There were 10 more girls than boys in the gymnasium. Excluding 3 boys, the number of boys was ³₄ of the number of girls. When another 12 girls joined, what fraction of the pupils were girls in the end? Give your answer in its simplest form.

	Girls	Boys
Comparing girls  and boys at first	10 more
Before	4 u	3 u + 3
Change	+ 12
After	4 u + 12	3 u + 3

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

4 u = 3 u + 3 + 10 
4 u - 3 u = 3 + 10

1 u = 13

Total number of girls in the end
= 4 u + 12
= 4 x 13 + 12
= 52 + 12
= 64

Total number of pupils in the end
= 4 u + 12 + 3 u + 3
= 7 u + 15
= 7 x 13 + 15
= 106

Fraction of the pupils who were girls in the end
= ⁶⁴₁₀₆
= ³²₅₃

Answer(s): ³²₅₃