.(a) In a class of 80 students,\(\frac{3}{4}\) study Biology and \(\frac{3}{5}\) study Physics. If each student studies at least one of the subjects: (i) dr...

Answer Details

(a)

(i) Here is a Venn diagram to represent the information given:

    B
   / \
  /   \
P/_____\
 |     |
80   ---

where P represents the students who study Physics and B represents those who study Biology. The region inside the circle P but outside the intersection represents students who only study Physics, while the region inside the circle B but outside the intersection represents students who only study Biology. The intersection represents students who study both subjects.

(ii) To find the number of students who study both subjects, we need to find the size of the intersection. Let x be the number of students who study both subjects. Then we can write two equations based on the given information:

• \(\frac{3}{4}\times80\) students study Biology.
• \(\frac{3}{5}\times80\) students study Physics.

Using these equations, we can solve for x:

• Biology students: \(\frac{3}{4}\times80 = 60\)
• Physics students: \(\frac{3}{5}\times80 = 48\)
• Total students: 80
• Students who study both subjects: x

From the diagram, we can see that the total number of students is 80, so we have:

60 + x + 48 = 80

Simplifying this equation, we get:

x = 12

Therefore, 12 students study both subjects.

(iii) To find the fraction of the class that study Biology but not Physics, we need to find the size of the region inside the circle B but outside the intersection, and divide by the total number of students. Let y be the number of students who study Biology but not Physics. Then we can write another equation based on the given information:

• \(\frac{3}{4}\times80\) students study Biology.
• x students study both subjects.

Using these equations, we can solve for y:

• Biology students: \(\frac{3}{4}\times80 = 60\)
• Students who study both subjects: x = 12
• Students who study Biology but not Physics: y

From the diagram, we can see that:

y + x = 60

Substituting x = 12, we get:

y + 12 = 60

Simplifying this equation, we get:

y = 48

Therefore, 48 students study Biology but not Physics. The fraction of the class that study Biology but not Physics is:

\(\frac{48}{80} = \frac{3}{5