There are 100 students in a class. It is found that 60 students study Mathematics, 45
students study Physics and 35 students study Chemistry. Moreover, 20 students study
Mathematics and Physics, 15 students study Physics and Chemistry, 25 students study
Chemistry and Mathematics and 10 students study none of these subjects. Then
(A) 95 students study at least one of Mathematics, Physics and Chemistry.
(B) 10 students study each of Mathematics, Physics and Chemistry.
(C) 70 students study either Mathematics or Chemistry.
(D) 35 students study Mathematics along with either Physics or Chemistry.
so, have you made your Venn diagram yet?
To solve this problem, we can use the principle of inclusion-exclusion. We need to find the number of students who study at least one of the three subjects: Mathematics, Physics, and Chemistry.
Let's break the problem down step by step:
Step 1: Find the total number of students studying at least one subject.
We know that 100 students are in the class, and 10 students study none of the three subjects. Therefore, the number of students studying at least one subject is 100 - 10 = 90.
Step 2: Find the number of students studying each subject individually.
We know that 60 students study Mathematics, 45 students study Physics, and 35 students study Chemistry.
Step 3: Subtract the students who study multiple subjects.
We know that 20 students study both Mathematics and Physics, 15 students study both Physics and Chemistry, and 25 students study both Chemistry and Mathematics.
To find the number of students who study exactly two subjects, we need to subtract those students who study all three subjects twice. This is because when we added up the students studying each subject individually, we included the students studying all three subjects twice.
Therefore, the number of students studying exactly two subjects is (20 + 15 + 25) - 2*(10) = 60.
Step 4: Add back the students who study all three subjects.
We subtracted the students studying all three subjects twice in the previous step, so we need to add them back once to get an accurate count.
We know that 10 students study all three subjects.
Step 5: Calculate the final answer for each option.
(A) 95 students study at least one of Mathematics, Physics, and Chemistry.
The actual count we found in step 1 is 90, which is less than 95. Therefore, this option is false.
(B) 10 students study each of Mathematics, Physics, and Chemistry.
We found that 10 students learn all three subjects, so this option is true.
(C) 70 students study either Mathematics or Chemistry.
To get this count, we need to add the number of students studying Mathematics and the number of students studying Chemistry, and then subtract the number of students studying both subjects.
This gives us 60 (Mathematics) + 35 (Chemistry) - 25 (both subjects) = 70.
Therefore, this option is true.
(D) 35 students study Mathematics along with either Physics or Chemistry.
We know that 20 students study both Mathematics and Physics, 15 students study both Physics and Chemistry, and 25 students study both Chemistry and Mathematics.
Therefore, the total count of students studying Mathematics along with either Physics or Chemistry is 20 + 25 = 45.
Therefore, this option is false.
In summary:
(A) False
(B) True
(C) True
(D) False