Bryan had ¹₂ as many buttons as Liam. Liam had ²₃ as many buttons as Joelle. After Bryan and Joelle gave 28 buttons to Liam, Liam had four times as many buttons as Bryan. Joelle then had the same number of buttons as Liam.

1. What was the ratio of the number of buttons Bryan had to the number of buttons Joelle had in the beginning? Answer in simplest form.
2. How many buttons did Liam have in the end?

Bryan	Liam	Joelle
1	2
	2	3
1	2	3

(a)
The number of buttons that Liam is repeated.  

Ratio of the number of buttons that Bryan had to the number of buttons that Joelle had in the beginning = 1 : 3

	Bryan	Liam	Joelle	Total
Before	1x3 = 3 u	2x3 = 6 u	3x3 = 9 u	6x3 = 18 u
Change	- 28	+ 56	- 28
After	1x2 = 2 u	4x2 = 8 u	4x2 = 8 u	9x2 = 18 u

The total number of buttons at first and in the end remains unchanged. Make the total number of buttons the same. LCM of 6 and 9 is 18.

Number of buttons that Liam received from Bryan and Joelle
= 2 x 28
= 56

Number of buttons that Liam received from Bryan and Joelle
= 2 x 1 u
= 2 u

2 u = 28

1 u = 28 ÷ 2 = 14

Number of buttons that Ashley had in the end
= 8 u
= 8 x 14
= 112