There were purple, blue and white marbles in a bottle. Sean put another 9 purple marbles and 12 white marbles in the bottle, then the ratio of the number of blue marbles to the number of white marbles became 5 : 3. Then, he doubled the number of purple marbles and removed 18 white marbles. The ratio of the number of purple marbles to blue marbles became 2 : 1. He counted and found that there were a total of 54 marbles left in the bottle. Find the number of purple marbles that he had in the end.

	Purple marbles	Blue marbles	White marbles	Total
Before	5 u - 9	5 u	3 u - 12
Change 1	+ 9		+ 12
After 1	1x5 = 5 u	5 u	3 u
Change 2	+ 1x5 = 5 u		- 18
After 2	2x5 = 10 u	1x5 = 5 u	3 u - 18	54

The number of blue marbles remains unchanged. Make the number of blue marbles the same. LCM of 1 and 5 is 5.

Total number of marbles in the end
= 10 u + 5 u + 3 u - 18
= 18 u - 18

18 u - 18 = 54
18 u = 54 + 18
18 u = 72

1 u = 72 ÷ 18 = 4

Number of purple marbles in the end
= 10 u 
= 10 x 4
= 40

Answer(s): 40