There are 900 students in a college, divided into categories 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 in only B & C (but not A) and only A & C (but not B) is 155. The number of students in A & B (but not C) is 30 less than the number of students in only B.

If 30% of students in only category A are boys, find the total number of girls in only category A.

Question

There are 900 students in a college, divided into categories 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 in only B & C (but not A) and only A & C (but not B) is 155. The number of students in A & B (but not C) is 30 less than the number of students in only B.

If 30% of students in only category A are boys, find the total number of girls in only 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:
   $$2x + 3x + 5x + 	ext{(students in other categories)} = 900$$

3. Students in all three categories = 50% of only C = $0.5 	imes 5x = 2.5x$

4. Sum of students in only B & C (not A) and only A & C (not B) = 155

5. Students in A & B (not C) = 30 less than only B = $3x - 30$

6. From the above, the total students in all categories can be expressed as:
   $$2x + 3x + 5x + 2.5x + 155 + (3x - 30) = 900$$

7. Simplify:
   $$2x + 3x + 5x + 2.5x + 155 + 3x - 30 = 900$$
   $$15.5x + 125 = 900$$
   $$15.5x = 775$$
   $$x = \frac{775}{15.5} = 50$$

8. Number of students in only A = $2x = 2 	imes 50 = 100$

9. 30% of only A are boys, so boys in only A = $30\%$ of 100 = 30

10. Therefore, girls in only A = $100 - 30 = 70$

**Answer:** 70

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