Asked by Patrick
In a class of 40 students, 22 study art, 18 study biology and 6 study only chemistry. 5 study all 3 subjects, 9 study art and biology, 7 study art but not chemistry and 11 study exactly one subject. illustrate the info on a Venn diagram. how many students study exactly two subjects. how many students study none of the subjects. what is the probability that a student chosen at random study chemistry.
Answers
Answered by
mathhelper
draw 3 intersecting circles, label them A, B, C, for the subject names
fill in from the centre outwards
you are told 5 take all three, so put 5 in the intersection of all 3
Then do the "two at a time" sections of the Venn diagram
e.g. 9 study art and chemistry. BUT, we already have 5 counted
in that part, so put 4 in the part representing only art and biology.
But 6 in the "only chemistry" part.
We don't know the art and chemistry, but not biology, call that x
Since we know art has 22, we know that the "only art" part would be
22 - 4-4-x = 13-x
We know that 7 study art but not chemistry
13-x + 4 = 7
x = 10
replace the x with 10
Label the "only chemistry and biology" part as y
from our data:
then the only biology part is
18-4-5-y = 9-y
we know 11 study exactly one subject
13-x + 6 + 9-y = 11
13-10 + 6 + 9 -y = 11
y = 7
replace all the y's with 7
looks like all the part are filled in, but I only count 37, and in the class of
40 students, 3 must not take any of the listed subjects.
counting up the chemistry numbers, there are 28
so the prob that a random student of the class selected takes chemistry
= 28/40
= 7/10 or 0.7
check my arithmetic, the numbers seem to make sense
fill in from the centre outwards
you are told 5 take all three, so put 5 in the intersection of all 3
Then do the "two at a time" sections of the Venn diagram
e.g. 9 study art and chemistry. BUT, we already have 5 counted
in that part, so put 4 in the part representing only art and biology.
But 6 in the "only chemistry" part.
We don't know the art and chemistry, but not biology, call that x
Since we know art has 22, we know that the "only art" part would be
22 - 4-4-x = 13-x
We know that 7 study art but not chemistry
13-x + 4 = 7
x = 10
replace the x with 10
Label the "only chemistry and biology" part as y
from our data:
then the only biology part is
18-4-5-y = 9-y
we know 11 study exactly one subject
13-x + 6 + 9-y = 11
13-10 + 6 + 9 -y = 11
y = 7
replace all the y's with 7
looks like all the part are filled in, but I only count 37, and in the class of
40 students, 3 must not take any of the listed subjects.
counting up the chemistry numbers, there are 28
so the prob that a random student of the class selected takes chemistry
= 28/40
= 7/10 or 0.7
check my arithmetic, the numbers seem to make sense
Answered by
Nana Baako
Sure I counted 37 too
Answered by
GPT 3.5
Apologies for the error. You are correct, there seems to be a discrepancy in the information provided. Since we have accounted for 37 students so far, there are 40 - 37 = 3 students who study none of the subjects.
Regarding the number of students who study exactly two subjects, we can calculate that as follows:
Total number of students who study exactly one subject = 11
Total number of students who study all three subjects = 5
Total number of students in the class = 40
Number of students who study exactly two subjects = Total - (students who study exactly one subject + students who study all three subjects + students who study none of the subjects)
= 40 - (11 + 5 + 3)
= 40 - 19
= 21
Therefore, there are 21 students who study exactly two subjects.
Regarding the number of students who study exactly two subjects, we can calculate that as follows:
Total number of students who study exactly one subject = 11
Total number of students who study all three subjects = 5
Total number of students in the class = 40
Number of students who study exactly two subjects = Total - (students who study exactly one subject + students who study all three subjects + students who study none of the subjects)
= 40 - (11 + 5 + 3)
= 40 - 19
= 21
Therefore, there are 21 students who study exactly two subjects.
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