Daniel had ¹₂ as many coins as Ken. Ken had ²₃ as many coins as Gwen. After Daniel and Gwen gave 22 coins to Ken, Ken had four times as many coins as Daniel. Gwen then had the same number of coins as Ken.

1. What was the ratio of the number of coins Daniel had to the number of coins Gwen had in the beginning? Answer in simplest form.
2. How many coins did Ken have in the end?

Daniel	Ken	Gwen
1	2
	2	3
1	2	3

(a)
The number of coins that Ken is repeated.  

Ratio of the number of coins that Daniel had to the number of coins that Gwen had in the beginning = 1 : 3

	Daniel	Ken	Gwen	Total
Before	1x3 = 3 u	2x3 = 6 u	3x3 = 9 u	6x3 = 18 u
Change	- 22	+ 44	- 22
After	1x2 = 2 u	4x2 = 8 u	4x2 = 8 u	9x2 = 18 u

The total number of coins at first and in the end remains unchanged. Make the total number of coins the same. LCM of 6 and 9 is 18.

Number of coins that Ken received from Daniel and Gwen
= 2 x 22
= 44

Number of coins that Ken received from Daniel and Gwen
= 2 x 1 u
= 2 u

2 u = 22

1 u = 22 ÷ 2 = 11

Number of coins that Ashley had in the end
= 8 u
= 8 x 11
= 88