There were 10 more girls than boys in the auditorium. Excluding 6 boys, the number of boys was ⁴₅ of the number of girls. When another 12 girls joined, what fraction of the students were girls in the end? Give your answer in its simplest form.

	Girls	Boys
Comparing girls  and boys at first	10 more
Before	5 u	4 u + 6
Change	+ 12
After	5 u + 12	4 u + 6

Number of girls at first was 10 more than the number of boys. If another 10 boys entered the auditorium at first, the number of girls and boys at first would be the same.

5 u = 4 u + 6 + 10 
5 u - 4 u = 6 + 10

1 u = 16

Total number of girls in the end
= 5 u + 12
= 5 x 16 + 12
= 80 + 12
= 92

Total number of students in the end
= 5 u + 12 + 4 u + 6
= 9 u + 18
= 9 x 16 + 18
= 162

Fraction of the students who were girls in the end
= ⁹²₁₆₂
= ⁴⁶₈₁

Answer(s): ⁴⁶₈₁