Daniel had ¹₂ as many marbles as Perry. Perry had ²₃ as many marbles as Kathy. After Daniel and Kathy gave 12 marbles to Perry, Perry had four times as many marbles as Daniel. Kathy then had the same number of marbles as Perry.

1. What was the ratio of the number of marbles Kathy had to the number of marbles Daniel had in the beginning? Answer in simplest form.
2. How many marbles did Perry have in the end?

Daniel	Perry	Kathy
1	2
	2	3
1	2	3

(a)
The number of marbles that Perry is repeated.  

Ratio of the number of marbles that Kathy had to the number of marbles that Daniel had in the beginning = 3 : 1

	Daniel	Perry	Kathy	Total
Before	1x3 = 3 u	2x3 = 6 u	3x3 = 9 u	6x3 = 18 u
Change	- 12	+ 24	- 12
After	1x2 = 2 u	4x2 = 8 u	4x2 = 8 u	9x2 = 18 u

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

Number of marbles that Perry received from Daniel and Kathy
= 2 x 12
= 24

Number of marbles that Perry received from Daniel and Kathy
= 2 x 1 u
= 2 u

2 u = 12

1 u = 12 ÷ 2 = 6

Number of marbles that Ashley had in the end
= 8 u
= 8 x 6
= 48