Tommy had ¹₂ as many buttons as Lee. Lee had ²₃ as many buttons as Olivia. After Tommy and Olivia gave 6 buttons to Lee, Lee had four times as many buttons as Tommy. Olivia then had the same number of buttons as Lee.

1. What was the ratio of the number of buttons Tommy had to the number of buttons Olivia had in the beginning? Answer in simplest form.
2. How many buttons did Lee have in the end?

Tommy	Lee	Olivia
1	2
	2	3
1	2	3

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

Ratio of the number of buttons that Tommy had to the number of buttons that Olivia had in the beginning = 1 : 3

	Tommy	Lee	Olivia	Total
Before	1x3 = 3 u	2x3 = 6 u	3x3 = 9 u	6x3 = 18 u
Change	- 6	+ 12	- 6
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 Lee received from Tommy and Olivia
= 2 x 6
= 12

Number of buttons that Lee received from Tommy and Olivia
= 2 x 1 u
= 2 u

2 u = 6

1 u = 6 ÷ 2 = 3

Number of buttons that Ashley had in the end
= 8 u
= 8 x 3
= 24