At first, the ratio of the number of boys to the number of girls at a concert hall was 4 : 5. When 42 girls and some boys entered the concert hall, the number of girls increased by 10% and the total number of children at the concert hall increased by 25%.

1. How many boys were there at first?
2. What percentage of the children at the concert hall were boys in the end? Express your answer as a mixed number.

(a)

Boys	Girls	Total
4 u	5 u	9 u
(+ 147)	+ 0.5 u (+ 42)	+ 2.25 u (+ 189)
4 u + 147	5.5 u	11.25 u

Number of girls that entered the concert hall
= 10% x 5 u
= ¹⁰₁₀₀ x 5 u
= 0.5 u

0.5 u = 42

1 u = 42 ÷ 0.5 = 84

Number of boys at first
= 4 u
= 4 x 84
= 336

(b)
Increase in the number of children
= 25% x 9 u
= ²⁵₁₀₀ x 9 u
= 2.25 u

Total number of children at the concert hall in the end
= 9 u + 2.25 u
= 11.25 u
= 11.25 x 84
= 945

Increase in the number of boys
= 189 - 42
= 147

Total number of boys in the end
= 4 u + 147
= 4 x 84 + 147
= 483

Percentage of children who were boys
= ⁴⁸³₉₄₅ x 100
= 51¹₉%

Answer(s): (a) 336; (b) 51¹₉%