Charlie had ¹₂ as many buttons as Japheth. Japheth had ²₃ as many buttons as Kylie. After Charlie and Kylie gave 10 buttons to Japheth, Japheth had four times as many buttons as Charlie. Kylie then had the same number of buttons as Japheth.

1. What was the ratio of the number of buttons Charlie had to the number of buttons Kylie had in the beginning? Answer in simplest form.
2. How many buttons did Japheth have in the end?

Charlie	Japheth	Kylie
1	2
	2	3
1	2	3

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

Ratio of the number of buttons that Charlie had to the number of buttons that Kylie had in the beginning = 1 : 3

	Charlie	Japheth	Kylie	Total
Before	1x3 = 3 u	2x3 = 6 u	3x3 = 9 u	6x3 = 18 u
Change	- 10	+ 20	- 10
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 Japheth received from Charlie and Kylie
= 2 x 10
= 20

Number of buttons that Japheth received from Charlie and Kylie
= 2 x 1 u
= 2 u

2 u = 10

1 u = 10 ÷ 2 = 5

Number of buttons that Ashley had in the end
= 8 u
= 8 x 5
= 40