Liam had ¹₂ as many buttons as Riordan. Riordan had ²₃ as many buttons as Tammy. After Liam and Tammy gave 4 buttons to Riordan, Riordan had four times as many buttons as Liam. Tammy then had the same number of buttons as Riordan.

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

Liam	Riordan	Tammy
1	2
	2	3
1	2	3

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

Ratio of the number of buttons that Liam had to the number of buttons that Tammy had in the beginning = 1 : 3

	Liam	Riordan	Tammy	Total
Before	1x3 = 3 u	2x3 = 6 u	3x3 = 9 u	6x3 = 18 u
Change	- 4	+ 8	- 4
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 Riordan received from Liam and Tammy
= 2 x 4
= 8

Number of buttons that Riordan received from Liam and Tammy
= 2 x 1 u
= 2 u

2 u = 4

1 u = 4 ÷ 2 = 2

Number of buttons that Ashley had in the end
= 8 u
= 8 x 2
= 16