Archie bought 176 gold and purple balloons for a party. The number of purple balloons was ⁴₇ times as many as the number of gold balloons. Archie used ¹₃ times as many purple balloons as gold balloons and was left with twice as many purple balloons as gold balloons. How many balloons did Archie have in the end?

	Purple (1)	Gold (2)	Make b the same. (2)x2 = (3)
Before	64	112	224
Change	- 1 p	- 3 p	- 6 p
After	2 b	1 b	2 b

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

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

11 u = 176

1 u = 176 ÷ 11 = 16

Number of purple balloons at first
= 4 u
= 4 x 16
= 64

Number of gold balloons at first
= 7 u
= 7 x 16
= 112

64 - 1 p = 2 b --- (1)
112 - 3 p = 1 b --- (2)

(2)_x2
224 - 6 p = 2 b --- (3)

(3) = (1)
224 - 6 p = 64 - 1 p
6 p - 1 p = 224 - 64 
5 p = 160

1 p = 160 ÷ 5 = 32

Number of balloons that Archie used
= 1 p + 3 p
= 4 p
= 4 x 32
= 128

Number of balloons that Archie had in the end
= 176 - 128
= 48

Answer(s): 48