Question

In a class of 25 students, 12 have taken Mathematics, 8 have taken Mathematics but not Biology. If each student has taken at least one subject, then find the number of students who have taken
i) Biology but not Mathematics
ii) Both Mathematics and Biology

Answer 1

- Hint: In this question it is given that in a class of 25 students, 12 have taken Mathematics, 8 have taken Mathematics but not Biology. If each student has taken at least one subject, then find the number of students who have taken Biology but not Mathematics and Both Mathematics and Biology. So to find the solution we need to know that among those 25 students some students have taken maths only and some of them take biology and the others taken both of the subjects. So by a simple addition and subtraction method we will get our required solution.

Complete step-by-step solution -
Let us consider M to be the number of students who have taken maths only and B be the number of students who have taken Biology only.
Also let C be the number of students who have taken Mathematics as well as Biology.
Therefore total students, T=M+B+C…………(1)
Here it is given that in a class there are 25 students,
$$\therefore T =25$$..................(2)
And 8 students took Mathematics only (i.e they did not took biology)
$$\therefore M =8$$...................(3)
Also it is given that 12 students took Mathematics, i.e, among them some students have taken biology.
So we can say that this is the combination of students who took mathematics (not Biology) and both the subjects.
Therefore we can write,
$$M+C =12$$
$$\Rightarrow 8+C =12$$ [by equation (3)]
$$\Rightarrow C =12-8$$
$$\Rightarrow C =4$$
Therefore only 4 students are there who have taken both of the subjects (Mathematics and Biology)
Now we have to find those students who have taken only Biology.
Since we know that 12 students are there who have taken biology,
Therefore we can write from equation (1)
$$ T= M+B+C$$
$$\Rightarrow 25=8+B+4$$
$$\Rightarrow 25=12+B$$
$$\Rightarrow 12+B=25$$
$$\Rightarrow B=25-12$$
$$\Rightarrow B=13$$
Therefore 13 students are there who have taken only Biology.

Note: While solving this type of question you need to know that, when you have given that any number of students take mathematics that doesn’t mean that these students only took mathematics, among them some of the students might take other subjects also.