Pierre had ¹₂ as many marbles as Tom. Tom had ²₃ as many marbles as Kathy. After Pierre and Kathy gave 26 marbles to Tom, Tom had four times as many marbles as Pierre. Kathy then had the same number of marbles as Tom.

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

Pierre	Tom	Kathy
1	2
	2	3
1	2	3

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

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

	Pierre	Tom	Kathy	Total
Before	1x3 = 3 u	2x3 = 6 u	3x3 = 9 u	6x3 = 18 u
Change	- 26	+ 52	- 26
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 Tom received from Pierre and Kathy
= 2 x 26
= 52

Number of marbles that Tom received from Pierre and Kathy
= 2 x 1 u
= 2 u

2 u = 26

1 u = 26 ÷ 2 = 13

Number of marbles that Ashley had in the end
= 8 u
= 8 x 13
= 104