There were 6 more girls than boys in the music room. Excluding 12 boys, the number of boys was ²₃ of the number of girls. When another 6 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	6 more
Before	3 u	2 u + 12
Change	+ 6
After	3 u + 6	2 u + 12

Number of girls at first was 6 more than the number of boys. If another 6 boys entered the music room at first, the number of girls and boys at first would be the same.

3 u = 2 u + 12 + 6 
3 u - 2 u = 12 + 6

1 u = 18

Total number of girls in the end
= 3 u + 6
= 3 x 18 + 6
= 54 + 6
= 60

Total number of students in the end
= 3 u + 6 + 2 u + 12
= 5 u + 18
= 5 x 18 + 18
= 108

Fraction of the students who were girls in the end
= ⁶⁰₁₀₈
= ⁵₉

Answer(s): ⁵₉