John bought 105 purple and black balloons for a party. The number of black balloons was ²₅ times as many as the number of purple balloons. John used ¹₃ times as many black balloons as purple balloons and was left with twice as many black balloons as purple balloons. How many balloons did John have in the end?

	Black (1)	Purple (2)	Make b the same. (2)x2 = (3)
Before	30	75	150
Change	- 1 p	- 3 p	- 6 p
After	2 b	1 b	2 b

Number of black balloons that John bought at first = 2 u
Number of purple balloons that John bought at first = 5 u

Total number of balloons at first
= 2 u + 5 u
= 7 u

7 u = 105

1 u = 105 ÷ 7 = 15

Number of black balloons at first
= 2 u
= 2 x 15
= 30

Number of purple balloons at first
= 5 u
= 5 x 15
= 75

30 - 1 p = 2 b --- (1)
75 - 3 p = 1 b --- (2)

(2)_x2
150 - 6 p = 2 b --- (3)

(3) = (1)
150 - 6 p = 30 - 1 p
6 p - 1 p = 150 - 30 
5 p = 120

1 p = 120 ÷ 5 = 24

Number of balloons that John used
= 1 p + 3 p
= 4 p
= 4 x 24
= 96

Number of balloons that John had in the end
= 105 - 96
= 9

Answer(s): 9