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.
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
Sure I counted 37 too
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.
Sure, let's break down the information step by step and illustrate it on a Venn diagram.
22 students study art, so we put that number in the "Art" circle.
18 students study biology, so we put that number in the "Biology" circle.
6 students study only chemistry, so we put that number outside all the circles, representing only chemistry.
5 students study all 3 subjects, so we put that number in the overlapping region of all three circles.
9 students study art and biology, but not chemistry, so we put that number in the overlapping region of the art and biology circles, but outside the chemistry circle.
7 students study art, but not biology or chemistry, so we put that number in the region of the art circle, but outside the other circles.
11 students study exactly one subject, so we put that number in each of the non-overlapping regions of the circles.
Now, let's find the missing information:
To calculate how many students study exactly two subjects, we add up the numbers in the overlapping regions, excluding the overlapping region of all three circles. So, 9 + 5 = 14 students study exactly two subjects.
To calculate how many students study none of the subjects, we take all the students outside the circles, which is 6.
To find the probability that a student chosen at random studies chemistry, we add up the number of students who study chemistry and divide it by the total number of students (40). So, (6 + 5) / 40 = 11/40.
In summary:
- Students studying exactly two subjects: 14
- Students studying none of the subjects: 6
- Probability of a student studying chemistry: 11/40
To illustrate the given information on a Venn diagram, we need to represent three overlapping circles, one for each subject: art, biology, and chemistry. Let's start by representing the information in the diagram:
The total number of students in the class is 40.
The number of students studying art is 22.
The number of students studying biology is 18.
The number of students studying only chemistry is 6.
The number of students studying all three subjects is 5.
The number of students studying art and biology is 9.
The number of students studying art but not chemistry is 7.
The number of students studying exactly one subject is 11.
Now, let's illustrate this information on a Venn diagram:
```
_____________
/ \
/ Art \
\_______________/
\
| \
| \_______________
| / Biology
/ \ /
/ \____________/
\___ /
\
\___
\
Chemistry
```
In this Venn diagram, the regions are labeled with the given numbers:
A = 7 (students studying art but not chemistry)
B = 9 (students studying art and biology)
C = 11 (students studying exactly one subject)
D = 5 (students studying all three subjects)
E = 6 (students studying only chemistry)
Now, let's answer your questions:
1. How many students study exactly two subjects?
To calculate the number of students studying exactly two subjects, we need to find the sum of the students studying art and biology (B) and the students studying all three subjects (D). So, B + D = 9 + 5 = 14 students study exactly two subjects.
2. How many students study none of the subjects?
To find the number of students studying none of the subjects, we should consider the complement set of all subjects, which is the total number of students minus the sum of students studying at least one subject. So, the complement set = Total students - (A + B + C + D + E) = 40 - (7 + 9 + 11 + 5 + 6) = 40 - 38 = 2 students study none of the subjects.
3. What is the probability that a student chosen at random studies chemistry?
To calculate the probability, we need to find the number of students studying chemistry (E) divided by the total number of students. So, the probability = E/Total students = 6/40 = 0.15 (or 15%).
I hope this explanation helps you understand how to approach such questions and how to utilize Venn diagrams to visualize the information.