There were 5 more boys than girls in the gymnasium. Excluding 6 girls, the number of girls was ⁴₅ of the number of boys. When another 7 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	5 more
Before	5 u	4 u + 6
Change	+ 7
After	5 u + 7	4 u + 6

Number of boys at first was 5 more than the number of girls. If another 5 girls entered the gymnasium at first, the number of boys and girls at first would be the same.

5 u = 4 u + 6 + 5 
5 u - 4 u = 6 + 5

1 u = 11

Total number of boys in the end
= 5 u + 7
= 5 x 11 + 7
= 55 + 7
= 62

Total number of students in the end
= 5 u + 7 + 4 u + 6
= 9 u + 13
= 9 x 11 + 13
= 112

Fraction of the students who were boys in the end
= ⁶²₁₁₂
= ³¹₅₆

Answer(s): ³¹₅₆