In a certain class,12 students study history,13 study government and 15 study Akan.Each student in the class study at least one of the three subjects,6 students study two of the three subjects and 3 students study All three subjects. Find 1. The number of students who are in the class 2. Those who study only one subject .3. illustrate the information on the Venn diagram.
1. Total number of students in the class = 12 + 13 + 15 - 6 - 3 = 31
2. Number of students who study only one subject = (12-3) + (13-6) + (15-3) = 8+7+12 = 27.
3. Venn diagram:
[Note: The Venn diagram will not be accurately represented since we do not have access to a visual diagram.]
* There will be three circles, one for each subject.
* Twelve students study History, so put 12 inside the circle labeled "History".
* Thirteen students study Government, so put 13 inside the circle labeled "Government".
* Fifteen students study Akan, so put 15 inside the circle labeled "Akan".
* Six students study both History and Government, so draw a line connecting the History and Government circles and put 6 inside the overlapping area.
* Three students study all three subjects, so put 3 inside the small area where all three circles overlap.
* To find the number of students who study only one subject, subtract the numbers in the overlapping areas from the numbers in the individual circles. This gives us:
* Number of students who study only History = 12 - 3 - 6 = 3
* Number of students who study only Government = 13 - 6 - 3 = 4
* Number of students who study only Akan = 15 - 3 - 6 = 6
To solve this problem, we can use the principle of inclusion-exclusion and create a Venn diagram.
Let's start by drawing a Venn diagram representing the three subjects: history, government, and Akan.
1. Number of students in the class:
From the information given, we know that 12 students study history, 13 study government, and 15 study Akan.
Let's denote:
H = the set of students who study history (12 students)
G = the set of students who study government (13 students)
A = the set of students who study Akan (15 students)
Now, to find the total number of students in the class, we can add up the individual sets and subtract the overlaps:
|H ∪ G ∪ A| = |H| + |G| + |A| - |H ∩ G| - |H ∩ A| - |G ∩ A| + |H ∩ G ∩ A|
|H ∪ G ∪ A| = 12 + 13 + 15 - 6 - x - y + 3
Since 6 students study two subjects and 3 students study all three subjects, we subtract those overlaps and add back the ones who study all three subjects.
Simplifying the equation:
|H ∪ G ∪ A| = 37 - x - y
We need additional information to determine the exact value of |H ∪ G ∪ A|.
2. Number of students who study only one subject:
To find the number of students who study only one subject, we need to subtract the students who study more than one subject from the total number of students in each subject.
Let's denote:
H' = the set of students who study only history
G' = the set of students who study only government
A' = the set of students who study only Akan
H' = |H| - |H ∩ G| - |H ∩ A| + |H ∩ G ∩ A|
G' = |G| - |H ∩ G| - |G ∩ A| + |H ∩ G ∩ A|
A' = |A| - |H ∩ A| - |G ∩ A| + |H ∩ G ∩ A|
Substituting the given values:
H' = 12 - 6 - x + 3
G' = 13 - 6 - y + 3
A' = 15 - x - y + 3
Again, we need additional information to determine the exact values of H', G', and A'.
3. Illustrating the information on the Venn diagram:
Based on the information given, you can now draw a Venn diagram with three circles representing history, government, and Akan. Label these circles with the given values (12, 13, and 15, respectively).
Add the overlaps (6 students study two subjects, and 3 students study all three subjects) based on the information given.
I hope this helps! Let me know if you have any further questions.