In a group of 105 students, 70 students passed Mathematics, 60 students passed History and 45 students passed Geography; 30 students passed Mathematics and History, 35 students passed History and Geography, 25 passed Mathematics and Geography, and 15 passed all three subjects.

Draw a Venn diagram to illustrate this information.
Find the number of students who
a) passed at least one subject
b) passed exactly two subject
c) passed Geography and failed Mathematics
d) passed all three subjects given that they passed two
e) failed Mathematics given that they passed History

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12

Please can the solution be shown

Temitayo

Answers

To draw the Venn diagram, we'll start by listing the given information:

- A total of 105 students in the group:
- 70 students passed Mathematics (M)
- 60 students passed History (H)
- 45 students passed Geography (G)
- 30 students passed both Mathematics and History (M ∩ H)
- 35 students passed both History and Geography (H ∩ G)
- 25 students passed both Mathematics and Geography (M ∩ G)
- 15 students passed all three subjects (M ∩ H ∩ G)

Now, let's create the Venn diagram step by step:

1. Draw three overlapping circles to represent Mathematics, History, and Geography.
2. Label the regions as follows:
- The area inside the left circle represents Mathematics (M).
- The area inside the middle circle represents History (H).
- The area inside the right circle represents Geography (G).
- The overlapping regions represent the intersections (M ∩ H, H ∩ G, M ∩ G).
- The region where all three circles overlap represents students who passed all three subjects (M ∩ H ∩ G).

Now, let's answer the questions using this Venn diagram:

a) To find the number of students who passed at least one subject, we need to add up the number of students in each region: M, H, G, M ∩ H, H ∩ G, M ∩ G, and M ∩ H ∩ G.

b) To find the number of students who passed exactly two subjects, we need to add up the number of students in each intersection (M ∩ H, H ∩ G, M ∩ G).

c) To find the number of students who passed Geography and failed Mathematics, we need to subtract the number of students who passed both subjects (M ∩ G) from the total number of students who passed Geography (G).

d) To find the number of students who passed all three subjects given that they passed two, we can directly use the number of students who passed all three subjects (M ∩ H ∩ G).

e) To find the number of students who failed Mathematics given that they passed History, we need to subtract the number of students who passed both subjects (M ∩ H) from the total number of students who passed History (H).

By following these steps, you should be able to find the answers to the given questions. If you have any additional questions or need further clarification, feel free to ask!

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