There were 12 more boys than girls in the music room. Excluding 7 girls, the number of girls was ²₃ of the number of boys. When another 6 boys joined, what fraction of the students were boys in the end? Give your answer in its simplest form.

	Boys	Girls
Comparing boys  and girls at first	12 more
Before	3 u	2 u + 7
Change	+ 6
After	3 u + 6	2 u + 7

Number of boys at first was 12 more than the number of girls. If another 12 girls entered the music room at first, the number of boys and girls at first would be the same.

3 u = 2 u + 7 + 12 
3 u - 2 u = 7 + 12

1 u = 19

Total number of boys in the end
= 3 u + 6
= 3 x 19 + 6
= 57 + 6
= 63

Total number of students in the end
= 3 u + 6 + 2 u + 7
= 5 u + 13
= 5 x 19 + 13
= 108

Fraction of the students who were boys in the end
= ⁶³₁₀₈
= ⁷₁₂

Answer(s): ⁷₁₂