There are 900 students in a college, categorized into groups A, B, and C. The ratio of students in only A, only B, and only C is $2 : 3 : 5$ respectively. The number of students in all three categories is 50\% of the number of students in only C. The sum of students who are in only B and C but not A, and those who are in only A and C but not B is 155. The number of students in A and B but not C is 30 less than the number of students in only B.

Find the difference between the total number of students in all three categories and the total number of students in category A.

Question

There are 900 students in a college, categorized into groups A, B, and C. The ratio of students in only A, only B, and only C is $2 : 3 : 5$ respectively. The number of students in all three categories is 50\% of the number of students in only C. The sum of students who are in only B and C but not A, and those who are in only A and C but not B is 155. The number of students in A and B but not C is 30 less than the number of students in only B.

Find the difference between the total number of students in all three categories and the total number of students in category A.

Accepted Answer

**Solution:**

1. Let the number of students in only A, only B, and only C be $2x$, $3x$, and $5x$ respectively.

2. Total students = 900, so the sum of all categories and their intersections must equal 900.

3. Number of students in all three categories (A \& B \& C) = 50\% of only C = $0.5 	imes 5x = 2.5x$.

4. Let the number of students in only B and C but not A be $y$, and those in only A and C but not B be $z$. Given, $y + z = 155$.

5. Number of students in A and B but not C is 30 less than only B, so it is $3x - 30$.

6. Total students in category A = only A + (A \& B but not C) + (A \& C but not B) + (A \& B \& C) = $2x + (3x - 30) + z + 2.5x = 7.5x + z - 30$.

7. The problem does not provide enough information to find values of $x$, $y$, or $z$ individually.

8. Without exact values of $x$, $y$, and $z$, the difference between students in all three categories and students in category A cannot be determined.

**Answer:** Can't be determined

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