There were red, black and purple balls in a bottle. Sean put another 4 red balls and 6 purple balls in the bottle, then the ratio of the number of black balls to the number of purple balls became 5 : 3. Then, he doubled the number of red balls and removed 13 purple balls. The ratio of the number of red balls to black balls became 2 : 1. He counted and found that there were a total of 59 balls left in the bottle. Find the number of red balls that he had at first.

	Red balls	Black balls	Purple balls	Total
Before	5 u - 4	5 u	3 u - 6
Change 1	+ 4		+ 6
After 1	1x5 = 5 u	5 u	3 u
Change 2	+ 1x5 = 5 u		- 13
After 2	2x5 = 10 u	1x5 = 5 u	3 u - 13	59

The number of black balls remains unchanged. Make the number of black balls the same. LCM of 1 and 5 is 5.

Total number of balls in the end
= 10 u + 5 u + 3 u - 13
= 18 u - 13

18 u - 13 = 59
18 u = 59 + 13
18 u = 72

1 u = 72 ÷ 18 = 4

Number of red balls at first
= 5 u - 4
= 5 x 4 - 4
= 20 - 4
= 16

Answer(s): 16