Luis had ¹₂ as many buttons as Daniel. Daniel had ²₃ as many buttons as Raeann. After Luis and Raeann gave 26 buttons to Daniel, Daniel had four times as many buttons as Luis. Raeann then had the same number of buttons as Daniel.

1. What was the ratio of the number of buttons Luis had to the number of buttons Raeann had in the beginning? Answer in simplest form.
2. How many buttons did Daniel have in the end?

Luis	Daniel	Raeann
1	2
	2	3
1	2	3

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

Ratio of the number of buttons that Luis had to the number of buttons that Raeann had in the beginning = 1 : 3

	Luis	Daniel	Raeann	Total
Before	1x3 = 3 u	2x3 = 6 u	3x3 = 9 u	6x3 = 18 u
Change	- 26	+ 52	- 26
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 Daniel received from Luis and Raeann
= 2 x 26
= 52

Number of buttons that Daniel received from Luis and Raeann
= 2 x 1 u
= 2 u

2 u = 26

1 u = 26 ÷ 2 = 13

Number of buttons that Ashley had in the end
= 8 u
= 8 x 13
= 104