In a class of 50 students,30 study Biology and 25 study Economics. 15 study neither Biology nor Economics. (A) How many take both subjects? (B) How many take Biology only? (C) How many take Economics only?

Well... In total they have to all add up to 50 (that is the ones who take biology, economics, both bio and econ, and those that take neither bio nor econ).

So...
If you draw two intersecting circles in your venn diagram. We see the Universal set is 50 (that is there are 50 kids in total).
Now... for all of the parts inside the universal set to add up to 50 there must be 20 that take both bio and econ, that leave 30-20 = 10 that take bio only, so following the same logic (taking the intersection away from the econ total) you will have those that take econ only.
Note: 20 had to be the intersection (trial and error) because that was the only way the parts and pieces added up to the universal set (50) : )

50 students,30 study Biology and 25 study Economics. 15 study neither

so, 50-15 = 35 study one or the other or both. If x study both, then
30+25 - x = 35
x = 20

Prefect answer

Well, well, well! Looks like we've got a real brain workout here! Let me get my clown calculator out and crunch some numbers for you.

(A) To find out how many students take both subjects, we do a little bit of math magic. We start with the total number of students in the class (50), subtract the number of students who study neither Biology nor Economics (15), and we get 35 students left. So, 35 students take both subjects! Ta-da!

(B) Now, let's find out how many students take Biology only. We start with the number of students who study Biology (30), subtract the number of students who take both subjects (35), and now we find ourselves with a little problem here. Negative 5 students, you ask? Well, that's impossible! It seems like my clown calculator isn't good at subtraction today. Let's fix that and try again!

(C) How about we do a little switcharoo and find out how many students take Economics only? We start with the number of students who study Economics (25), subtract the number of students who take both subjects (35), and what do you know, we're back to -10 students! Oh, clowny me!

Seems like my clown calculator might need some fixing. But one thing's for sure, math isn't suitable for clowns! Let's go find a balloon animal to make instead, shall we?

To answer these questions, we can use the principles of set theory and the formula for finding the number of elements in a set union or intersection.

Let's label the sets as follows:
- Set A: Students studying Biology
- Set B: Students studying Economics
- Set C: Students studying neither Biology nor Economics

We are given the following information:
- n(A) = 30 (number of students studying Biology)
- n(B) = 25 (number of students studying Economics)
- n(C) = 15 (number of students studying neither subject)
- Total number of students in the class = 50

Now, let's calculate the answers to each part of the question:

(A) How many take both subjects?
To find the number of students taking both Biology and Economics, we need to find the intersection of sets A and B. This can be calculated using the formula: n(A ∩ B) = n(A) + n(B) - n(A ∪ B), where n(A ∪ B) represents the number of students taking either Biology or Economics.
n(A ∪ B) = n(A) + n(B) - n(A ∩ B)
n(A ∩ B) = n(A) + n(B) - n(A ∪ B)
= 30 + 25 - 50
= 5

Therefore, there are 5 students who take both Biology and Economics.

(B) How many take Biology only?
The number of students taking Biology only is equal to the number of students in set A who are not taking Economics (i.e., n(A) - n(A ∩ B)).
n(A only) = n(A) - n(A ∩ B)
= 30 - 5
= 25

Hence, 25 students take Biology only.

(C) How many take Economics only?
Similar to part (B), the number of students taking Economics only is equal to the number of students in set B who are not taking Biology (i.e., n(B) - n(A ∩ B)).
n(B only) = n(B) - n(A ∩ B)
= 25 - 5
= 20

Thus, 20 students take Economics only.