Tommy, Vincent and Sean had some buttons in the ratio of 4 : 5 : 9. Sean gave 50% of his buttons to Tommy and Vincent. As a result, Tommy had the same number of buttons as Vincent and Sean had 36 fewer buttons than Tommy.

1. What was the percentage increase in Tommy's buttons after receiving from Sean?
2. How many buttons did Vincent have at first?

Tommy	Vincent	Sean	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 Sean gave away
= 50% x 9
= ⁵⁰₁₀₀ x 9
= 4.5 u

Number of buttons that Tommy and Vincent 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 Tommy had
= 6.75 u - 4 u
= 2.75 u

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

(b)
6.75 u = 4.5 u + 36
6.75 u - 4.5 u = 36
2.25 u = 36

1 u = 36 ÷ 2.25 = 16

Number of buttons that Vincent had at first
= 5 u
= 5 x 16
= 80

Answer(s): (a) 68.75%; (b) 80