There were 10 more boys than girls in the school hall. Excluding 8 girls, the number of girls was ⁴₅ of the number of boys. When another 12 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	10 more
Before	5 u	4 u + 8
Change	+ 12
After	5 u + 12	4 u + 8

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

5 u = 4 u + 8 + 10 
5 u - 4 u = 8 + 10

1 u = 18

Total number of boys in the end
= 5 u + 12
= 5 x 18 + 12
= 90 + 12
= 102

Total number of students in the end
= 5 u + 12 + 4 u + 8
= 9 u + 20
= 9 x 18 + 20
= 182

Fraction of the students who were boys in the end
= ¹⁰²₁₈₂
= ⁵¹₉₁

Answer(s): ⁵¹₉₁