Answer to Question #160209 in Discrete Mathematics for Maryrose

Question

In set using venn diagram: The are 60 sciences students in ss3, in a secondary school, 35 of whom study chemistry, 30 study technical drawing, 12 out of those students study biology and chemistry but not technical drawing, 10 study chemistry but neither biology nor technical drawing, 11 study only technical drawing and neither biology or chemistry, 10 also study chemistry and technical drawing only. How many students study all the 3 subjects? How many students study biology and technical drawing but not chemistry? How many students study biology only? How many students study biology all together?

Answer 1

Using Venn diagram, we have:

Let $a$ be the number of students that study all the 3 subjects, $b$ be the number of students that study biology and technical drawing but not chemistry, $c$ be the number of students that study biology only.

We know that 35 students study chemistry. It is $10+10+12+a=35$ . So, $a=35-10-10-12=3$ .

Also it is known that 30 students study technical drawing. That is $10+11+a+b=30$ . So, $b=30-10-11-a=30-10-11-3=6$ .

There are 60 students. Therefore, $10+10+11+12+a+b+c=60$ .

We have $c=60-10-10-11-12-a-b=60-10-10-11-12-3-6=8$ .

We can find the number of students that study biology. That is $12+a+b+c=12+3+6+8=29$ .

Answer: 3 students study all the 3 subjects; 6 students study biology and technical drawing but not chemistry; 8 students study biology only; 29 students study biology.