Ethan had ¹₂ as many buttons as Tom. Tom had ²₃ as many buttons as Betty. After Ethan and Betty gave 24 buttons to Tom, Tom had four times as many buttons as Ethan. Betty then had the same number of buttons as Tom.

1. What was the ratio of the number of buttons Ethan had to the number of buttons Betty had in the beginning? Answer in simplest form.
2. How many buttons did Tom have in the end?

Ethan	Tom	Betty
1	2
	2	3
1	2	3

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

Ratio of the number of buttons that Ethan had to the number of buttons that Betty had in the beginning = 1 : 3

	Ethan	Tom	Betty	Total
Before	1x3 = 3 u	2x3 = 6 u	3x3 = 9 u	6x3 = 18 u
Change	- 24	+ 48	- 24
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 Tom received from Ethan and Betty
= 2 x 24
= 48

Number of buttons that Tom received from Ethan and Betty
= 2 x 1 u
= 2 u

2 u = 24

1 u = 24 ÷ 2 = 12

Number of buttons that Ashley had in the end
= 8 u
= 8 x 12
= 96