1. in a class of 60 student, 22 offered french, 36 offered history , 8 french and english, 10 english and history, 12 french and history while 6 clid not offer any of the three subjects. (a) draw avenn diagram to represent the data? (b) use the diagram in (a) to the numbers of students who offered (i)all the three subjects (ii) history only (iii) french and history only?
I think you forgot to say how many offer English. Using what you have given, all we know is that if
E took English
x took all three, then
(22+36+E)-(8+10+12)+x = 60-6
E+x = 26
So there are lots of possible solutions.
(a) I'm sorry, I am a text-based bot and cannot draw diagrams. However, I can explain how to create a Venn diagram to represent the given data.
Start by drawing three overlapping circles. Label one circle as "French," another as "History," and the third as "English."
Next, place the numbers given in the appropriate regions of the Venn diagram. In this case, we have:
- 22 students offered French, so place 22 in the French circle.
- 36 students offered History, so place 36 in the History circle.
- 8 students offered both French and English, so place 8 in the overlapping region of French and English circles.
- 10 students offered both English and History, so place 10 in the overlapping region of English and History circles.
- 12 students offered both French and History, so place 12 in the overlapping region of French and History circles.
- 6 students did not offer any of the three subjects, so place 6 outside all three circles.
(b) Using the diagram from (a), we can determine the numbers of students who offered:
(i) All three subjects: Since there is no specific information given about this intersection, we can't determine the exact number of students who offered all three subjects.
(ii) History only: To find this number, we need to count the students located in the History circle but not in any other circle. Based on the diagram, it would be 36 - (10 + 12), which equals 14 students who offered History only.
(iii) French and History only: To find this number, we need to count the students located in the overlapping region of French and History circles but not in any other circle. Based on the diagram, it would be 12 students.
(a) Here is a Venn diagram representation of the given data:
|--------------------|--------------------|------------------------|
| French | History | English |
|--------------------|--------------------|------------------------|
| | | |
| 8 | 12 | 10 |
| (French & | (French & | (English & |
| English) | History) | History) |
| | | |
| | | |
| 22 | 36 | |
| (French) | (History) | |
| | | |
| | | |
|---------------------------------------|----------------|
The number in each section represents the count of students who offered the respective subjects.
(b) Using the Venn diagram from (a), we can determine the numbers of students who offered:
(i) All three subjects: This would be the overlap of all three circles, which is empty in this case. Therefore, the number of students who offered all three subjects is 0.
(ii) History only: The number of students who offered history only is the count in the History circle excluding the overlap with other circles. So, it would be 36 - 12 = 24.
(iii) French and history only: The number of students who offered French and history only would be the overlap between the French and History circles, excluding any overlap with the English circle. So, it would be 12.
The numbers of students who offered:
(i) all three subjects is 0
(ii) history only is 24
(iii) French and history only is 12.
To solve this problem, we can use a Venn diagram to visually represent the data. A Venn diagram consists of overlapping circles or sets to show the relationships between different groups or categories.
(a) Drawing the Venn diagram:
To create the Venn diagram, follow these steps:
Step 1: Start by drawing three overlapping circles. Label them as "French," "History," and "English."
Step 2: Place the given numbers inside the circles and their intersections. Based on the information provided, we know that 22 students offered French, 36 offered History, and 8 offered both French and English. Place these numbers accordingly.
Step 3: Place the remaining numbers, such as 10 students offering English and History, 12 students offering French and History, and 6 students not offering any of the three subjects, outside the circles.
In the Venn diagram, the overlapping regions represent the students who offered multiple subjects.
(b) Using the Venn diagram to find the numbers:
(i) To find the number of students who offered all three subjects, we need to look at the intersection of all the circles. In this case, the intersection of all three circles is empty since there are no students who offered all three subjects. Therefore, the answer is 0.
(ii) To find the number of students who offered History only, we look specifically at the region that is within the History circle but outside the French and English circles. Based on the diagram, this region contains 36 students.
(iii) To find the number of students who offered French and History only, we focus on the region that is within both the French and History circles but outside the English circle. Based on the diagram, this region contains 12 students.
By analyzing the Venn diagram, we can answer these questions:
(i) 0 students offered all three subjects.
(ii) 36 students offered History only.
(iii) 12 students offered French and History only.
The question is incomplete. 22 also offered English.
Therefore
22 + x = 26
x = 26 - 22
x = 4
History only = 14 + x = 14 + 4 = 18
French and history only = 12 - x = 12 - 4 = 8