Archie bought 209 purple and silver balloons for a party. The number of silver balloons was ⁴₇ times as many as the number of purple balloons. Archie used ¹₃ times as many silver balloons as purple balloons and was left with twice as many silver balloons as purple balloons. How many balloons did Archie have in the end?

	Silver (1)	Purple (2)	Make b the same. (2)x2 = (3)
Before	76	133	266
Change	- 1 p	- 3 p	- 6 p
After	2 b	1 b	2 b

Number of silver balloons that Archie bought at first = 4 u
Number of purple balloons that Archie bought at first = 7 u

Total number of balloons at first
= 4 u + 7 u
= 11 u

11 u = 209

1 u = 209 ÷ 11 = 19

Number of silver balloons at first
= 4 u
= 4 x 19
= 76

Number of purple balloons at first
= 7 u
= 7 x 19
= 133

76 - 1 p = 2 b --- (1)
133 - 3 p = 1 b --- (2)

(2)_x2
266 - 6 p = 2 b --- (3)

(3) = (1)
266 - 6 p = 76 - 1 p
6 p - 1 p = 266 - 76 
5 p = 190

1 p = 190 ÷ 5 = 38

Number of balloons that Archie used
= 1 p + 3 p
= 4 p
= 4 x 38
= 152

Number of balloons that Archie had in the end
= 209 - 152
= 57

Answer(s): 57