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
Thank you so, so much for huge help, I'm really appreciate it. Thank you:))
To find the answers to the questions, we can start by drawing a Venn diagram based on the given information.
Step 1: Draw three overlapping circles to represent Mathematics, History, and Geography, respectively.
Step 2: Label the circles accordingly. Let's label them as "M" for Mathematics, "H" for History, and "G" for Geography.
Step 3: Fill in the information we have. Based on the given information, we know that there are 70 students who passed Mathematics (M), 60 students who passed History (H), and 45 students who passed Geography (G). Additionally, there are 30 students who passed both Mathematics and History (M ∩ H), 35 students who passed both History and Geography (H ∩ G), and 25 students who passed both Mathematics and Geography (M ∩ G). Finally, 15 students passed all three subjects (M ∩ H ∩ G).
Now, based on this Venn diagram, we can answer the questions.
a) To find the number of students who passed at least one subject, we need to add up the total number of students in all the circles (including those in the overlaps). We have:
M: 70
H: 60
G: 45
M ∩ H: 30
H ∩ G: 35
M ∩ G: 25
M ∩ H ∩ G: 15
To find the total number of students who passed at least one subject, we can add the numbers together:
Total = M + H + G - (M ∩ H) - (H ∩ G) - (M ∩ G) + (M ∩ H ∩ G)
Total = 70 + 60 + 45 - 30 - 35 - 25 + 15
Therefore, the number of students who passed at least one subject is 100.
b) To find the number of students who passed exactly two subjects, we simply need to add up the numbers in the overlapping areas:
Number of students who passed exactly two subjects = (M ∩ H) + (H ∩ G) + (M ∩ G)
Number of students who passed exactly two subjects = 30 + 35 + 25
Therefore, the number of students who passed exactly two subjects is 90.
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):
Number of students who passed Geography and failed Mathematics = G - (M ∩ G)
Number of students who passed Geography and failed Mathematics = 45 - 25
Therefore, the number of students who passed Geography and failed Mathematics is 20.
d) To find the number of students who passed all three subjects given that they passed two, we can use the number of students who passed exactly two subjects:
Number of students who passed all three subjects = (M ∩ H) + (H ∩ G) + (M ∩ G)
Number of students who passed all three subjects = 30 + 35 + 25
Therefore, the number of students who passed all three subjects is 90.
e) To find the number of students who failed Mathematics given that they passed History, we can subtract the number of students who passed both subjects (M ∩ H) from the total number of students who passed History (H):
Number of students who failed Mathematics given that they passed History = H - (M ∩ H)
Number of students who failed Mathematics given that they passed History = 60 - 30
Therefore, the number of students who failed Mathematics given that they passed History is 30.
I hope this helps! Let me know if you have any more questions.
Solution
Workings
a) To find the number of students who passed at least one subject, we need to sum up the number of students who passed each subject individually and subtract the students who passed all three subjects (as they would be counted multiple times).
Passed Mathematics: 70
Passed History: 60
Passed Geography: 45
Passed all three subjects: 15
So, the number of students who passed at least one subject is:
70 + 60 + 45 - 15 = 160.
b) To find the number of students who passed exactly two subjects, we need to sum up the number of students who passed each combination of two subjects and subtract the students who passed all three subjects (as they would be counted twice).
Passed Mathematics and History: 30
Passed History and Geography: 35
Passed Mathematics and Geography: 25
Passed all three subjects: 15
So, the number of students who passed exactly two subjects is:
30 + 35 + 25 - 15 = 75.
c) To find the number of students who passed Geography and failed Mathematics, we need to subtract the students who passed both subjects from the number who passed Geography.
Passed Geography: 45
Passed Mathematics and Geography: 25
So, the number of students who passed Geography and failed Mathematics is:
45 - 25 = 20.
d) Given that the students passed two subjects, the number of students who passed all three subjects is already provided: 15.
e) Given that the students passed History, we need to subtract the students who passed both Mathematics and History from the number who passed History.
Passed History: 60
Passed Mathematics and History: 30
So, the number of students who failed Mathematics given that they passed History is:
60 - 30 = 30.
I hope this helps! Let me know if you have any more questions.
A student has passed 70 percent of the 20 quizzes he has written so far successfully. If this student takes 60 quizzes during the school year, and passes 85 percent of the remaining quizzes successfully, what will his average score be?
a)105 students
b)90 students
c)35 students
d)15 students
e)35 students
Hope this helps Lucas.(: