Liam, Paul and Oscar had some buttons in the ratio of 4 : 5 : 9. Oscar gave 50% of his buttons to Liam and Paul. As a result, Liam had the same number of buttons as Paul and Oscar had 27 fewer buttons than Liam.

1. What was the percentage increase in Liam's buttons after receiving from Oscar?
2. How many buttons did Paul have at first?

Liam	Paul	Oscar	Total
4 u	5 u	9 u	18 u
+ 4.5 u		- 4.5 u
6.75 u	6.75 u	4.5 u	18 u

(a)
Number of buttons that Oscar gave away
= 50% x 9
= ⁵⁰₁₀₀ x 9
= 4.5 u

Number of buttons that Liam and Paul each had in the end
= (18 u - 4.5 u) ÷ 2
= 13.5 u ÷ 2
= 6.75 u

Increase in the number of buttons that Liam had
= 6.75 u - 4 u
= 2.75 u

Percentage increase in Liam's buttons after receiving from Oscar
= ^2.75₄ x 100%
= 68.75%

(b)
6.75 u = 4.5 u + 27
6.75 u - 4.5 u = 27
2.25 u = 27

1 u = 27 ÷ 2.25 = 12

Number of buttons that Paul had at first
= 5 u
= 5 x 12
= 60

Answer(s): (a) 68.75%; (b) 60