,\quad I\text{ only}=5,\quad Y\text{ only}=7.\]
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